Rank One Higgs Bundles and Representations of Fundamental

Groups of Riemann Surfaces

William M. Goldman

Eugene Z. Xia

Author address:

Department of Mathematics, University of Maryland, College Park, MD 20742 USA

E-mail address: wmg@math.umd.edu

Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

E-mail address: ezxia@ncku.edu.tw

To the memory of Hsieh Po-Hsun

Contents

Introduction 1

1. Equivalences of deformation theories 4

2. The Betti and de Rham deformation theories and their moduli spaces 5 2.1. The Betti groupoid 6 2.2. The de Rham groupoid 8 2.3. Equivalence of de Rham and Betti groupoids 14

3. The Dolbeault groupoid 17 3.1. Holomorphic line bundles 17 3.2. The moduli spaces 21 3.3. Geometric structure of the Dolbeault moduli space 23

4. Equivalence of de Rham and Dolbeault groupoids 25 4.1. Construction of the equivalence 25 4.2. Higgs coordinates 33 4.3. Involutions 36

5. Hyperkähler geometry on the moduli space 37 5.1. The quaternionic structure 37 5.2. The Riemannian metric 39 5.3. Complex-symplectic structure 40 5.4. Quaternionization 42

6. The twistor space 43 6.1. The complex projective line 43 6.2. The twistor space as a smooth vector bundle 48 6.3. A holomorphic atlas for the twistor space 49 6.4. The twistor lines 51 6.5. The real structure on the twistor space 52 6.6. Symplectic geometry of the twistor space 53 6.7. The lattice quotient 55 6.8. Functions and flows 56 7. The moduli space and the Riemann period matrix 59

7.1. Coordinates for the Betti moduli space 59 7.2. Abelian differentials and their periods 60 7.3. Flat connections 62 7.4. Higgs fields 64 7.5. The ℂ^{∗}-action in terms of the period matrix 65 7.6. The ℂ^{∗}-action and the real points 65

Bibliography 67

Abstract

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface X and their relation to representations of the fundamental group of X. We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of X. We describe the moduli spaces and their geometry in terms of the Riemann period matrix of X.

This is the simplest case of the theory developed by Hitchin, Simpson and others. We emphasize its formal aspects that generalize to higher rank Higgs bundles over higher dimensional Kähler manifolds.

Received by the editor September 30, 2005.

Goldman gratefully acknowledges partial support by NSF grants DMS-9504764, DMS-9803518, DMS-0103889 and a Semester Research Award from the General Research Board of the University of Maryland in Fall 1998. Xia gratefully acknowledges partial support by National Science Council Taiwan grant NSC 91-2115-M-006-022, 93-2115-M-006-002.

Rank One Higgs Bundles

Introduction

The set of equivalence classes of representations of the fundamental group π of a closed Riemann surface X into a Lie group G is a basic object naturally associated to π and G. Powerful analytic techniques have been employed by Hitchin, Simpson, Corlette and Donaldson et al [H1, Si1, Si2, Si3] to understand the global topology and geometry of this object. Rank one Higgs bundles provide a toy model with a more explicit form than general Higgs bundles. This paper expounds this case, emphasizing its formal aspects to isolate and clarify the main ideas and motivate its generalization to higher rank.

We consider three moduli spaces of seemingly different objects: complex characters of the fundamental group π of a closed and orientable surface Σ, flat connections on a trivial line bundle over Σ, and pairs consisting of a holomorphic line bundle over a Riemann surface and a holomorphic 1-form on X. These three moduli problems or deformation theories arise in different contexts, but are nonetheless equivalent. Simpson [Si3] names these deformation theories Betti, de Rham, and Dolbeault respectively. In rank one, the Betti moduli space is the collection of ordered 2k-tuples of nonzero complex numbers (k is the genus of Σ). The de Rham moduli space is the quotient

H^{1}(Σ, ℂ)/ H^{1}(Σ, ℤ) ≅ H^{1}(Σ, ℂ/ℤ).

The Dolbeault moduli space is the cotangent bundle T^{∗}Jac(X) of the Jacobian of the Riemann surface X.

This paper assumes a basic knowledge of topology (for example, as in Fulton [Fu]) and differential, and complex geometry. Our perspective is differential geometric and gauge-theoretic. We assume basic facts about symplectic geometry and Hamiltonian flows (as in Weinstein [Wei]). We hope this exposition of the simplest part of deep ideas in algebraic geometry will be useful to topologists and differential geometers interested in moduli spaces of representations of surface groups.

The paper is organized as follows. §1 is a brief introduction of groupoids and their equivalences. §2 constructs the Betti, de Rham, groupoids and moduli spaces and the various structures on these moduli spaces. The main player in the Betti groupoid is a surface group π, whereas the main player in the de Rham groupoid is a closed smooth surface Σ with fundamental group π. §3 develops the Dolbeault groupoid; here Σ is given a conformal structure, making it a Riemann surface X over which we consider various holomorphic objects. The inherent complex structure J on the Betti and de Rham moduli spaces are isomorphic, but J differs from the inherent complex structure I on the Dolbeault moduli space. In §5, these two different structures give rise to a hyperkähler structure on the underlying moduli space. §6 explicitly constructs the twistor space, a holomorphic object containing both moduli spaces. All of these constructions apply to the cotangent bundle of an arbitrary principally polarized abelian variety (not necessarily the Jacobian of a curve). In particular they can all be expressed in terms of the Riemann period matrix Π of X.

Our story ends (§7) by explicitly describing the moduli spaces in terms of the period matrix. We revisit the Betti moduli space which appears as a product of the real torus

Hom(π, U(1)) ≅ T^{2k}

with the real symplectic vector space

Hom(π, ℝ^{+}) ≅ H^{1}(Σ, ℝ) ≅ ℝ^{2k}.

The new structure as the (Dolbeault) moduli space identifies the torus Hom(π, U(1)) as the quotient of ℂ^{k} with the lattice ℤ^{k} + Πℤ^{k} corresponding to the columns of Π. The conformal structure of the Riemann surface X determines a complex structure 𝕁_{Π} on ℝ^{2k} which is easily expressed (7.5.1) in terms of Π. The resulting ℂ^{∗}-action on the moduli space, which determines the full structure of the Dolbeault moduli space and is equivalent to the Hodge structure on the moduli space.

Although many features of the rank one case generalize to higher rank, several important technical issues are altogether absent in rank one. We warn the reader of these simplifications. First, since the structure group is abelian, the moduli spaces are naturally groups (or torsors over groups). This strong structure is missing for higher rank Higgs bundles. Endomorphism bundles of line bundles are trivial, but endomorphism bundles are generally nontrivial. In rank one, moduli spaces exist for all objects. One need not remove from consideration “unstable” objects in the sense of Geometric Invariant Theory. Furthermore the main differential equation —- Hitchin’s self-duality equation — decouples, leaving the Higgs field and the holomorphic structure completely independent of each other. This arises from the direct product decomposition ℂ^{∗} ≅ U(1) × ℝ^{+}, which does not generalize to higher rank.

We do not discuss the hyperkähler moment map and quotient construction in this paper, although all the moduli spaces described here can be constructed as complex-symplectic and hyperkähler quotients. The constructions in rank one are particularly simple and familiar. As we do not need this machinery, we do not discuss them here.

For other points of view and related topics, see [A, ABCKT, C2, D3, Si1, Si2, Si3, Si4, Si5, T] and references cited therein.

We thank Nigel Hitchin, Taejung Kim, Weiping Li, Michael Thaddeus and Mike Wolf for their critical reading of this manuscript and numerous helpful suggestions. We are also grateful to Steve Bradlow, Robert Bryant, Kevin Corlette, Simon Donaldson, Lawrence Ein, Elisha Falbel, Charlie Frohman, Oscar Garcia-Prada, Steve Kudla, François Labourie, John Loftin, John Millson, Niranjan Ramachandran, Jonathan Rosenberg, Carlos Simpson, Domingo Toledo, Richard Wentworth, and Scott Wolpert for helpful conversations.

Notation and terminology. To emphasize the different contexts, we reserve Σ for the smooth surface, and X for the Riemann surface “diffeomorphic to Σ.” That is, X is Σ with a conformal structure (which for us is the Hodge ⋆-operator on 1-forms). For constructions involving the differential structure, whether we use Σ or X is a decision on the context. For example, both A^{∗}(Σ) and A^{∗}(X) are correct notations for the de Rham algebra of smooth differential forms on the surface.

Differential forms and cohomology classes are complex-valued, unless otherwise stated. Tensor products of modules are over ℤ unless otherwise stated. If V is a complex vector space, then we denote the real vector space underlying V by V_{ℝ}. The complex vector space V then consists of the pair (V_{ℝ}, I) where I : V_{ℝ} ⟶ V_{ℝ} is the complex structure, the ℝ-linear automorphism I satisfying I^{2} = −1 corresponding to scalar multiplication by \sqrt{−1}. The complex vector space V̄ complexconjugate to V is (V_{ℝ}, −I) where −I is the opposite complex structure. Thus an anti-linear map V ⟶ W into a complex vector space W is a linear map V̄ ⟶ W.

The multiplicative groups of nonzero complex and real numbers are denoted by ℂ^{∗} and ℝ^{∗} respectively. The multiplicative group of positive real numbers is denoted ℝ^{+}. The multiplicative group of unit complex numbers is denoted U(1). We denote the k × k identity matrix by 𝕀_{k}.

By a vector bundle we mean a smooth complex vector bundle. If E ⟶ M is a vector bundle and N is a submanifold with inclusion map f : N ↪ M, we denote by E|_{N} the restriction of E to N, that is, the vector bundle over N defined as pullback f^{∗}E of E by f.

1. Equivalences of deformation theories

A moduli problem seeks to classify a class of objects up to an equivalence relation, often defined by a group of transformations of the set of objects.

A deformation theory (or transformation groupoid) (S, G) consists of a category C defined by a group action as follows. Let α : G×S ⟶ S be a left action of a group G on a set S. The deformation theory (S, G) consists of the category C whose objects form a set Obj(C) = S with morphisms

x ⟶^{g} y.

corresponding to triples (g, x, y) ∈ G × S × S such that α(g, x) = y.

The identity element e ∈ G determines, for each object x ∈ S the identity morphism

x ⟶^{e} x.

The inverse of the morphism

x ⟶^{g} y

is

y ⟶^{g^{−1}} x

and the composition of morphisms

x ⟶^{g} y ⟶^{h} z

equals

x ⟶^{hg} z.

In particular every morphism is an isomorphism.

The moduli set corresponding to such a deformation theory is the set Iso(C) of isomorphism classes of objects. The isotropy group of an object x ∈ Obj(C) is the set Mor(x, x) consisting of morphisms x ⟶ x, which has the structure of a group. An equivalence of categories is a functor F : A ⟶ B such that there exists a functor H : B ⟶ A and natural transformations from the compositions F ∘ H and H ∘ F to the identity functors of B and A respectively. (See Jacobson [Ja] or Gelfand-Manin [GM], p.28 for discussion of this notion and Goldman-Millson [GoMi] for an application closely related to this one.) An equivalence of categories induces a bijection Iso(A) ⟶ Iso(B), although in general Obj(A) and Obj(B) may be enormously different. For example, each groupoid arising from a group G operating on itself by left-multiplication is equivalent to the groupoid with one object and one morphism.

Equivalent deformation theories yield equivalent moduli sets. However the finer notion of equivalence has further implications— for example isotropy groups of corresponding points in the moduli spaces are isomorphic.

Often the sets Obj(A) admit additional algebraic or geometric structures, which induce additional structures on Iso(A). For the examples discussed here, these moduli sets are Lie groups, and the equivalences of deformation theories induces isomorphisms of (real) Lie groups.

Equivalent deformation theories may have different structures. An equivalence of a deformation theory A with another deformation theory may provide additional structures to Iso(A). For example, Hom(π, U(1)) inherits the structure of a complex abelian variety from every Riemann surface with fundamental group π.

The following criterion is a useful tool for proving that a functor is an equivalence of categories. A functor F : A ⟶ B is an equivalence if and only if:

• Surjective on isomorphism classes: The induced map

F_{∗} : Iso(A) ⟶ Iso(B)

is surjective;

• Full: For x, y ∈ Obj(A), the map

F (x, y) : Mor(x, y) ⟶ Mor(F (x), F (y))

is surjective;

• Faithful: For x, y ∈ Obj(A), the map

F (x, y) : Mor(x, y) ⟶ Mor(F (x), F (y))

is injective.

2. The Betti and de Rham deformation theories and their moduli spaces

This section describes the Betti and de Rham deformation theories. Fix a structure group G. Although much of what is here generalizes to the case that G is a linear algebraic group, we restrict in this paper to the cases of G being the group ℂ^{∗} of nonzero complex numbers, the group U(1) of unit complex numbers, or the group ℝ^{∗} of nonzero real numbers. In what follows, Σ is a compact smooth oriented surface with fundamental group π.

• The Betti groupoid whose objects are representations π ⟶ G, with morphisms G;

• The de Rham groupoid whose objects are flat connections on a trivial complex line bundle over Σ, with morphisms gauge transformations;

2.1. The Betti groupoid. Denote by Hom(π, G) the set of representations from π to G. The group G acts on representations by conjugation. The Betti groupoid is the category having Hom(π, G) as the set of objects and morphisms

g : ρ_{1} ⟶ ρ_{2}

where g ∈ G, ρ_{1}, ρ_{2} ∈ Hom(π, G) and

ρ_{2} = ι_{g} ∘ ρ_{1}

where ι_{g} : G ⟶ G is the inner automorphism defined by conjugation by g. The Betti groupoid is (Hom(π, G), G).

The surface group π admits a presentation

(2.1.1)  ⟨A_{1}, B_{1}, …, A_{k}, B_{k} | [A_{1}, B_{1}] … [A_{k}, B_{k}] = 1⟩

where [A, B] denotes ABA^{−1}B^{−1}. The map

Hom(π, G) ⟶ G^{2k}
ρ ⟼ (ρ(A_{1}), ρ(B_{1}), …, ρ(A_{k}), ρ(B_{k}))

embeds Hom(π, G) as the Zariski-closed subset of G^{2k} defined by

(2.1.2)  [α_{1}, β_{1}] … [α_{k}, β_{k}] = 1.

Since G is abelian, it acts trivially on Hom(π, G). Furthermore (2.1.2) is trivially satisfied and

(2.1.3)  Hom(π, G)/G ≅ Hom(π, G) ≅ G^{2k}.

Hence the moduli space Hom(π, ℂ^{∗}) identifies with (ℂ^{∗})^{2k} with a natural complex structure J.

Each ρ ∈ Hom(π, G) provides 𝔤 with a π-module structure via the adjoint action of G on 𝔤. Since G is abelian, this π-module structure is trivial. The tangent space of Hom(π, G) at ρ identifies with the group cohomology H^{1}(π, 𝔤). Cup product

H^{1}(π, 𝔤) × H^{1}(π, 𝔤) ⟶ ℂ,

defined by

(2.1.4)  Ω(u, v) := ∑_{i=1}^{g} u(A_{i})v(B_{i}) − u(B_{i})v(A_{i}),

provides a closed nondegenerate exterior 2-form on Hom(π, G). When G = ℂ^{∗}, this closed exterior 2-form Ω is holomorphic with respect to J. Thus (J, Ω) defines a complex-symplectic structure on Hom(π, ℂ^{∗}).

Each element C ∈ π defines a function

f_{C} : Hom(π, G) ⟶ ℂ
(2.1.5)  ρ ⟼ ρ(C).

which determines a Hamiltonian flow on Hom(π, G) (Goldman [G2]) as follows. Since G is abelian, the function f_{C} depends only on the homology class of C. By applying an automorphism of π, we may assume that C = (A_{1})^{n} for some n ≥ 0. The corresponding flow

Φ_{t} : Hom(π, G) ⟶ Hom(π, G)

for t ∈ ℝ is defined on the generators as follows:

(2.1.6)  Φ_{t}(ρ) : \begin{cases}A_{1} & ⟼ ρ(A_{1}) \\ B_{1} & ⟼ ρ(B_{1})e^{int} \\ A_{j} & ⟼ ρ(A_{j}) if j > 1 \\ B_{j} & ⟼ ρ(B_{j}) if j > 1 \end{cases}

Hom(π, ℂ^{∗}) is a complex Lie group under pointwise multiplication of homomorphisms. Namely, if ρ_{1}, ρ_{2} are homomorphisms, then

(2.1.7)  π ⟶ ℂ^{∗}
γ ⟼ ρ_{1}(γ)ρ_{2}(γ)

is a homomorphism, defining a group structure on Hom(π, ℂ^{∗}) isomorphic to (ℂ^{∗})^{2k}.

Two real forms of ℂ^{∗} are U(1) and ℝ^{∗}. These subgroups are the fixed point sets of anti-involutions of ℂ^{∗} defined by

ι_{U} : z ⟼ (z̄)^{−1}, ι_{ℝ} : z ⟼ z̄

respectively. These induce real structures on Hom(π, ℂ^{∗}) whose sets of fixed points are Hom(π, U(1)) and Hom(π, ℝ^{∗}) respectively. Note that the composition of the anti-involutions is the involution

ι_{U} ∘ ι_{ℝ} : z ⟼ z^{−1}.

2.2. The de Rham groupoid. The second deformation theory concerns flat connections on a smooth complex line bundle E over a smooth surface Σ. The morphisms are gauge transformations, that is, smooth maps Σ ⟶ G. A flat connection defines a notion of a smooth section being locally constant, or parallel. The connection is described as a differential operator which vanishes precisely on locally constant sections.

We restrict to the case when E is the trivial line bundle. As the Betti moduli space, the de Rham moduli space is a complex Lie group of dimension 2k, where k is the genus of Σ. It furthermore enjoys an invariant complex-symplectic structure as well as two real structures (anti-automorphisms) corresponding to the real forms of ℂ^{∗}.

We first review exterior differential forms, connections, and gauge transformations. For background on fiber bundles and their differential geometry, we refer the reader to Steenrod [S] and Kobayashi-Nomizu [KN].

Exterior differential calculus. Let A^{∗}(Σ) denote the de Rham algebra of Σ, that is, the differential graded algebra of ℂ-valued smooth exterior differential forms on Σ. The operations are wedge product

A^{k}(Σ) × A^{l}(Σ) ⟶ A^{k+l}(Σ)

and exterior derivative

d : A^{k}(Σ) ⟶ A^{k+1}(Σ)

satisfying:

• d(ξ ∧ η) = dξ ∧ η + (−1)^{k}ξ ∧ dη if ξ ∈ A^{k}(Σ);
• d ∘ d = 0.

Complex-conjugation η ⟼ η̄ defines an anti-involution of the complex vector space A^{∗}(Σ). Its fixed-point set is the subalgebra A^{∗}(Σ; ℝ) of real differential forms. Its −1-eigenspace consists of the subspace A^{∗}(Σ; iℝ) of purely imaginary differential forms.

Let E be a (smooth complex) vector bundle over Σ. Let A^{k}(Σ; E) denote the collection of E-valued exterior k-forms over Σ; then the graded vector space

A^{∗}(Σ; E) = ⨁_{k≥0} A^{k}(Σ; E)

is a graded module over A^{∗}(Σ).

Let E be a trivial line bundle. A trivialization τ is a nonvanishing section of E. Then A^{k}(Σ; E) identifies with A^{k}(Σ) via

A^{k}(Σ) ⟶ A^{k}(Σ; E)
(2.2.1)  η ⟼ ητ.

Gauge transformations. A linear gauge transformation of E is a smooth bundle automorphism ξ : E ⟶ E covering the identity map of Σ. Thus for each x ∈ S, the gauge transformation ξ acts on the fiber E_{x} by scalar multiplication of some scalar g(x) ∈ ℂ^{∗}. In terms of τ,

ξ(τ) = g · τ

but the smooth map g : Σ ⟶ ℂ^{∗} is independent of τ since ℂ^{∗} is abelian. The group G_{l}(E) of linear gauge transformations of E identifies with the space Map(Σ, ℂ^{∗}) of smooth maps Σ ⟶ ℂ^{∗}. Similarly, the U(1)-gauge group is the subgroup Map(Σ, U(1)) of smooth U(1)-valued maps.

The fundamental groups of G = U(1) and ℂ^{∗} are infinitely cyclic, so a smooth map g ∈ Map(Σ, G) induces a homomorphism

π_{1}(g) : π ⟶ ℤ,

determining an element of

Hom(π, ℤ) ≅ H^{1}(Σ, ℤ).

The resulting group homomorphism Map(Σ, G) ⟶ H^{1}(Σ, ℤ) induces an isomorphism of the group of connected components

π_{0}(Map(Σ, G)) ≅ H^{1}(Σ, ℤ).

Its kernel is the identity component Map(Σ, G)^{0} consisting of nullhomotopic maps Σ ⟶ G. Since

ℤ ↪ ℂ ⟶^{E} ℂ^{∗}
z ⟼ exp(2πiz),

is a universal covering space, Map(Σ, ℂ^{∗})^{0} identifies with

Map(Σ, ℂ) = A^{0}(Σ)

via the exact sequence

Map(Σ, ℂ) ⟶^{exp_{∗}} Map(Σ, ℂ^{∗}) ⟶^{S} H^{1}(Σ, ℤ).

Similarly the identity component of Map(Σ, U(1)) identifies with

Map(Σ, iℝ) = A^{0}(Σ, iℝ).

Connections on vector bundles. A connection on E is an operator

D : A^{0}(Σ; E) ⟶ A^{1}(Σ; E),

such that

(2.2.2)  D(fs) = fD(s) + df ∧ D(s).

(D(s) is called the covariant differential of s with respect to D.) Such a map extends to

D : A^{p}(Σ; E) ⟶ A^{p+1}(Σ; E)

by enforcing the identity

D(α ∧ β) = D(α) ∧ β + (−1)^{k}α ∧ D(β)

for any α ∈ A^{k}(Σ), β ∈ A^{l}(Σ; E). We denote the space of all connections on E by 𝔄(E).

A trivialization τ determines a connection

D_{0} : A^{0}(Σ; E) ⟶ A^{1}(Σ; E) :

by the rule

(2.2.3)  D_{0}(fτ) := df ∧ τ.

(where f ∈ A^{0}(Σ) is a smooth function determining a section fτ by (2.2.1)). This is the unique connection for which τ is parallel (has covariant differential zero).

With respect to this connection, an arbitrary connection D has the form

D = D_{0} + η

where η ∈ A^{1}(Σ) and η acts by exterior multiplication. That is, the covariant differential of a section s = fτ ∈ A^{0}(Σ; E) with respect to D is

D(s) = df ∧ τ + η ∧ fτ ∈ A^{1}(Σ; E).

In particular the 1-form η is given by

D(τ) = η ∧ τ.

The curvature of a connection D is the End(E)-valued exterior 2form F (D) ∈ A^{2}(Σ, End(E)) such that for any section s of E,

D(D(s)) = F (D) ∧ s.

If E is a line bundle, then End(E) is canonically trivial, so F (D) identifies with an ordinary exterior 2-form.

With respect to the trivialization τ, the curvature is:

F (D) = dη ∈ A^{2}(Σ).

Hence a connection D is flat, that is, its curvature vanishes, if and only if dη = 0. Thus the space F_{l}(E) of flat connections identifies with the subspace Z^{1}(Σ) ⊂ A^{1}(Σ) of closed 1-forms. A line bundle (E, D), where D ∈ F_{l}(E) is a flat line bundle.

Gauge transformations and the de Rham groupoid. Let τ be a trivialization and let D_{0} be the corresponding connection. If ξ ∈ G_{l}(E) corresponds to a map g ∈ Map(S, ℂ^{∗}), then the action of ξ on a connection D_{0} + η is given by:

(2.2.4)  ξ · (D_{0} + η) := D_{0} + η + g^{−1}dg

where η ∈ A^{1}(Σ).

To see this, let ξ ·τ be the trivialization obtained by transforming τ by ξ. That is, ξ ·τ = gτ (scalar multiplication). An arbitrary section of E is given by a scalar multiple s = fτ. Let D = D_{0} +η be an arbitrary connection. The effect of the transformed connection ξ ·D on s is given by:

(ξ · D)(s) = D(fgτ) = df ∧ gτ + fdg ∧ τ + fgD(τ)
= df ∧ (ξ · τ) + fg^{−1}dg ∧ (gτ) + fgη ∧ τ
= (df + f(g^{−1}dg + η)) ∧ (ξ · τ)

(since ℂ^{∗} is abelian). In particular this action is independent of τ. Naturality of curvature under gauge transformations

F (ξ · D) = ξ^{∗}(F (D))

follows, in our context, from the closedness of exact forms and the triviality of the action of G_{l}(E) on A^{2}(Σ):

F (ξ · D) = d(η + g^{−1}dg) = dη = F (D) = ξ^{∗}F (D).

Hence the G_{l}(E)-action preserves curvature. The de Rham groupoid is (F_{l}(E), G_{l}(E)).

The de Rham moduli space. The space 𝔄(E) of all connections is an affine space modeled on the space A^{1}(Σ) comprising smooth ℂvalued 1-forms on Σ. Let D_{0} be the connection corresponding to the trivialization τ. Every connection on E is of the form D_{0} + η, where η ∈ A^{1}(Σ). The objects of the de Rham groupoid — flat connections — comprise an affine subspace of 𝔄(E).

With τ, the space F_{l}(E) of flat connections on E identifies with the vector space Z^{1}(Σ) of closed 1-forms. We shall always assume a fixed trivialization τ_{0} for E, which will provide a basepoint D_{0} (an origin) for the affine space 𝔄(E) of connections.

The gauge group G_{l}(E) identifies with Map(Σ, ℂ^{∗}) and its action on 𝔄(E) is given by (2.2.4).

The G_{l}(E)-action on 𝔄(E) decomposes into the action of the identity component G_{l}(E)^{0} = Map(Σ, ℂ^{∗})^{0} on 𝔄(E) and the action of π_{0}(G_{l}(E)) on the quotient. If g ∈ G_{l}(E)^{0} then g = exp f for some f ∈ A^{0}(Σ), and the action on 𝔄(E) is given by:

D_{0} + η ⟼^{g} D_{0} + η + df.

Thus the quotient F_{l}(E)/(G_{l}(E)^{0}) is an affine space whose underlying vector space is the cohomology

H^{1}(Σ) := Z^{1}(Σ)/B^{1}(Σ).

The action of π_{0}(G_{l}(E)) ≅ H^{1}(Σ; ℤ) corresponds to the action of H^{1}(Σ, ℤ) on H^{1}(Σ) via the monomorphism

(2.2.5)  i^{∗} : H^{1}(Σ, ℤ) ⟶ H^{1}(Σ)

induced by the coefficient inclusion ℤ ↪^{i} ℂ. In summary:

Proposition 2.2.1. The de Rham moduli space F_{l}(E)/G_{l}(E) identifies with the cokernel of the map H^{1}(Σ, ℤ) ⟶ H^{1}(Σ) induced by ℤ ↪ ℂ.

Therefore the moduli space inherits a complex structure J arising from the operation

(2.2.6)  J(η) := iη

on 1-forms. Cup product

H^{1}(Σ) × H^{1}(Σ) ⟶ ℂ,

defined on the level of 1-forms by

(2.2.7)  Ω(α, β) := ∫_{Σ} α ∧ β,

induces a nondegenerate exterior 2-form Ω on F_{l}(E)/G_{l}(E). Since Ω is parallel on this affine space, it is closed. Furthermore, this closed exterior 2-form Ω is holomorphic with respect to J. Thus(J, Ω), defines a complex-symplectic structure on F_{l}(E)/G_{l}(E).

Every C ∈ π defines a function

f_{C} : F_{l}(E) ⟶ ℂ
ω ⟼ ∫_{C} ω

corresponding to the function (2.1.5) on the Betti moduli space, and with a Hamiltonian flow corresponding to (2.1.6).

The identity

Ω(Jα, β) = Ω(α, Jβ) = iΩ(α, β)

means that Ω is complex-bilinear with respect to J (that is, has type (2, 0)).

Group structure. The de Rham moduli space H^{1}(Σ)/ H^{1}(Σ, ℤ) is a Lie group as a quotient of a vector space by a discrete subgroup. Addition corresponds to tensor product of flat line bundles as follows. The trivialization τ_{0} of E determines a trivialization τ_{0} ⊗ τ_{0} of E which we henceforth identify with τ_{0}. Let (E, D_{1}) and (E, D_{2}) be two flat line bundles with local sections s_{1}, s_{2} respectively. In terms of the trivialization, write s_{i} = f_{i}τ_{0} where s_{i} ∈ A^{0}(Σ) are functions. The tensor product flat bundle (E, D_{1}) ⊗ (E, D_{2}) is the unique flat bundle (E, D_{1} ⊗ D_{2}) for which s_{1} ⊗ s_{2} is parallel with respect to D_{1} ⊗ D_{2}, where s_{1} and s_{2} are local sections of E parallel with respect to D_{1} and D_{2} respectively.

Write

D_{1} = D_{0} + η_{1}
D_{2} = D_{0} + η_{2}
D_{1} ⊗ D_{2} = D_{0} + η

where η_{1}, η_{2}, η ∈ Z^{1}(Σ). Then s_{i} is D_{i}-parallel if and only if

0 = D_{i}s_{i} = df_{i} + η_{i}f_{i}

for i = 1, 2 whence

(D_{1} ⊗ D_{2})(s_{1} ⊗ s_{2}) = D_{1}(s_{1}) ⊗ s_{2} + s_{1} ⊗ D_{2}(s_{2})
= ((df_{1} + η_{1}f_{1})f_{2} + f_{1}(df_{2} + η_{2}f_{2}))τ_{0}
= D_{0}(s_{1} ⊗ s_{2}) + (η_{1} + η_{2})s_{1} ⊗ s_{2}.

Thus the 1-form η corresponding to the tensor product D_{1} ⊗ D_{2} equals the sum η_{1} + η_{2}.

Real structure. Corresponding to the real structures ι_{ℝ} and ι_{U} on ℂ^{∗} are real structures on the de Rham moduli space. Their fixed point sets are real forms of the de Rham moduli space, corresponding to subdeformation theories for the real forms U(1) and ℝ^{∗} of ℂ^{∗}. On the Lie algebra ℂ of ℂ^{∗} the anti-involutions ι_{U}, ι_{ℝ} respectively induce maps

η ⟼^{ι_{U}} −η̄
(2.2.8)  η ⟼^{ι_{ℝ}} η̄

which are related by the composition

ι_{U} ∘ ι_{ℝ} : η ⟼ −η.

These anti-involutions on the Lie algebras are coefficient maps for antiinvolutions on the moduli spaces, denoted by (ι_{U})_{∗} and (ι_{ℝ})_{∗} respectively. The fixed point set of (ι_{U})_{∗} is the moduli space F_{u}(E)/G_{u}(E) of flat unitary connections. The fixed point set of (ι_{ℝ})_{∗} is the moduli space of flat real connections.

Using the real structure, the complex-symplectic form decomposes into real and imaginary parts

(2.2.9)  Ω = Ω^{Re} + iΩ^{Im},

each of which is a real symplectic structure on the de Rham moduli space. They satisfy:

Ω^{Re}(Jα, Jβ) = −Ω^{Re}(α, β)
Ω^{Im}(Jα, Jβ) = −Ω^{Im}(α, β)
Ω^{Im}(α, β) = Ω^{Re}(Jα, β).

2.3. Equivalence of de Rham and Betti groupoids. Let G be ℂ^{∗}. The cases of ℝ^{∗}, ℝ^{+} and U(1) are similar. Let D be a flat connection on the trivial line bundle E over Σ. Let τ_{0} denote a trivialization, and write

Dτ_{0} = ητ_{0}

for a closed 1-form η. Over any smooth path

[a, b] ⟶^{γ} Σ

parallel transport defines a linear map between the fibers

Π^{−1}(γ(a)) ⟶ Π^{−1}(γ(b))

as follows. The induced line bundle γ^{∗}E over [a, b] has an induced flat connection γ^{∗}D as well as an induced trivialization τ_{0} ∘ γ. Thus

γ^{∗}η = g(t)dt

for a unique function g on [a, b]. Similarly, every section s is

s(t) = f(t)(τ_{0} ∘ γ)

for a unique function f(t). The section s is parallel if and only if:

0 = (γ^{∗}D)s = (f′(t) + g(t)f(t)) dt ⊗ (τ_{0} ∘ γ),

that is,

0 = f′(t) + g(t)f(t).

This differential equation has unique solution

f(t) = exp ( − ∫_{a}^{t} g(s)ds)f(a)

for an arbitrary initial condition f(a). Starting from

s(a) = f(a)τ_{0}(γ(a)),

the corresponding section s(t) = f(t)τ_{0}(γ(t)) is the parallel transport of s(a) along γ. Flatness of D implies that the parallel transport from Π^{−1}γ(a) to Π^{−1}γ(b) along γ depends only on the homotopy class of γ relative to its endpoints.

If γ is a based loop, that is, γ(a) = γ(b) = x_{0}, then holonomy defines a homomorphism

(2.3.1)  hol_{p}(D) : π_{1}(Σ; x_{0}) ⟶ G.

(Compare [KN].)

Let ξ ∈ G_{l}(E) be a gauge transformation. Evaluation of ξ at x_{0} gives a homomorphism of groups G_{l}(E) → G, for which (2.3.1) is equivariant. Together these maps give the holonomy functor between the de Rham and the Betti groupoid

Theorem 2.3.1. The holonomy functor

hol : (F_{l}(E), G_{l}(E)) ⟶ (Hom(π, G), G)

is an equivalence of deformation theories.

Proof. Let Σ be a smooth manifold with basepoint x_{0} ∈ Σ. Let π = π_{1}(Σ, x_{0}) be the corresponding fundamental group and Σ̃ ⟶^{Π} Σ the corresponding universal covering space. Corresponding to a representation ρ ∈ Hom(π, G) is a flat line bundle ℂ_{ρ} ⟶ Σ, defined as follows. The group π acts on the total space Σ̃ × ℂ of the trivial line bundle over Σ̃ by deck transformations on the first factor and via ρ on the second factor:

(2.3.2)  (s̃, x) ⟼^{γ} (γ · s̃, ρ(γ)x)

where x ∈ G. The quotient (Σ̃ × ℂ)/π is the total space of a smooth line bundle

ℂ_{ρ} ⟶^{Π} Σ

which carries a natural flat structure 𝔉_{ρ}, a foliation of ℂ_{ρ} transverse to the fibration such that the restriction of Π to each leaf of 𝔉_{ρ} is a covering space of Σ. The leaves are the projections of the leaves Σ̃×{v} of the product foliation of Σ̃ × ℂ under the quotient map

Σ̃ × ℂ ⟶ ℂ_{ρ}.

The representation ρ is the holonomy representation of ℂ_{ρ}.

The exterior derivative

d̃: A^{k}(Σ̃) ⟶ A^{k+1}(Σ̃)

induces the trivial connection D̃ on the bundle (Σ̃ × ℂ)

D̃ : A^{k}(Σ̃, Σ̃ × ℂ) ⟶ A^{k+1}(Σ̃, Σ̃ × ℂ)

by considering sections as ℂ-valued functions and then taking exterior derivative. The π-action via ρ is equivariant with respect to D̃. Since flatness is a local condition, D̃ descends to a flat connection D on (Σ̃ × ℂ)/π

A local section s is parallel if and only if D(s) = 0 if and only if its image lies in a leaf of 𝔉_{ρ}. Equivalently, a lift of s to Σ̃ has constant projection under the Cartesian projection

Σ̃ × ℂ ⟶ ℂ

defining the foliation.

The covering homotopy theorem for fiber bundles implies that any continuous path ρ_{t} ∈ Hom(π, G), where 0 ≤ t ≤ 1, induces an isomorphism of line bundles

ℂ_{ρ_{0}} ⟶ ℂ_{ρ_{1}}.

Since Hom(π, G) is an ℝ-algebraic set, its connected components are path-connected and the topological type of the bundle ℂ_{ρ} depends only on the connected component of Hom(π, G) containing ρ. If Hom(π, G) is connected, then any representation can be connected to the trivial representation by a continuous path, and therefore defines the trivial bundle. The existence of such a D for each ρ ∈ Hom(π, G) shows that hol_{p} is surjective on isomorphism classes.

Next we show hol_{p} is faithful and full. Recall that (2.3.1) is equivariant with respect to homomorphism G_{l}(E) → G. Hence G_{l}(E)- equivalent connections give rise to G-equivalent representations. Suppose

D_{1}, D_{2} ∈ F_{l}(E).

There are two cases, depending on whether or not D_{1}, D_{2} are G_{l}(E)- equivalent. If D_{1}, D_{2} are not G_{l}(E)-equivalent, then Mor(D_{1}, D_{2}) is empty. Otherwise, there exists ξ ∈ G_{l}(E) corresponding to a map g ∈ Map(X, G) such that

D_{2} = D_{1} + g^{−1}dg

and if ξ_{1} is another gauge transformation corresponding to a map g_{1} ∈ Map(X, G) such that

D_{2} = D_{1} + g_{1}^{−1}dg_{1},

then g_{1} = gc, where c is a constant map. We denote the subgroup of G_{l}(E) corresponding to constant maps X ⟶ G by G. Hence Mor(D_{1}, D_{2}) corresponds to the coset g · G.

Now suppose that D_{1}, D_{2} ∈ F_{l}(E) and hol_{p}(D_{i}) = ρ_{i} for i = 1, 2. If D_{1} is not G_{l}(E)-equivalent to D_{2}, then then Mor(ρ_{1}, ρ_{2}) = ∅. Otherwise, ρ_{1} = gρ_{1}g^{−1} = ρ_{2} where g ∈ G. Mor(ρ_{1}, ρ_{2}) = gGg^{−1} = G. Then we have an isomorphism

g^{−1} : Mor(D_{1}, D_{2}) ⟶ Mor(ρ_{1}, ρ_{2}).

In both cases, hol_{p} induces an isomorphism

Mor(D_{1}, D_{2}) ⟶ Mor(hol_{p}(D_{1}), hol_{p}(D_{2}))

as desired. □

For a careful treatment of holonomy for higher ranks and in general, see Kobayashi-Nomizu [KN] and Goldman-Millson [GoMi]. Theorem 2.3.1 is stated and proved in [GoMi].

The holonomy representation of a tensor product of flat line bundles is the product of the holonomy representations, as defined in (2.1.7). Therefore the corresponding isomorphism of moduli spaces is an isomorphism of groups.

If D = D_{0} + η is a flat connection, then its holonomy is the homomorphism

π ⟶ G
γ ⟼ exp ∫_{γ} η

which depends holomorphically on D. Thus the isomorphism of moduli spaces is an isomorphism of complex Lie groups. This isomorphism is also symplectomorphic with regard to the complex-symplectic structures Ω defined above.

3. The Dolbeault groupoid

Let X be a Riemann surface diffeomorphic to Σ. With this extra structure is associated a third deformation theory, the Dolbeault groupoid. In this section we define this deformation theory and relate its moduli space to the Jacobian Jac(X).

3.1. Holomorphic line bundles. We begin by reviewing differential calculus on a Riemann surface. A holomorphic structure is defined as an extension D′′ of the \overline{∂}-operator to sections of a line bundle. For line bundles, a Higgs field is a holomorphic 1-form on X. A Higgs pair is a holomorphic structure together with a Higgs field.

Exterior calculus on Riemann surface. The Hodge ⋆-operator on 1-forms

⋆ : A^{1}(X) ⟶ A^{1}(X)

describes the conformal structure on X as follows. Let 𝕁 : T X ⟶ T X denote the complex structure on T X: for each x ∈ X, multiplication by \sqrt{−1} is an ℝ-linear automorphism 𝕁_{x} of the (real) tangent space (T_{x}X)_{ℝ} with

𝕁_{x} ∘ 𝕁_{x} = −1.

A 1-form η ∈ A^{1}(X) defines an ℝ-linear map T_{x}X ⟶ ℂ. The Hodge ⋆- operator is the map on A^{1}(X) induced by 𝕁, that is, define ⋆η ∈ A^{1}(X) as the composition

T_{x}X ⟶^{𝕁^{−1}} T_{x}X ⟶^{η} ℂ.

(When X is given a conformal Riemannian metric, this operator agrees with the Riemannian ⋆-operator. For 1-forms on a 2-manifold, ⋆ is independent of the Riemannian metric.)

Thus ⋆ defines an endomorphism of the complex vector space A^{1}(X) with ⋆⋆ = −1, so its eigenvalues are ±i. Furthermore A^{1}(X) decomposes as a direct sum of its i-eigenspace and (−i)-eigenspace respectively:

A^{1}(X) = A^{1,0}(X) ⊕ A^{0,1}(X)

and the summands are complex-conjugates of each other.

Composing the exterior derivative d : A^{0}(X) ⟶ A^{1}(X) with projections of A^{1}(X) onto A^{1,0}(X) and A^{0,1}(X) defines operators

∂ : A^{0}(X) ⟶ A^{1,0}(X)
\overline{∂} : A^{0}(X) ⟶ A^{0,1}(X)

with d = ∂ + \overline{∂}.

A holomorphic 1-form is a (1, 0)-form which is closed; a closed (0, 1)- form is an anti-holomorphic 1-form. A harmonic 1-form is the sum of a holomorphic 1-form and an antiholomorphic 1-form. The spaces of holomorphic 1-forms, antiholomorphic 1-forms, and harmonic 1-forms, denoted by ℋ^{1,0}(X), ℋ^{0,1}(X), and ℋ^{1}(X) respectively, satisfy:

ℋ^{1}(X) = ℋ^{0,1}(X) ⊕ ℋ^{1,0}(X).

where the two summands are complex-conjugates of each other.

Later we need the following basic facts, which can be found in, for example, Farkas-Kra [FK], Forster [Fo] (Theorem 19.9, p.156, Theorem 19.12, p.159) Griffiths-Harris [Gr], Jost [Jo],§§3.3,5.2, Wells [Wel].

• Let η ∈ A^{0,1}(X). Then there exists a unique antiholomorphic 1-form η_{0} ∈ ℋ^{0,1}(X) and a function f ∈ A^{0}(X) such that

η = η_{0} + \overline{∂}f

• Let η ∈ Z^{1}(X) be a closed 1-form. Then there exists a unique harmonic 1-form η_{0} and a function f ∈ A^{0}(X) such that

η = η_{0} + df.

These standard facts form the analytic foundation for the theory expounded here.

Holomorphic structures. Just as a connection on a vector bundle provides a differential criterion characterizing sections to be locally constant, or parallel, a holomorphic structure on a vector bundle over a complex manifold provides a notion of a section to be holomorphic.

From a preferred class of holomorphic local sections, one can find an atlas of local trivializations of the vector bundle such that the transition functions on overlapping coordinate patches are holomorphic. We do not take that approach here, instead referring to Gunning[Gu1], Kobayashi [K].

The trivialization τ_{0} determines a holomorphic structure, such that s = fτ_{0} is a holomorphic section defined on an open set U if and only if f is a holomorphic function on U. In particular holomorphic local sections solve the Cauchy-Riemann equation.

Holomorphic structures are conveniently described by differential operators of degree 1

D′′ : A^{p,q}(X; E) ⟶ A^{p,q+1}(X; E)

which satisfy

D′′(f · s) = \overline{∂}f ∧ s + f · D′′(s)

for f ∈ A^{0}(X). Locally every holomorphic structure admits holomorphic sections, and are thus equivalent to the standard one D′′_{0} (which arises from \overline{∂} and the trivialization). This follows from the solvability of the inhomogeneous Cauchy-Riemann equation. See Atiyah-Bott [AB], §5 (pp. 554–55) or Kobayashi [K], Chapter 1, Propositions 3.5–3.6, p.9 for further discussion.

Denote by D′_{0} and D′′_{0} the (1,0)- and (0,1)- parts of D_{0} respectively:

D′_{0} = (D_{0})^{1,0}
D′′_{0} = (D_{0})^{0,1}

so that

D_{0} = D′_{0} + D′′_{0}.

With respect to the trivialization τ_{0}, a holomorphic structure on E is an operator

D′′ = D′′_{0} + Ψ

(where Ψ ∈ A^{0,1}(X)), which operates on smooth sections by:

A^{0,0}(X, E) ⟶^{D′′} A^{0,1}(X, E)
fτ_{0} ⟼ (\overline{∂}f + fΨ)τ_{0}.

(When dim(X) > 1, holomorphic structures must satisfy an extra integrability condition. Compare Kobayashi [K].)

Denote the space of all holomorphic structures on E by Hol(E). The linear gauge group G_{l}(E) acts on Hol(E) by

(3.1.1)  D′′_{0} + Ψ ⟼^{ξ} D′′_{0} + Ψ + g^{−1}\overline{∂}g.

Denote the resulting deformation theory by (Hol(E), G_{l}(E)).

Higgs fields. If (E, D′′) is a holomorphic vector bundle over X, then a Higgs field on (E, D′′) is a (1,0)-form Φ on X taking values in the endomorphism bundle End(E). The Higgs field Φ is required to be holomorphic with respect to the holomorphic structure on T^{∗}M ⊗ End(E) induced by the complex structure on X and the holomorphic structure D′′ on E. Thus a Higgs field is a holomorphic bundle map T X ⟶ End(E). (In higher dimensions, the Higgs field is required to satisfy the integrability condition Φ ∧ Φ = 0, which automatically holds when X is 1-dimensional.)

A Higgs pair (or Higgs bundle) is a pair ((E, D′′), Φ) where (E, D′′) is a holomorphic vector bundle and Φ is a Higgs field on (E, D′′).

When E is a holomorphic line bundle, End(E) is trivial and a Higgs field is a holomorphic 1-form. Thus a rank one Higgs pair is just a holomorphic structure D′′ together with a holomorphic 1-form. Thus the space of Higgs pairs (E, D′′, Φ) is

(3.1.2)  Higgs(E) = Hol(E) × ℋ^{1,0}(X).

The linear gauge group G_{l}(E) acts on Hol(E) by (3.1.1) and on Higgs fields by conjugation. Since multiplication in End(E) is commutative, G_{l}(E) acts trivially on ℋ^{1,0}(X). Denote the resulting groupoid by (Higgs(E), G_{l}(E)).

The groupoid (Higgs(E), G_{l}(E)) contains the subgroupoid of holomorphic line bundles (Hol(E), G_{l}(E)) as Higgs pairs whose Higgs fields are identically zero.

3.2. The moduli spaces. Now we describe the Dolbeault moduli spaces of rank one holomorphic bundles and Higgs bundles over a compact Riemann surface X. These moduli spaces will identify with spaces of unitary and linear representations of π_{1}(X) respectively.

Moduli of holomorphic structures. The space Hol(E) of all holomorphic structures is an affine space modeled on A^{0,1}(X). If D′′_{0} is the holomorphic structure corresponding to a trivialization, then every holomorphic structure is of the form D′′_{0} + Ψ, where Ψ ∈ A^{0,1}(X). As for connections on a trivial line bundle E, we fix a trivialization and its corresponding holomorphic structure. The affine spaces have natural vector space structures, in which vector addition corresponds to tensor product of holomorphic line bundles.

The G_{l}(E)-action on Hol(E) is given by

(3.2.1)  Ψ ↦ Ψ + g^{−1}\overline{∂}g,

where Ψ ∈ A^{0,1}(X) and g ∈ Map(X, ℂ^{∗}).

The G_{l}(E)-action on Hol(E) decomposes into the action of the identity component

G_{l}(E)^{0} = Map(X, ℂ^{∗})^{0}

on Hol(E) and the action of π_{0}(G_{l}(E)) on Hol(E)/(G_{l}(E)^{0}). If g ∈ G_{l}(E)^{0} then g = exp f for some f ∈ A^{0}(X), and the action on Hol(E) is given by:

D′′_{0} + Ψ ⟼^{g} D′′_{0} + Ψ + \overline{∂}f.

Thus the quotient Hol(E)/(G_{l}(E)^{0}) is an affine space whose underlying vector space is the quotient A^{0,1}(X)/\overline{∂}A^{0}(X), the Dolbeault cohomology group. The Hodge decomposition into antiholomorphic forms and \overline{∂}- exact forms (§3)

A^{0,1}(X) = ℋ^{0,1}(X) ⊕ \overline{∂}A^{0}(X)

gives an isomorphism of A^{0,1}(X)/\overline{∂}A^{0}(X) with the space ℋ^{0,1}(X) of antiholomorphic 1-forms on X, which we henceforth denote as V. Thus Hol(E)/(G_{l}(E)^{0}) identifies with the complex vector space V = ℋ^{0,1}(X). The dimension of V equals the genus k of X.

The action of π_{0}(G_{l}(E)) ≅ H^{1}(X; ℤ) corresponds to the action of H^{1}(X, ℤ) on ℋ^{0,1}(X) by translation via the composition map

(3.2.2)  H^{1}(X, ℤ) ⟶^{i^{∗}} H^{1}(X) ⟶^{p_{1}} ℋ^{0,1}(X) = V

where i^{∗} is as in (2.2.5) and p_{1} is the projection to antiholomorphic 1-forms. The image of H^{1}(X, ℤ) is a lattice L ⊂ V of rank

2k = dim(V_{ℝ}).

The quotient Jac(X) := V/L is a complex torus, the Jacobian of X.

The Dolbeault isomorphism identifies H^{1}(X, O) with ℋ^{0,1}(X), where O is the sheaf of germs of holomorphic functions on X. Under the Dolbeault isomorphism, the above homomorphism

H^{1}(X, ℤ) ⟶ ℋ^{0,1}(X)

identifies with the map induced by the inclusion ℤ ⟶ O of the sheaf of germs of locally constant ℤ-valued functions. Its cokernel then identifies with the kernel of the homomorphism H^{1}(X, O^{∗}) ⟶ H^{2}(X, ℤ) induced by the sheaf homomorphism

O ⟶^{E} O^{∗}
f ⟼ exp(2πif),

by the exactness of

0 ⟶ ℤ ⟶ O ⟶^{E} O^{∗}

where O^{∗} is the multiplicative sheaf of germs of nonvanishing holomorphic functions.

Just as the tensor product of flat line bundles corresponds to addition of closed 1-forms, the tensor product of holomorphic line bundles corresponds to addition of antiholomorphic 1-forms, that is, to addition in V. Namely if D′′_{i} = D′′_{0} + Ψ_{i}, then the tensor product holomorphic structure is

D′′_{1} ⊗ D′′_{2} = D′′_{0} + Ψ_{1} + Ψ_{2}.

This multiplicative structure agrees with the multiplicative structure induced by the multiplicative structure on O^{∗}.

Moduli of Higgs pairs. Let (D′′, Φ) ∈ Higgs(E) be a Higgs pair. Its equivalence class is recorded by the equivalence class of the holomorphic bundle (E, D′′) in Jac(X) and the Higgs field Φ in ℋ^{1,0}(X).

Proposition 3.2.1. The moduli space Higgs(E)/G_{l}(E) of Higgs pairs identifies with the product

Jac(X) × ℋ^{1,0}(X).

Complex-conjugation maps the space ℋ^{1,0}(X) of Higgs fields to

ℋ^{0,1}(X) = V̄ .

Thus we identify the space of Higgs fields with V̄ and Proposition 3.2.1 implies that Higgs(E)/G_{l}(E) is the cokernel of

(3.2.3)  L ⟶ V × V̄ ,

that is, V/L × V̄.

We denote by I the complex structure on Higgs(E)/G_{l}(E) arising from the complex structures on V and V̄. In terms of the parameters Ψ ∈ V and Φ ∈ V̄ this complex structure is:

Ψ ⟶^{I} iΨ
(3.2.4)  Φ ⟶^{I} iΦ.

Similarly T^{∗}Jac(X) is an abelian group via a tensor product construction on Higgs pairs. Namely define the tensor product (D′′_{1}, Φ_{1}) ⊗ (D′′_{2}, Φ_{2}) to have holomorphic structure D′′_{1}⊗D′′_{2} and Higgs field Φ_{1}+Φ_{2}.

3.3. Geometric structure of the Dolbeault moduli space. The conformal structure of X induces a rich geometry to the Dolbeault moduli space Higgs(E)/G_{l}(E). This space is a complex manifold under the complex structure I induced from X, which is the total space of a vector bundle over Jac(X). This vector bundle is naturally isomorphic to the cotangent bundle T^{∗}Jac(X), which has a natural complex-symplectic structure. Furthermore the Riemannian metric of Jac(X) defines a proper exhaustion function vanishing on the zero-section Jac(X), and whose gradient flow defines a deformationretraction to the zero-section. In rank two, Hitchin [H1] used a similar technique to compute the Betti numbers of the Higgs bundle moduli space.

Cotangent bundles. We identify Higgs(E)/G_{l}(E) with the cotangent bundle T^{∗}Jac(X) using a Riemannian metric on Higgs(E)/G_{l}(E) associated to X as follows.

The natural Hermitian form on A^{1}(X) defined by

(3.3.1)  ⟨α, β⟩ := ∫_{X} α ∧ ⋆β̄

is nondegenerate, positive definite on A^{0,1}(X) and negative definite on A^{1,0}(X). Its restriction to V = ℋ^{0,1}(X) defines an isomorphism V̄ ⟶ V^{∗} of complex vector spaces. The tangent space of Jac(X) = V/L at any point identifies with V. By the isomorphism defined by ⟨, ⟩, the cotangent space of Jac(X) identifies with V̄. Thus

V/L × V̄ ≅ V/L × V^{∗} ≅ T^{∗}Jac(X).

The real part of ⟨, ⟩ defines Riemannian metrics on V and V^{∗} ≅ V̄ which in turn defines a Riemannian metric g on T^{∗}Jac(X).

The cotangent bundle T^{∗}M of any complex manifold M enjoys a natural complex-symplectic structure Ω_{T^{∗}}, In fact, Ω_{T^{∗}} is exact, that is, there exists a holomorphic 1-form α_{T^{∗}} on M such that Ω_{T^{∗}} = −dα_{T^{∗}}. Let Π : T^{∗}M ⟶ M denote projection and dΠ : T (T^{∗}M) ⟶ T M denote its differential. Let u ∈ T^{∗}M be a covector at Π(u) ∈ M. The value of α_{T^{∗}} at u is defined as:

(α_{T^{∗}})_{(u)} : v ⟼ u(dΠ(v))

for a tangent vector v ∈ T_{u}(T^{∗}M). Then

Ω_{T^{∗}} = −dα_{T^{∗}}

is a complex-symplectic structure on T^{∗}M. This construction is natural, so any biholomorphism of M preserves α_{T^{∗}} and Ω_{T^{∗}}.

For a complex torus, the complex-symplectic-structure Ω_{T^{∗}} is parallel. If q = (q_{1}, …, q_{n}) are local coordinates on M, with corresponding coordinates p = (p_{1}, …, p_{n}) on the fibers, then

α_{T^{∗}} = ∑_{j=1}^{n} p_{i}dq_{i}

(3.3.2)  Ω_{T^{∗}} = ∑_{j=1}^{n} dq_{i} ∧ dp_{i}.

The space T^{∗}Jac(X) supports a holomorphic completely integrable Hamiltonian system. The Hitchin map

Higgs(E)/G_{l}(E) ⟶^{H} ℋ^{1,0}(X)
(3.3.3)  [(D′′, Φ)] ⟼ Φ

associates to a pair (D′′, Φ) its Higgs field. In terms of (3.3.2), this is the projection

(q, p) ⟼ p.

The complex Hamiltonian vector field with potential p_{j} is the parallel vector field \frac{∂}{∂q_{j}} which generates translation in the q_{j}-coordinate. The Hitchin map is a holomorphic moment map for a holomorphic Hamiltonian ℂ^{k}-action, whose orbits are sections of T^{∗}Jac(X) corresponding to the trivialization H, where k is the genus of X. Equivalently these parallel copies of Jac(X) are the preimages of H. Although in this case this construction is elementary, its generalization to higher rank is highly nontrivial and revealing. See Hitchin [H4].

On T^{∗}Jac(X) = V/L × V̄, this structure arises from the linear complex-symplectic structure on the vector space

V ⊕ V̄ = ℋ^{0,1}(X) ⊕ ℋ^{1,0}(X)

defined by

(3.3.4)  Ω_{T^{∗}}((Ψ_{1}, Φ_{1}), (Ψ_{2}, Φ_{2})) := ⟨Ψ_{1}, \overline{Φ_{2}}⟩ − ⟨Ψ_{2}, \overline{Φ_{1}}⟩.

Functions and flows on the Dolbeault moduli space. The energy function

T^{∗}Jac(X) ⟶^{e} ℝ
(3.3.5)  (q, p) ⟼ \frac{1}{2}⟨p, p⟩

vanishes precisely on the zero-section Jac(X), which contains the only critical points of e. These critical points are minima. Let ∇e denote the gradient vector field of e with respect to the Riemannian metric g. Then the flow of −∇(e) defines a deformation retraction of T^{∗}Jac(X) to its zero-section Jac(X), from which one can determine the homotopy type of the moduli space. Hitchin [H1] uses a similar technique to determine the topology of some moduli spaces in rank two. However, in higher rank, not all critical points are minima, and the calculation is considerably more complicated. For further applications see [BGG1, BGG2, Go1, Go2, Go2, X0, X1, X2].

The vector field I∇(e) is the Hamiltonian vector field having potential e with respect to the Kähler form on T^{∗}Jac(X). It generates the Hamiltonian circle action:

(3.3.6)  S^{1} × T^{∗}Jac(X) ⟶ T^{∗}Jac(X)
(3.3.7)  (λ, (q, p)) ⟼ (q, λp)

which evidently extends to a holomorphic ℂ^{∗}-action on T^{∗}Jac(X). The fixed point set of each action equals the zero section p = 0, which is precisely the critical set of e. Furthermore the ℂ^{∗}-action extends to an action of the multiplicative semigroup ℂ; the limit as λ ⟶ 0 lies in the critical set. We revisit this geometry in §6.8.

4. Equivalence of de Rham and Dolbeault groupoids

The Betti, de Rham, and Dolbeault deformation theories of the previous sections are equivalent. Hence the three moduli sets are isomorphic. However these moduli sets have natural differentiable structures (in fact they are complex Lie groups). With this extra structure, we refer to the moduli sets as moduli spaces. The equivalence between the de Rham groupoid and the Dolbeault groupoid induces an isomorphism of real Lie groups between their moduli spaces.

4.1. Construction of the equivalence. Passage from de Rham to Dolbeault groupoids requires the construction of a holomorphic structure and a Higgs field for each flat connection D. To this end, we introduce a Hermitian metric H on E to decompose D into a unitary connection and a 1-form ϕ. The unitary connection determines a unique holomorphic structure D′′. The condition that ϕ determines a Higgs field is equivalent to a harmonicity condition on H, which follows from the standard theory of harmonic forms.

Hermitian metrics. A Hermitian metric H on E is a family of positive definite Hermitian forms ⟨, ⟩_{H}

E_{x} × E_{x} ⟶ ℂ

on the fibers E_{x} smoothly varying with x ∈ Σ. For any vector space V, let Her(V) denote the space of positive definite Hermitian forms V × V ⟶ ℂ. The space Her(E) of Hermitian metrics on the vector bundle E is the space of sections of the Her(ℂ^{r})-bundle associated to E. In terms of a basis, a Hermitian form is represented by a Hermitian matrix h:

H(u, v) = u^{†}hv̄

where † indicates transpose. The action of a linear transformation g ∈ GL(r, ℂ) on a Hermitian form defined by a matrix h is then:

g : h ⟼ g^{†}hḡ.

If E is a flat vector bundle with holonomy ϕ : π ⟶ GL(r, ℂ), then a Hermitian metric H ∈ Her(E) corresponds to a ϕ-equivariant map

(4.1.1)  Σ̃ ⟶^{h} Her(ℂ^{r}).

In particular End(E) is a complex line bundle with a canonical everywhere nonzero section, the identity endomorphism 𝕀 : E_{x} ⟶ E_{x}. Hence End(E) identifies canonically with the trivial line bundle ℂ over X.

The associated bundles of Hermitian metrics are trivial, but not canonically trivial.

However, in the important special case when the holonomy lies in the subgroup U(1), the standard Hermitian form on ℂ

⟨z, w⟩ := zw̄

defines a Hermitian metric on E. Gauge transformations act on Hermitian metrics by:

⟨u, v⟩_{ξ·H} := ⟨g^{−1} · u, g^{−1} · v⟩_{H}

and, in rank one, the corresponding positive function h transforms by:

h ⟼ |g|^{−2}h.

Two Hermitian metrics H_{1}, H_{2} ∈ Her(E) on a complex line bundle E relate by their ratio, the positive function h : X ⟶ ℝ^{+} defined by:

⟨u, v⟩_{H_{1}} = h(x)⟨u, v⟩_{H_{2}}

where u, v ∈ E_{x}.

Lemma 4.1.1. The linear gauge group G_{l}(E) acts transitively on the space Her(E) of Hermitian metrics.

Proof. If h_{1}, h_{2} are positive functions representing Hermitian metrics on E, then

g(z) = \sqrt{h_{1}(z)/h_{2}(z)}

defines a gauge transformation g ∈ G_{l}(E) with g · h_{2} = h_{1}. □

The unique Hermitian metric H_{0} for which

⟨τ, τ⟩_{H_{0}} = 1

corresponds to the constant function 1 and its stabilizer in G_{l}(E) is the unitary gauge group G_{u}(E), corresponding to functions X ⟶ U(1).

A Hermitian metric H on E induces Hermitian pairings over A^{∗}(X)

A^{k}(X; E) × A^{l}(X; E) ⟶ A^{k+l}(X).

A connection D is unitary with respect to a metric H if and only if

d⟨s_{1}, s_{2}⟩_{H} = ⟨Ds_{1}, s_{2}⟩_{H} + ⟨s_{1}, Ds_{2}⟩_{H},

for sections s_{1}, s_{2} ∈ A^{0}(X, E). Equivalently we say that H is parallel with respect to D. With respect to τ_{0}, the connection D = D_{0} + η is unitary with respect to H if and only if:

(4.1.2)  η + η̄ = 2 Re(η) = h^{−1}dh = d log h.

Thus D = D_{0} + η ∈ F_{l}(E) is unitary with respect to some Hermitian metric on E if and only if the 1-form Re(η) is exact.

In particular a connection D = D_{0} + η is unitary with respect to the Hermitian metric H_{0} corresponding to the trivialization τ_{0} (that is, the Hermitian metric with h ≡ 1) if and only if η ∈ A^{1}(X; iℝ). We denote the subset of F_{l}(E) consisting of flat connections unitary with respect to H_{0} by F_{u}(E). Since G_{l}(E) acts transitively on Her(E), a connection D ∈ F_{l}(E) is G_{l}(E)-equivalent to a connection in F_{u}(E) if and only if D is unitary with respect to some Hermitian metric on E.

The following well known result relates holomorphic structures, Hermitian structures and unitary connections:

Proposition 4.1.2. Let E be a complex line bundle over X with Hermitian metric H, and let D′′ be a holomorphic structure on E. Then there exists a unique connection D on E such that

• The (0, 1)-part D^{0,1} of D equals D′′;

• H is parallel with respect to D.

We call D the connection compatible with D′′ and H.

Proof. Write D′′ = D′′_{0} + μ, where μ ∈ A^{0,1}(X). Let h be the equivariant positive function on X̃ corresponding to H as in (4.1.1). Let D = D_{0} + η where

η = μ − μ̄ + h^{−1}∂h.

Since η + η̄ = h^{−1}dh, (4.1.2) implies that D is unitary with respect to H. Since η^{0,1} = μ, the holomorphic structure defined by D equals D′′ as desired.

For uniqueness, let D_{1}, D_{2} be two U(1)-connections with the same (0, 1)-part. By Lemma 4.1.1, we may assume H = H_{0}. With respect to the trivialization τ_{0}, write D_{i} = D_{0} + ψ_{i} where D_{0} is the trivial connection. Then ψ_{1} and ψ_{2} are 1-forms with values in iℝ. Their difference ψ_{1} − ψ_{2} has values in iℝ with Hodge type (1,0). Since a purely imaginary 1-form of Hodge type (1,0) is necessarily zero, ψ_{1}−ψ_{2} is zero. □

(Although we only proved Lemma 4.1.1 and Proposition 4.1.2 for line bundles, these results hold for vector bundles of arbitrary rank.)

The unitary case. Suppose D is a connection. Its composition

A^{0}(X; E) ⟶^{D} A^{1}(X; E) ⟶ A^{0,1}(X; E)

with (0, 1)-projection is a holomorphic structure, denoted D^{0,1}. If

D = D_{0} + η,

then the corresponding holomorphic structure is

D^{0,1} = D′′_{0} + Ψ

where Ψ = η^{0,1}.

The correspondence between holomorphic structures on E and flat unitary connections is almost an equivalence of deformation theories. Define a functor

(F_{u}(E), G_{u}(E)) ⟶^{S} (Hol(E), G_{l}(E)),

which on objects is the map assigning to a flat U(1)-connection D on E the holomorphic structure D^{0,1}, and on morphisms is the inclusion G_{u}(E) ↪ G_{l}(E).

Unfortunately S is not full. Consider the connection D_{0} on E and D′′_{0} the trivial holomorphic structure. Then

Mor(D_{0}, D_{0}) ⟶ Mor(D′′_{0}, D′′_{0})

is not surjective: Mor(D_{0}, D_{0}) consists of scalar multiplication by unit complex numbers, while Mor(D′′_{0}, D′′_{0}) corresponds to scalar multiplication by ℂ^{∗}.

To overcome this difficulty, rigidify the groupoid (Hol(E), G_{l}(E)) by including a Hermitian metric as an extra piece of data. Therefore G_{l}(E) acts on Hol(E) × Her(E).

Suppose D′′ ∈ Hol(E) and H ∈ Her(E). Let D be the unique connection compatible with D′′ and H (Proposition 4.1.2).

Definition 4.1.3. A Hermitian metric H is adapted to D′′ if and only if the connection D compatible with D′′ and H is flat, that is, F (D) = 0.

(In Corlette [C2], such metrics are called harmonic, with respect to a holomorphic line bundle. To avoid confusion, we chose to restrict the terminology “harmonic metric” to the context of flat bundles and refer to the corresponding notion for holomorphic bundles as “adapted metric”.)

Let Hol_{u}(E) be the subset of Hol(E) × Her(E) consisting of all (D′′, H) such that H is adapted to D′′ in the sense of Definition 4.1.3. Then G_{l}(E) acts on Hol_{u}(E), and we denote the corresponding deformation theory (Hol_{u}(E), G_{l}(E)).

Proposition 4.1.4. The natural map

F_{u}(E) ⟶ Hol_{u}(E)
D ⟼ (D^{0,1}, 1)

is equivariant with respect to the natural G_{l}(E)-action. The corresponding functor

(F_{u}(E), G_{u}(E)) ⟶^{T} (Hol_{u}(E), G_{l}(E))

is an equivalence of deformation theories.

Proof. Let (D′′, h) ∈ Hol_{u}(E). By applying a linear gauge transformation, we may assume that h = 1. Apply Proposition 4.1.2, to obtain a U(1)-connection D satisfies D^{0,1} = D′′. By definition of Hol_{u}(E), the connection D is unitary, whence D ∈ F_{u}(E). Thus T_{∗} is surjective.

Since the subgroup of G_{l}(E) preserving H_{0} is G_{u}(E), the induced map

Mor(D_{1}, D_{2}) ⟶^{T (D_{1},D_{2})} Mor(D′′_{1}, D′′_{2})

is an isomorphism. Thus T is faithful and full. □

Proposition 4.1.5. For each D′′ ∈ Hol(E), there exists h ∈ Her(E) such that (D′′, h) ∈ Hol_{u}(E).

Proof. Let D′′ ∈ Hol(E). One must find a flat unitary connection which is equivalent by a gauge transformation in G_{l}(E) to a connection whose (0, 1)-part equals D′′. Write D′′ = D′′_{0} + Ψ where Ψ ∈ A^{0,1}(X). Now

A^{0,1}(X) = ℋ^{0,1}(X) ⊕ \overline{∂}A^{0}(X)

and Ψ = Ψ_{0} + \overline{∂}s, where Ψ_{0} is antiholomorphic. (§3). In particular Ψ_{0} is closed. Let

D = D_{0} + Ψ_{0} − Ψ̄_{0}

and g = exp(s). Then D is flat (since dΨ_{0} = dΨ̄_{0} = 0), unitary with respect to 1 (since Ψ_{0} − Ψ̄_{0} is purely imaginary). Furthermore the gauge transformation g takes

D^{0,1} = D′′_{0} + Ψ_{0}

to D′′. The metric H_{0} corresponds to the function ḡg. The desired adapted metric is g · H_{0}. □

Together, Propositions 4.1.4 and 4.1.5 imply:

Corollary 4.1.6. The induced map

F_{u}(E)/G_{u}(E) ⟶ Hol(E)/G_{l}(E)

is an isomorphism of sets.

That every topologically trivial holomorphic line bundle over X arises from a unitary character of π_{1}(X) is a standard result; see Weyl[Wey], §18, Gunning [Gu1], §8a, pp. 131–135, or Farkas-Kra [FK], §III.9, pp. 119–129. In the terminology of [FK], a character is a homomorphism π_{1}(X) ⟶ ℂ^{∗} and a normalized character is a unitary character. A multiplicative function in the sense of [Wey, FK] corresponds to a holomorphic section of a flat complex line bundle.

The linear case. Now we extend the preceding theory from U(1)- representations and flat unitary connections to ℂ^{∗}-representations and flat linear connections.

Suppose D ∈ F_{l}(E). With respect to τ_{0}, write D = D_{0} + η, where η ∈ Z^{1}(X). Let ϕ, ψ be the real and imaginary parts of η respectively. Since D is flat,

dϕ + dψ = 0.

Since dψ is purely imaginary and dϕ is real, this single equation is equivalent to the pair of equations:

dϕ = dψ = 0.

Thus D is flat if and only if ϕ and ψ are closed.

For any Hermitian metric H on E, decompose the connection D into its skew-adjoint part (a connection D_{H} compatible with H) and its self-adjoint part, a 1-form ϕ_{1} which is self-adjoint with respect to H:

(4.1.3)  D = D_{H} + ϕ_{1},

where

D_{H} = D_{0} + ψ + \frac{1}{2}h^{−1}dh

and

ϕ_{1} = ϕ − \frac{1}{2}h^{−1}dh

where h is the positive function corresponding to H as in §4.1.

Then D_{H} is flat (h^{−1}dh is exact) and compatible with H (by (4.1.2)). Since D_{H} is flat, ϕ_{1} is a closed 1-form. Let Φ = (ϕ_{1})^{1,0}.

Harmonic metrics. We also recall basic facts about harmonic functions. A smooth function f is harmonic if and only if d ⋆ df = 0. In particular holomorphic functions are harmonic. A harmonic 1-form locally is the differential of a harmonic function. Every harmonic function is locally the real part of a holomorphic function. A smooth function f : X ⟶ ℝ^{+} is multiplicatively harmonic if and only if its logarithm

log f : X̃ ⟶ ℝ

is a harmonic function.

Definition 4.1.7. Let D be a flat connection on the line bundle E over X. A Hermitian metric H ∈ Her(E) is harmonic with respect to D if and only if the equivariant map

X̃ ⟶^{h̃} ℝ^{+}

corresponding to H is a multiplicatively harmonic function on X̃, that is, its logarithm log h̃ is harmonic function X ⟶ ℝ.

This definition is independent of the trivialization of D over the universal covering space X̃ ⟶ X used to define h̃.

H is a harmonic metric with respect to D if and only if the 1-form ϕ_{1} = ϕ − \frac{1}{2}h^{−1}dh is harmonic, or equivalently \overline{∂}Φ = 0.

Lemma 4.1.8. Let D be a flat connection. Then there exists a Hermitian metric H harmonic with respect to D. Furthermore H is unique up to multiplication by a positive constant.

Proof. Write D = D_{0} + ψ + ϕ as above, where ϕ ∈ Z^{1}(X, ℝ). Then ϕ is cohomologous to a unique harmonic form ϕ_{1} ∈ ℋ^{1}(X; ℝ), that is,

ϕ = ϕ_{1} + df.

for some smooth function f ∈ A^{0}(X, ℝ) (§3). The desired harmonic metric is h = e^{2f}. Since f is unique up to addition of a real constant, the metric H is unique up to multiplication by a positive constant. □

In higher rank, these facts generalize to deeper technical results. The existence and uniqueness of flat unitary connections on a topologically trivial holomorphic line bundle (Proposition 4.1.5) generalizes to Donaldson’s proof of the Narasimhan-Seshadri theorem [D1], Hitchin’s proof of solutions of the self-duality equations in rank two [H1] and Simpson’s proof in general [Si1, Si2]. Lemma 4.1.8 generalizes to the existence of harmonic metrics on flat bundles with reductive holonomy (Donaldson [D2] in rank two, and Corlette [C1] in general). The existence of a harmonic metric generalizes the existence and uniqueness of harmonic maps of compact Riemannian manifolds into complete Riemannian manifolds of nonnegative curvature, due to Eells-Sampson [ES], and has been further generalized by Labourie [L]).

Let D ∈ F_{l}(E) and H its harmonic metric and D = D_{H} +ϕ_{1}, where D_{H} is compatible with H. Let D′′ be the (0,1)-part of D_{H} and Φ be the (1,0)-part of ϕ_{1}. Since ϕ_{1} is harmonic, \overline{∂}Φ = 0. Thus (E, D′′, Φ) is a Higgs pair.

Theorem 4.1.9. The resulting functor

(F_{l}(E), G_{l}(E)) ⟶^{S} (Higgs(E), G_{l}(E)).

is an equivalence of deformation theories.

Proof. First we show that S is surjective on isomorphism classes. Let (D′′, Φ) ∈ Higgs(E). By Proposition 4.1.5, D′′ has an adapted metric H corresponding to a function h : X → ℝ^{+}. Let D_{H} be the unique connection compatible with H and (D_{H})^{0,1} = D′′ (Proposition 4.1.2). Write D_{H} = D_{0} + ψ. Then ξ · H = H_{0} for some linear gauge transformation ξ ∈ G_{l}(E). Let

D = ξ · (D_{H} + Φ + Φ̄) = D_{0} + Φ + Φ̄.

The curvature of D is

F (D) = F (ξ · D_{H}) + d(Φ + Φ̄).

Since H is adapted to D′′, the curvature F (D_{H}) = 0, so F (ξ · D_{H}) = 0. Since Φ is holomorphic,

d(Φ + Φ̄) = 0.

Hence F (D) = 0 and D ∈ F_{l}(E). Then S(D) is a Higgs pair differing from (D′′, Φ) by the gauge transformation ξ.

Next we show S is faithful and full. Suppose

(D′′_{1}, Φ_{1}), (D′′_{2}, Φ_{2}) ∈ Higgs(E).

There are two cases, depending on whether or not (D′′_{1}, Φ_{1}), (D′′_{2}, Φ_{2}) are G_{l}(E)-equivalent. If (D′′_{1}, Φ_{1}), (D′′_{2}, Φ_{2}) are not G_{l}(E)-equivalent, then

Mor((D′′_{1}, Φ_{1}), (D′′_{2}, Φ_{2}))

is empty. Otherwise, there exists ξ ∈ G_{l}(E) corresponding to a map g ∈ Map(X, ℂ^{∗}) such that

D′′_{2} = D′′_{1} + g^{−1}\overline{∂}g, Φ_{2} = Φ_{1}

and if ξ_{1} is another gauge transformation corresponding to a map g_{1} ∈ Map(X, ℂ^{∗}) such that

D′′_{2} = D′′_{1} + g_{1}^{−1}\overline{∂}g_{1},

then g_{1} = gc, where c is a constant map. We denote the subgroup of G_{l}(E) corresponding to constant maps X ⟶ ℂ^{∗} by ℂ^{∗}. Hence Mor((D′′_{1}, Φ_{1}), (D′′_{2}, Φ_{2})) corresponds to the coset g · ℂ^{∗}.

Now suppose that D_{1}, D_{2} ∈ F_{l}(E) and S(D_{i}) = (D′′_{i}, Φ_{i}) for i = 1, 2. If D_{1} is not G_{l}(E)-equivalent to D_{2}, then then Mor(D_{1}, D_{2}) = ∅ since ℂ^{∗} acts trivially on F_{l}(E). Otherwise, D_{1} = g · D_{2} where g ∈ G_{l}(E). Then Mor(D_{1}, D_{2}) = g · ℂ^{∗}.

In both cases, S induces an isomorphism

Mor(D_{1}, D_{2}) ⟶ Mor(S(D_{1}), S(D_{2}))

as desired. □

Corollary 4.1.10. The induced map

F_{l}(E)/G_{l}(E) ⟶^{S_{∗}} Higgs(E)/G_{l}(E)

is an isomorphism of sets.

4.2. Higgs coordinates. To see that the bijection S_{∗} between the de Rham and Dolbeault moduli spaces is an isomorphism of real Lie groups, we examine in detail the constructions in the proof of Theorem 4.1.9. We explicitly describe how to pass from the equivalence class of a flat connection D = D_{0} + η to an equivalence class of a Higgs pair (D′′, Φ), where D′′ = D′′_{0} + Ψ. We call η ∈ ℋ^{1}(X) the connection coordinates of the point in F_{l}(E)/G_{l}(E) and (Ψ, Φ) ∈ V ⊕ V̄ the Higgs coordinates.

Passing from connection coordinates η ∈ ℋ(X) to Higgs coordinates (Ψ, Φ) ∈ V ⊕ V̄ involves the following steps.

We assume that η is harmonic, since we are only interested in the equivalence class of D under G_{l}(E). Decompose η into its imaginary part ψ and its real part ϕ:

η = ψ + ϕ

where

ψ = \frac{1}{2}(η − η̄), ϕ = \frac{1}{2}(η + η̄).

The unitary connection D_{H} := D_{0} + ψ corresponds to a holomorphic structure D′′ = D′′_{0} + Ψ where the antiholomorphic 1-form Ψ is the (0, 1)-part of ψ:

Ψ = ψ^{0,1}
ψ = Ψ − Ψ̄.

The (1, 0)-part Φ of ϕ is a holomorphic 1-form:

Φ = ϕ^{1,0}
ϕ = Φ + Φ̄.

In summary, to pass from Higgs coordinates to connection coordinates:

(4.2.1)  η = Ψ − Ψ̄ + Φ + Φ̄

and to pass from connection coordinates to Higgs coordinates:

Ψ = (i Im η)^{0,1}
(4.2.2)  Φ = (Re η)^{1,0}

Holomorphic structures associated to flat connections. Since

η = ψ + ϕ
= (Ψ − Ψ̄) + (Φ + Φ̄)
= (Ψ + Φ̄) + (Φ − Ψ̄)

where Ψ ∈ ℋ^{0,1}(X) and Φ ∈ ℋ^{1,0}(X), the Hodge components of η are:

η^{0,1} = Ψ + Φ̄
η^{1,0} = Φ − Ψ̄.

The flat connection D determines a holomorphic structure:

D^{0,1} := D′′_{0} + η^{0,1} = D′′_{0} + Ψ + Φ̄ = D′′ + Φ̄.

The holomorphic structure D′′ in the Higgs pair (D′′, Φ) is:

D′′ := D′′_{0} + Ψ.

Thus, unless Φ = 0, the holomorphic structure in the Higgs pair is not the holomorphic structure determined by the flat connection.

Isomorphism of Lie groups. Now we strengthen Corollary 4.1.10:

Proposition 4.2.1. The induced map

F_{l}(E)/G_{l}(E) ⟶^{S_{∗}} Higgs(E)/G_{l}(E)

is an isomorphism of real Lie groups

Proof. Using harmonic decomposition, the moduli space F_{l}(E)/G_{l}(E) identifies with the quotient of ℋ^{1}(X) by the discrete subgroup corresponding to the subgroup H^{1}(X, ℤ) ↪ H^{1}(X). The moduli space Higgs(E)/G_{l}(E) corresponds to the quotient of V ⊕V̄ by a lattice L ⊂ V in the first factor. The map S_{∗} arises from the ℝ-linear map

ℋ^{1}(X) ⟶ V ⊕ V̄
η ⟼ (Ψ, Φ)

in Higgs coordinates. The induced bijection S_{∗} is an isomorphism of real Lie groups. □

Real structures. The anti-involutions defining the real forms U(1) and ℝ^{∗} were described in connection coordinates in (2.2.8). Using (4.2.1) and (4.2.2) to pass to Higgs coordinates, (ι_{U})_{∗} and (ι_{ℝ})_{∗} are:

(Ψ, Φ)|⟶^{(ι_{U})_{∗}} (Ψ, −Φ)
(4.2.3)  (Ψ, Φ)|⟶^{(ι_{ℝ})_{∗}} (−Ψ, Φ).

The fixed point set of (ι_{U})_{∗} equals F_{u}(E)/G_{u}(E). In the description of the moduli space as T^{∗}Jac(X), the involution ι_{U} is scalar multiplication by −1 on the fibers of T^{∗}Jac(X) ⟶ Jac(X).

In contrast, ι_{ℝ} leaves Φ unchanged, but replaces the holomorphic structure D′′ by its inverse. In the cotangent bundle description, ι_{ℝ} is the lift of inversion

Jac(X) ⟶ Jac(X)
L ⟼ L^{−1}

to T^{∗}Jac(X) which fixes the parallelism

T^{∗}Jac(X) ⟶ T_{L_{0}}^{∗}Jac(X) ≅ V^{∗} ≅ V̄ .

The differential of inversion induces the composition of this map with the map ι_{U} above given by scalar multiplication by −1 on the cotangent fibers.

The fixed points of inversion form the 2-torsion subgroup

Jac_{2}(X) ↪ Jac(X)

corresponding to

(\frac{1}{2}L)/L ⊂ V/L

which is isomorphic to (ℤ/2)^{2k} where k is the genus of X. Thus the fixed point set of ι_{ℝ} acting on Hom(π, ℂ^{∗}) is the subset Hom(π, ℝ^{∗}) of

Hom(π, ℂ^{∗}) ⟷ T^{∗}Jac(X)

corresponding to

T_{Jac(X)}^{∗}|_{Jac_{2}(X)} ≅ (ℤ/2)^{2k} × V̄ .

4.3. Involutions. The involution ι_{U} is holomorphic with respect to I, but anti-holomorphic with respect to J. The fixed point set of t = −1 in the ℂ^{∗}-action is holomorphic with respect to I, but is a purely real subset with respect to J. Hence this fixed point set is Hom(π, U(1)) which embeds into Higgs(E)/G_{l}(E) as a holomorphic submanifold by (4.2.3).

Similarly, the involution ι_{ℝ} is holomorphic with respect to I and antiholomorphic with respect to J. The fixed point set of ι_{ℝ} is Hom(π, ℝ^{∗}) which is holomorphic with respect to I.

The common fixed point set of ι_{U} and ι_{ℝ} is the intersection

Hom(π, U(1)) ∩ Hom(π, ℝ^{∗}) = Hom(π, ±𝕀)

which identifies with the 2-torsion subgroup Jac_{2}(X) of the Jacobian. (This is also the fixed point set of the composition ι_{U} ∘ι_{ℝ}.) In particular Hom(π, ℝ^{∗}) corresponds to Higgs pairs (D′′, Φ) where the holomorphic line bundle (E, D′′) has order two in the Jacobian. Its identity component Hom(π, ℝ^{+}) corresponds to Higgs bundles whose underlying holomorphic structure is trivial.

The holomorphic structure of the Higgs pair corresponding to an ℝ^{+}-representation is trivial. If ρ : π ⟶ ℝ^{+} is a nontrivial real character, then there exists a global holomorphic section to the corresponding flat line bundle but no global parallel section. This holomorphic section arises as follows. Under the identifications

Hom(π, ℝ^{+}) ⟶^{log} Hom(π, ℝ) ≅ H^{1}(X; ℝ)

ρ corresponds to a harmonic 1-form ϕ. On the universal cover, ϕ lifts to an exact 1-form which is the differential of a harmonic function f : X̃ ⟶ ℝ satisfying:

(4.3.1)  f ∘ γ − f = log ρ(γ).

There exists a holomorphic function F : X̃ ⟶ ℂ with Re F = f which necessarily satisfies (4.3.1). (The function F is unique up to an additive purely imaginary constant.) Then exp F is a holomorphic function on X̃ and defines a nonvanishing holomorphic section of the corresponding flat line bundle.

This holomorphic section is nonvanishing for the following general reason. Since every flat line bundle has degree zero, a nonzero holomorphic section has no zeroes. Compare the discussion in Gunning [Gu2], pp.237–238. The function f arose earlier in the construction of the harmonic metric in Lemma 4.1.8.

In general the direct product decomposition of Lie groups ℂ^{∗} ≅ U(1) × ℝ^{+} induces a decomposition of Betti moduli spaces

Hom(π, ℂ^{∗}) ≅ Hom(π, U(1)) × Hom(π, ℝ^{+}).

The corresponding decomposition of Dolbeault moduli spaces

Higgs(E)/G_{l}(E) ≅ Jac(X) × H^{1}(X; ℝ)

associates to a Higgs pair the underlying holomorphic structure and the class of the logarithmic differential of the harmonic metric in H^{1}(X; ℝ).

5. Hyperkähler geometry on the moduli space

Theorem 2.3.1 and Corollaries 4.1.6 and 4.1.10 establish bijections between the three moduli spaces

(ℂ^{∗})^{2k}, H^{1}(Σ)/ H^{1}(Σ, ℤ), T^{∗}Jac(X)

which are isomorphisms of real Lie groups. The first two — Betti and de Rham — were isomorphic as complex Lie groups with isomorphic complex-symplectic structures. We henceforth will only work with the de Rham and Dolbeault moduli spaces.

The complex structure J on the de Rham moduli space and the complex structure I on the Dolbeault space are quite different. Using the isomorphism S_{∗}, we superimpose these structures on a single object which we call M. The interaction between these structures leads to a rich hyperkähler geometry and group actions on M.

5.1. The quaternionic structure. In connection coordinates η ∈ ℋ^{1}(X) and Higgs coordinates (Ψ, Φ) ∈ V ⊕ V̄, the complex structure I on M is:

η ⟼^{I} − ⋆ η̄
(5.1.1)  (Ψ, Φ) ⟼ (iΨ, iΦ).

The formula in connection coordinates may be derived as follows. Let (Ψ, Φ) ∈ V ⊕ V̄ be the Higgs coordinates of η. By definition of I, the Higgs coordinates of I(η) are (iΨ, iΦ) so (4.2.1) implies:

I(η) = (iΨ) − \overline{(iΨ)} + (iΦ) + \overline{(iΦ)}
= iΨ + iΨ̄ + iΦ − iΦ̄
= ⋆Ψ − ⋆Ψ̄ − ⋆Φ − ⋆Φ̄
= − ⋆ \overline{(−Ψ̄ + Ψ + Φ̄ + Φ)}
= − ⋆ η̄.

Using the equivalence S_{∗} with the de Rham moduli space, the natural complex structure on the de Rham moduli space (2.2.6) defines the complex structure J on M:

η ⟼^{J} iη
(5.1.2)  (Ψ, Φ) ⟼ (iΦ̄, −iΨ̄).

To derive the expression in Higgs coordinates, decompose η as in (4.2.1). Then:

J(η) = iη = iΨ − iΨ̄ + iΦ + iΦ̄
= (iΦ̄) − \overline{(iΦ̄)} + (−iΨ̄) + \overline{(−iΨ̄)}

so that the Higgs coordinates of J(η) are (iΦ̄, −iΨ̄). Let K := IJ. In coordinates:

(5.1.3)  η ⟼^{K} i ⋆ η̄
(Ψ, Φ) ⟼ (−Φ̄, Ψ̄).

The complex structures I, J satisfy

I^{2} = J^{2} = −1, IJ = −JI

and together I, J, K define a left action of the algebra of quaternions ℍ on ℋ^{1}(X). Here ℍ is the ℝ-algebra with basis 1, I, J, K and multiplication satisfying:

(5.1.4)  I^{2} = J^{2} = K^{2} = −1
IJ = −JI = K, JK = −KJ = I, KI = −IK = J.

Furthermore every purely imaginary unit quaternion u = u_{I}I + u_{J}J + u_{K} (where u_{I}, u_{J}, u_{K} ∈ ℝ satisfy u_{I}^{2} + u_{J}^{2} + u_{K}^{2} = 1) satisfies u^{2} = −1, obtaining a whole S^{2} of complex structures. Opposite complex structures are related by the antipodal map:

S^{2} ⟶ S^{2}
(5.1.5)  u ⟼ −u.

5.2. The Riemannian metric. The real part of the Hermitian form ⟨, ⟩ on V in (3.3.1) and the real part of the corresponding Hermitian form on V̄ define a Riemannian metric g on M:

g((Ψ_{1}, Φ_{1}), (Ψ_{2}, Φ_{2})) = Re⟨Ψ_{1}, Ψ_{2}⟩ + Re⟨Φ_{1}, Φ_{2}⟩
(5.2.1)  g(η_{1}, η_{2}) = ∫_{X} η_{1} ∧ ⋆ \overline{η_{2}}.

The Riemannian metric is invariant under the complex structures I, J, K:

g(η_{1}, η_{2}) = g(Iη_{1}, Iη_{2}) = g(Jη_{1}, Jη_{2}) = g(Kη_{1}, Kη_{2})

and more generally g(uη_{1}, uη_{2}) = g(η_{1}, η_{2}) for u ∈ S^{2}. Thus, for each u ∈ S^{2}, the Riemannian metric g is Hermitian with respect to u, that is, of type (1, 1) with respect to u.

Each purely imaginary unit quaternion u ∈ S^{2} determines a realsymplectic structure, or Kähler form ω_{u}:

(5.2.2)  ω_{u}(η_{1}, η_{2}) := g(uη_{1}, η_{2}).

ω_{u} is the real-symplectic structure associated to a Kähler structure H for which the associated Riemannian structure is g, with respect to the complex structure u:

Im H = ω_{u}
Re H = g.

The general definition of a hyperkähler structure is a quadruple

(g, I, J, K)

where g is a Riemannian metric and I, J, K are integrable almost complex structures satisfying the quaternion identities (5.1.4), for which g is Kählerian with respect to each of them.

A hyperkähler structure also determines complex-symplectic structures. For each fixed u ∈ S^{2}, there is a family of complex-symplectic structures which are holomorphic with respect to u. Namely, extend u to a positively oriented orthonormal frame (u, u_{1}, u_{2}), and define:

(5.2.3)  Ω_{(u_{1},u_{2})} := ω_{u_{1}} + iω_{u_{2}},

where the Kähler forms ω_{u_{i}} are defined in (5.2.2). Then Ω_{(u_{1},u_{2})} is ℂbilinear with respect to the complex structure u. Furthermore every other positively oriented orthonormal frame (u, u′_{1}, u′_{2}) is obtained by rotation:

u′_{1} = cos(θ)u_{1} − sin(θ)u_{2}
u′_{2} = sin(θ)u_{1} + cos(θ)u_{2}

for some θ ∈ ℝ and

Ω_{(u′_{1},u′_{2})} = e^{iθ}Ω_{(u_{1},u_{2})}.

The unit vector u_{2} is determined by u and u_{1} as the cross-product u × u_{1}. Thus these complex-symplectic structures are parametrized by orthonormal pairs (u, u_{1}). Since the tangent space to S^{2} at u ∈ S^{2} is the orthogonal complement u^{⊥} ⊂ ℝ^{3}, the complex-symplectic structures are parametrized by the unit tangent bundle T_{1}S^{2}. Left-multiplication by u preserves u^{⊥} and defines an integrable almost complex structure on S^{2}. In §6.1, we describe how this complex manifold identifies with the complex projective line ℙ^{1} and the corresponding line bundle is the holomorphic tangent bundle of P^{1}. It has degree 2 and is denoted O(2). In §6 these complex-symplectic structures on the moduli space unify into one single holomorphic exterior 2-form on the twistor space which takes values in a line bundle induced from O(2).

5.3. Complex-symplectic structure. An alternative approach to hyperkähler geometry is to begin with a complex-symplectic manifold and refine the complex-symplectic structure to a hyperkähler structure. Suppose that (M, J, Ω) is a complex-symplectic manifold, that is, (M, J) is a complex manifold and Ω is closed nondegenerate complexvalued exterior 2-form of type (2, 0) with respect to J. A hyperkähler structure refining (J, Ω) consists of another complex structure I which is anti-linear with respect to J and for which the symmetric form defined by

(5.3.1)  g(α, β) := Re Ω(α, Iβ)

is Kählerian with respect to J and I.

This approach can be succinctly described in terms of classical Lie groups as follows. Let ℂ^{2n} be the standard 2n-dimensional complex symplectic vector space. The complex symplectic group Sp(2n, ℂ) is a subgroup of the automorphisms of ℂ^{2n}. The stabilizer of the standard Hermitian structure on ℂ^{2n} is the compact symplectic group

Sp(2n) = Sp(2n, ℂ) ∩ U(2n),

the maximal compact subgroup of Sp(2n, ℂ).

A reduction of structure group for the tangent bundle of a 4ndimensional real manifold from GL(4n, ℝ) to Sp(2n, ℂ) consists of a pair (Ω, J) where J is an almost complex structure and Ω a J-bilinear complex-valued nondegenerate exterior 2-form on M. This reduction of structure group is integrable, that is, corresponds to a complexsymplectic structure on M if and only if J is integrable and Ω is closed.

A further reduction of structure group from Sp(2n, ℂ) to its maximal compact subgroup Sp(2n) consists of another complex structure I which is J-anti-linear, and such that (5.3.1) defines a positive definite Hermitian metric with respect to I. Such a reduction of structure group gives tensor fields g, I, J, K etc. which satisfy the hyperkähler identities. For further examples and discussion of hyperkähler geometry, we refer the reader to Besse [B], Dancer [D], Hitchin [H3], Hitchin-Karlhede-Lindström-Roc̆ek [HKLR], Joyce [Joy], or Salamon [Sa].

Relation to the moduli spaces. These objects relate to the J-holo-morphic complex-symplectic structure Ω on the de Rham moduli space in §2.2. By the definitions ((2.2.7), (3.2.4), (5.2.1)) of Ω, I, g,

g(η_{1}, η_{2}) = Re Ω(Iη_{1}, η_{2}).

Thus

(5.3.2)  Ω = Ω_{(−I,K)} = −ω_{I} + iω_{K}

in the notation of (5.2.3), with real and imaginary parts discussed in (2.2.9).

Similarly the I-holomorphic complex-symplectic structure Ω_{T^{∗}} on the Dolbeault moduli space (3.3.4) is:

(5.3.3)  Ω_{T^{∗}} = 8Ω_{J,K}

in the notation of (5.2.3). We indicate the proof of (5.3.3) as follows: The projections onto Hodge components are:

ℋ^{1}(X) ⟶ ℋ^{0,1}(X)  ℋ^{1}(X) ⟶ ℋ^{0,1}(X)
η ⟼ \frac{1}{2}(η − i ⋆ η)  η ⟼ \frac{1}{2}(η + i ⋆ η).

The (0, 1)− and (1, 0)− components of ψ = 2i Im(η) and ϕ = 2 Re(η) respectively, are:

Ψ = \frac{1 − i⋆}{2} \frac{η − η̄}{2} = \frac{1}{4}(η − η̄ − i ⋆ η + i ⋆ η̄)
Φ = \frac{1 + i⋆}{2} \frac{η + η̄}{2} = \frac{1}{4}(η + η̄ + i ⋆ η + i ⋆ η̄).

Applying (4.2.1) to (3.3.4)

Ω_{T^{∗}}(η_{1}, η_{2}) = 2{(η_{1} ∧ η_{2} − η̄_{1} ∧ η̄_{2})

+ (η_{1} ∧ i ⋆ η̄_{2} − η̄_{1} ∧ i ⋆ η_{2}) + (−i ⋆ η_{1} ∧ η̄_{2} + i ⋆ η̄_{1} ∧ η_{2})
+ (⋆η_{1} ∧ ⋆η_{2} − ⋆η̄_{1} ∧ ⋆η̄_{2})}

Taking real parts and imaginary parts, (5.3.3) follows. The Kähler form on T^{∗}Jac(X) is ω_{I}.

5.4. Quaternionization. The hyperkähler structure identifies the vector space V ⊕ V̄ with the tensor product (over ℂ)

V_{ℍ} := ℍ ⊗_{ℂ} V.

Here the quaternion algebra ℍ is a complex vector space where the action of i is left-multiplication by I ∈ ℍ. This quaternionic action extends to V_{ℍ}, making V_{ℍ} a left ℍ-module.

Identify V ⊕ V̄ with V_{ℍ} as follows: Let a : V ⟶ V̄ be the antiisomorphism given by the identity map on V_{ℝ}. The ℝ-linear automorphism

(Ψ, Φ) ⟼^{J} (−a^{−1}(Φ), a(Ψ))

of V ⊕ V̄ is anti-linear and satisfies J^{2} = −1. The given complex structures I and J define a left ℍ-action on V ⊕ V̄. The resulting map

V_{ℍ} ⟶ V ⊕ V̄

is an isomorphism of left ℍ-modules.

The two complex structures ±I are special, since the ℝ-span of L is invariant under each of these structures. For those two complex structures, V_{ℍ}/L contains the complex torus V/L as a compact holomorphic submanifold. However, for all other complex structures u ∈ S^{2}, the resulting complex manifold is Stein:

Theorem 5.4.1. Let u ∈ S^{2}. Suppose that u ≠ ±I. Then the complex manifold (V_{ℍ}/L, 𝕁_{u}) is biholomorphic to (ℂ^{∗})^{2k} where k = dim_{ℂ}(V).

Proof. The lattice L spans V over ℝ, and V is I-invariant. Denote by ℂ_{u} the subalgebra ℝ1 + ℝu generated by u. Since u ≠ ±I, the two unit quaternions I and u generate ℍ. Thus V generates V_{ℍ} over the subalgebra ℂ_{u} ⊂ ℍ. Thus the ℝ-linear map

ℂ ⊗_{ℝ} V ⟶ V_{ℍ}
(x + yi) ⊗ v ⟼ (x + yu)v

is surjective. It is an isomorphism because

dim_{ℝ} (ℂ ⊗_{ℝ} V ) = 4k = dim_{ℝ} V_{ℍ}.

As abelian groups, V = L ⊗ ℝ, so

V_{ℍ} ≅ ℂ ⊗_{ℝ} V ≅ ℂ ⊗ L

Since L is free abelian of rank 2k,

V_{ℍ}/L ≅ (ℂ ⊗ L)/L
≅ (ℂ ⊗ L)/(ℤ ⊗ L)
≅ (ℂ/ℤ) ⊗ L
≅ (ℂ^{∗}) ⊗ L ≅ (ℂ^{∗})^{2k}

as desired □

6. The twistor space

A hyperkähler structure is equivalent to a holomorphic construction which embodies all the complex and complex-symplectic structures, due to Hitchin-Karlhede-Lindström-Roc̆ek [HKLR].

For each unit purely imaginary quaternion u ∈ S^{2}, left-multiplication by u describes a complex structure on V_{ℍ}/L. The collection of complex manifolds (V_{ℍ}/L, u) unifies in a single holomorphic fibration Π_{Z} over S^{2}, where S^{2} identifies with the complex projective line ℙ^{1} by stereographic projection (6.1.1). The corresponding holomorphic fibration Z ⟶^{Π_{Z}} ℙ^{1} is the twistor space of M.

In this section we explicitly describe this fibration and use it to describe Simpson’s ℂ^{∗}-action on M. We realize ℙ^{1} as a space of complex structures on ℍ by stereographic projection. The twistor space results, as well as a closed holomorphic exterior 2-form Ω on Z. This exterior 2-form defines the holomorphic complex-symplectic structure on the fibers of Π_{Z}, taking values in the pullback Π_{Z}^{∗}O(2) of the line bundle O(2) on ℙ^{1}. Differentiably, Z is the product bundle V_{ℍ}/L × ℙ^{1}, but holomorphically it is nontrivial. The rational curves l_{v} = {[v]} × ℙ^{1} define a family of holomorphic sections of Z, the twistor lines, whose normal bundles are isomorphic to O(1). Finally the antipodal map α on ℙ^{1} lifts to an anti-holomorphic involution on Z which satisfies compatibility relations with the twistor lines and Ω. The basic result (Theorem 3.3 of [HKLR]) is that the collection (Π_{Z}, {l_{v}}, Ω, α̃) is equivalent to a hyperkähler structure. We describe these objects explicitly in this special case (Theorem 6.7.1).

Finally the ℂ^{∗}-action is described in terms of the twistor construction.

6.1. The complex projective line. We briefly review the geometry of ℙ^{1}. For more details, see §1–§2 of Hitchin’s article in [HSW].

Points in ℙ^{1} correspond to complex lines (1-dimensional linear subspaces) in ℂ^{2}. Cover ℙ^{1} by two affine patches 𝔸_{1}, 𝔸_{2} with coordinate charts:

ℂ ⟶^{ψ^{(1)}} 𝔸_{1}  ℂ ⟶^{ψ^{(2)}} 𝔸_{2}
ζ ⟼ ℂ [\begin{matrix}ζ \\ 1 \end{matrix}] ⊂ ℂ^{2}  ξ ⟼ ℂ [\begin{matrix}1 \\ ξ \end{matrix}] ⊂ ℂ^{2}

where ζξ = 1 on the intersection ℂ^{∗} ≅ 𝔸_{1} ∩ 𝔸_{2}.

Stereographic projection. The unit sphere S^{2} of purely imaginary unit quaternions, with its complex structure defined in §5.1, identifies with ℙ^{1} as follows. We embed ℂ ⊂ ℍ as the subalgebra generated by I. The two affine patches 𝔸_{1}, 𝔸_{2} ⊂ ℙ^{1} map affinely into the multiplicative subgroup ℍ^{∗} of nonzero quaternions by:

𝔸_{1} ⟶ ℍ^{∗}
ψ^{(1)}(ζ) ⟼ 1 + ζK

and

𝔸_{2} ⟶ ℍ^{∗}
ψ^{(2)}(ξ) ⟼ ξ̄+ K

respectively. Use these maps to conjugate I ∈ ℍ to purely imaginary unit quaternions in S^{2}:

I_{ζ}^{(1)} := (1 + ζK)I(1 + ζK)^{−1}
(6.1.1)  I_{ξ}^{(2)} := (ξ̄+ K)I(ξ̄+ K)^{−1} = (ξ − K)^{−1}I(ξ − K),

where ζ, ξ ∈ ℂ. These two maps combine to define a map ℙ^{1} ⟶ S^{2} because:

Lemma 6.1.1. If ξζ = 1, then I_{ζ}^{(1)} = I_{ξ}^{(2)}.

Proof. Since ζ ∈ ℂ = ℝ1 + ℝI,

1 + ζK = 1 + Kζ̄ = (ζ̄^{−1} + K)ζ̄ = (ξ̄+ K)ζ̄

so

(1 + ζK)^{−1} = (ζ̄)^{−1}(ξ̄+ K)^{−1}

and

I_{ζ}^{(1)} = (1 + ζK)I(1 + ζK)^{−1}
= (ξ̄+ K)ζ̄I(ζ̄)^{−1}(ξ̄+ K)^{−1}
= (ξ̄+ K)I(ξ̄+ K)^{−1}
= I_{ξ}^{(2)}.
□

Evidently I_{0}^{(1)} = I , I_{0}^{(2)} = −I, and I^{(1)}(1) = I^{(2)}(1) = J. Explicitly, I_{ζ}^{(1)}, I_{ξ}^{(2)} are stereographic projections:

I_{ζ}^{(1)} = \frac{1 − ζζ̄}{1 + ζζ̄}I + \frac{2 Re(ζ)}{1 + ζζ̄}J + \frac{2 Im(ζ)}{1 + ζζ̄}K
(6.1.2)  I_{ξ}^{(2)} = −\frac{1 − ξξ̄}{1 + ξξ̄}I + \frac{2 Re(ξ)}{1 + ξξ̄}J − \frac{2 Im(ξ)}{1 + ξξ̄}K.

(This differs slightly from formulas (3.70) and (3.71) of [HKLR]. The formulas in [HKLR] relate to ours by replacing ζ with ζ̄. In our convention, stereographic projection is a map from ℙ^{1} to S^{2} with the orientation of S^{2} given by the outward normal vector.)

The antipodal map. Stereographic projection relates the antipodal map S^{2} ⟶ S^{2} defined in (5.1.5) to the map ℙ^{1} ⟶^{α} ℙ^{1} induced by the anti-linear involution of ℂ^{2}:

ℂ^{2} ⟶^{ᾱ} ℂ^{2}
[\begin{matrix}z_{1} \\ z_{2} \end{matrix}] ⟼ [−z̄_{2} ; z̄_{1}] .

In terms of the atlas, it is given by:

ψ^{(1)}(ζ) ⟷^{α} ψ^{(2)}(−ζ̄)
ψ^{(2)}(ξ) ⟷^{α} ψ^{(1)}(−ξ̄)

for ξ, ζ ∈ ℂ and in particular

ψ^{(1)}(0) ⟷^{α} ψ^{(2)}(0).

Lemma 6.1.2. The antipodal map takes a complex structure to its opposite. That is, if ζ ∈ ℂ is nonzero, then

I_{−ζ̄^{−1}}^{(1)} = I_{−ζ̄}^{(2)} = −I_{ζ}^{(1)}

Proof.

I_{−ζ̄^{−1}}^{(1)} = (1 − ζ̄^{−1}K)I(1 − ζ̄^{−1}K)^{−1}

= \frac{1}{1 + |ζ̄^{−1}|^{2}} (1 − ζ̄^{−1}K)I(1 + ζ̄^{−1}K)

= \frac{|ζ|^{2}}{|ζ|^{2} + 1} (1 − ζ̄^{−1}K)I(1 + ζ̄^{−1}K)

= \frac{1}{1 + |ζ|^{2}} (1 − Kζ^{−1})ζIζ̄(1 + ζ̄^{−1}K)

= \frac{1}{1 + |ζ|^{2}} (ζ − K)I(ζ + K)

= \frac{1}{1 + |ζ|^{2}} (−ζK − 1)KIK(1 − ζK)

= − \frac{1}{1 + |ζ|^{2}} (1 + ζK)I(1 − ζK)

= − I_{ζ}^{(1)}

as claimed. □

Line bundles. For each d ∈ ℤ there is a holomorphic line bundle O(d) of degree d over ℙ^{1}, unique up to isomorphism. Every holomorphic line bundle over ℙ^{1} is isomorphic to some O(d).

Explicitly, the total space of O(d) is defined in the above atlas as the identification space of the disjoint union

(ℂ × 𝔸_{1}) ∐ (ℂ × 𝔸_{2})

under the equivalence relation:

(6.1.3)  (z, ψ^{(1)}(ζ), z) ∼ (ξ^{d}z, ψ^{(2)}(ξ)),

where ζξ = 1. A section s of O(d) determines a pair (s^{(1)}, s^{(2)}) of maps ℂ ⟶ ℂ such that s(ψ^{(1)}(ζ)) corresponds to (ψ^{(1)}(ζ), s^{(1)}(ζ)) in 𝔸_{1}×ℂ and s(ψ^{(2)}(ξ)) corresponds to (ψ^{(2)}(ξ), s^{(2)}(ξ)) in 𝔸_{2} × ℂ. By (6.1.3), the pair (s^{(1)}, s^{(2)}) must satisfy:

(6.1.4)  s^{(1)}(ζ) = ξ^{−d}s^{(2)}(ξ)

whenever ζξ = 1. Conversely any pair (s^{(1)}, s^{(2)}) satisfying (6.1.4) corresponds to a section of O(d)

The trivial rank two bundle

ℂ^{2} × ℙ^{1} ⟶ ℙ^{1}

contains the (tautological) subbundle O(−1) whose fiber over [v] ∈ ℙ^{1} is the line ℂv ⊂ ℂ^{2}. The hyperplane bundle over ℙ^{1} is its dual O(1): the fiber over ℂv consists of linear functionals on ℂ_{v}. A linear functional

ℂ^{2} ⟶^{a} ℂ

defines a section of O(1) by assigning to [v] ∈ ℙ^{1} its restriction to ℂv. Every holomorphic section of O(1) arises from such a linear functional on ℂ^{2}. A nonzero holomorphic section has a single zero of order 1, corresponding to the kernel of a. For example, the linear functional

[\begin{matrix}z_{1} \\ z_{2} \end{matrix}] ⟼ z_{2}

determines the section σ of O(1) having

s^{(1)}(ζ) = 1, s^{(2)}(ξ) = ξ.

This section is holomorphic on 𝔸_{1} and has a single simple zero at ψ^{(2)}(0) ∈ 𝔸_{2}.

The canonical line bundle T^{∗}ℙ^{1} has degree −2 and is isomorphic to O(−2); the holomorphic 1-form

dζ = −ξ^{−2}dξ

is holomorphic on 𝔸_{1} with a double pole at ζ = ∞. In the atlas, this section is given by holomorphic functions

s^{(1)}(ζ) = 1
s^{(2)}(ξ) = ξ^{−2}

where the local nonvanishing sections over 𝔸_{1} and 𝔸_{2} are dζ and −dξ respectively.

The inverse of the canonical bundle, the tangent bundle T ℙ^{1} ≅ O(2), has a holomorphic section

(6.1.5)  δ = \frac{∂}{∂ζ} = −ξ^{2} \frac{∂}{∂ξ}

with a double zero at ξ = 0. In coordinate charts,

δ^{(1)}(ζ) = 1
δ^{(2)}(ξ) = ξ^{2}.

This section corresponds to the section −σ^{2} = −σ ⊗ σ of O(2) = O(1) ⊗ O(1).

Antiholomorphic line bundles. Given any holomorphic vector bundle E ⟶ M, the conjugate vector bundle Ē ⟶ M is defined as the holomorphic vector bundle with opposite complex structure on the fibers. Fiberwise complex-conjugation defines an isomorphism of smooth vector bundles between E and Ē which is antiholomorphic. A Hermitian structure on ℂ^{2} defines Hermitian structures on any O(d) which induces isomorphisms

\overline{O(d)} ≅ O(−d).

Lemma 6.1.3. Suppose that d ∈ ℤ is even. Then the antipodal map α of ℙ^{1} lifts to an antiholomorphic isomorphism

O(d) \underrightarrow{α̃} \overline{O(d)}.

Proof. In the coordinate atlas, define

(z, ψ^{(1)}(ζ)) α̃ ; ⟼ (z̄, ψ^{(2)}(−ζ̄))
(w, ψ^{(2)}(ξ)) α̃ ; ⟼ (w̄, ψ^{(1)}(−ξ̄)).

Suppose that z, ζ ∈ ℂ define a point (z, ψ^{(1)}(ζ)) in ℂ × 𝔸_{1}, whose image α̃(z, ψ^{(1)}(ζ)) corresponds to

(6.1.6)  ( (−ξ̄)^{d}z̄, ψ^{(1)}(−ξ̄) ) ∈ ℂ × 𝔸_{1},

where ζξ = 1. On the other hand, (z, ψ^{(1)}(ζ)) corresponds to the point (ξ^{d}z, ψ^{(2)}(ξ)) in ℂ×𝔸_{2}. This point maps under α̃ to ((\overline{ξ^{d}z}), ψ^{(1)}(−ξ̄)), which equals (6.1.6) when d is even. □

When d = 2, the image (α̃)^{∗}dζ = −\overline{dξ} is the image of the section defined above under the differential of α.

6.2. The twistor space as a smooth vector bundle. The triviality of the vector bundle ℂ^{2k} ⊗ O(1) as a smooth real vector bundle may be seen as follows. Isomorphism classes of bundles over ℙ^{1} with structure group G are classified by the homotopy class of their clutching function 𝔸_{1} ∩ 𝔸_{2} ⟶ G, which identifies with an element of π_{1}(G) (see for example Steenrod [S]). Namely, a fiber bundle over a contractible space is trivial, and writing a trivialization over 𝔸_{1} in terms of a trivialization over 𝔸_{2} defines a map 𝔸_{1} ∩ 𝔸_{2} ⟶ G, whose homotopy class is a complete invariant of the isomorphism type as a smooth fiber bundle with structure group G. Since 𝔸_{1} ∩ 𝔸_{2} is homotopy-equivalent to S^{1}, isomorphism classes of G-bundles are classified by π_{1}(G).

The invariant of the line bundle O(1) is a generator α ∈ π_{1}(U(1)). The invariant of ℂ^{2k} ⊗ O(1) is the image of

α ⊕ ⋯ ⊕ α ∈ π_{1}( U(1) × ⋯ × U(1))

under the homomorphism

π_{1}( U(1) × ⋯ × U(1)) ⟶ π_{1}( U(2k))
∥  ∥
ℤ^{2k}  ⟶  ℤ

induced by the inclusion of diagonal matrices U(1)×⋯×U(1) in U(2k). The isomorphism π_{1}(U(2k)) ⟶ ℤ is induced by

det : U(2k) ⟶ U(1).

Thus the invariant of ℂ^{2k}⊗O(1) as a complex vector bundle equals 2kα, where α denotes a generator of π_{1}( U(2k)) ≅ ℤ. The isomorphism type as a real vector bundle is the image of 2kα under the homomorphism

π_{1}(U(2k)) ⟶ π_{1}(O(4k))
∥  ∥
ℤ  ⟶ ℤ/2

which is evidently zero.

6.3. A holomorphic atlas for the twistor space. The twistor space of the quaternionic vector space V_{ℍ} is defined as the product Z̃ = V_{ℍ} × ℙ^{1} with coordinate projection:

Z̃ = V_{ℍ} × ℙ^{1} ⟶^{ΠZ̃} ℙ^{1}.

The complex structure on Z̃ is the unique complex structure such that ΠZ̃ is a holomorphic vector bundle, and the complex structure on the fiber over [v] ∈ ℙ^{1} is the complex structure defined by leftmultiplication by the purely imaginary unit quaternion corresponding to [v]. We shall see that the twistor space Z̃ of V_{ℍ} is the total space of the holomorphic vector bundle ℂ^{2k} ⊗ O(1), and describe the structures (holomorphic symplectic structures, real structure) on this space explicitly in terms of the atlas {(𝔸_{1}, ψ^{(1)}), (𝔸_{2}, ψ^{(2)})} in this and subsequent sections.

Represent complex vectors v ∈ ℂ^{2k} by the notation:

(6.3.1)  v = [\begin{matrix}q \\ p \end{matrix}]

where the components are vectors q, p ∈ ℂ^{k}. Thus a general quaternion vector in ℍ^{k} is:

q + pJ = x_{1} + y_{1}I + x_{2}J + y_{2}K,

where q = x_{1} + iy_{1} and p = x_{2} + iy_{2} for real vectors x_{1}, x_{2}, y_{1}, y_{2} ∈ ℝ^{k}. For ζ, ξ ∈ ℂ, v ∈ ℂ^{2k}, define

ℂ^{2k} ⟶^{f_{ζ}^{(1)}} ℍ^{k}
v = [\begin{matrix}q \\ p \end{matrix}] ⟼ \frac{1}{1 + ζζ̄}(1 + ζK)(q + pJ) = (1 − ζK)^{−1}(q + pJ)

and

ℂ^{2k} ⟶^{f_{ξ}^{(2)}} ℍ^{k}
v = [\begin{matrix}q \\ p \end{matrix}] ⟼ \frac{1}{1 + ξξ̄}(ξ̄+ K)(q + pJ) = (ξ − K)^{−1}(q + pJ).

For each ξ, ζ ∈ ℂ, the maps f_{ζ}^{(1)}, f_{ξ}^{(2)}, are ℝ-linear isomorphisms. Hence if we define charts

ℂ^{2k} × 𝔸_{1} ⟶^{F^{(1)}} ℍ^{k} × 𝔸_{1}
ℂ^{2k} × 𝔸_{2} ⟶^{F^{(2)}} ℍ^{k} × 𝔸_{2}

F^{(1)}(v, ζ) = (f_{ζ}^{(1)}(v), ψ^{(1)}(ζ))
(6.3.2)  F^{(2)}(v, ξ) = (f_{ξ}^{(2)}(v), ψ^{(2)}(ξ)),

we have

F^{(1)}(v, ψ^{(1)}(ζ)) = F^{(2)}(ξv, ψ^{(2)}(ξ)),

thus, obtaining a holomorphic rank 2k vector bundle

ΠZ̃ : Z̃ ⟶ ℙ^{1}.

The fiber of Z̃_{ζ} over ψ^{(1)}(ζ) is ℍ^{k} with the complex structure I_{ζ}^{(1)}. We denote the fiber Π−1 ; Z̃(ψ^{(2)}(0)) by Z̃_{∞}.

Lemma 6.3.1. Let ξζ = 1. Then f_{ζ}^{(1)}(v) = f_{ξ}^{(2)}(ξv).

Proof.

f_{ζ}^{(1)}(v) = \frac{1}{1 + ζζ̄}(1 + ζK)(q + pJ)

= \frac{ξξ̄}{ξξ̄+ 1}(1 + ξ^{−1}K)(q + pJ)

= \frac{1}{1 + ξξ̄}(1 + ξ^{−1}K)ξξ̄(q + pJ)

= \frac{1}{ξξ̄+ 1}(1 + Kξ̄^{−1})ξ̄ξ(q + pJ)

= \frac{1}{ξξ̄+ 1}(ξ̄+ K)ξ(q + pJ)

= f_{ξ}^{(2)}(ξv)

□

Thus {F^{(1)}, F^{(2)}} is an atlas for a holomorphic vector bundle Z̃ of rank 2k over ℙ^{1}. Lemma 6.3.1 implies that this vector bundle is ℂ^{2k} ⊗ O(1), a direct sum of 2k copies of O(1).

Finally, the complex structure on the fiber Z̃_{ζ} is I_{ζ}^{(1)} and I_{0}^{(2)} if ζ = ∞.

Lemma 6.3.2. Let ζ, ξ ∈ ℂ, ζξ = 1. Then

f_{ζ}^{(1)}(iv) = I_{ζ}^{(1)}f_{ζ}^{(1)}(v), f_{ξ}^{(2)}(iv) = I_{ξ}^{(2)}f_{ξ}^{(2)}(v).

Proof.

f_{ζ}^{(1)}(iv) = (1 + ζζ̄)^{−1}(1 + ζK)(iq + ipJ)
= (1 + ζζ̄)^{−1}(1 + ζK)I(q + pJ)
= (1 + ζζ̄)^{−1}I_{ζ}^{(1)}(1 + ζK)(q + pJ)
= I_{ζ}^{(1)}f_{ζ}^{(1)}(v)

and similarly for f_{ξ}^{(2)}. □

6.4. The twistor lines. By (6.3.2) and Lemma 6.3.1, each v ∈ ℂ^{2k} defines a nowhere vanishing holomorphic section of l_{v}, called the twistor line corresponding to v. The normal bundle to l_{v} is isomorphic to the vector bundle ΠZ̃ itself, that is, to ℂ^{2k} ⊗ O(1).

Here is how the the twistor lines appear in the holomorphic atlas. Write

v_{0} = q_{0} + p_{0}J

52 RANK ONE HIGGS BUNDLES

with q_{0}, p_{0} ∈ ℂ^{k}. The twistor line l_{v}(ψ^{(1)}(ζ)) intersects the fiber Z̃_{ζ} at

F^{(1)}( [\begin{matrix}q^{(1)}(ζ) \\ p^{(1)}(ζ) \end{matrix}] , ψ^{(1)}(ζ)),

where q^{(1)}(ζ), p^{(1)}(ζ) ∈ ℂ^{k} are defined by

(6.4.1)  [\begin{matrix}q^{(1)}(ζ) \\ p^{(1)}(ζ) \end{matrix}] := [\begin{matrix}q_{0} + iζp̄_{0} \\ p_{0} − iζq̄_{0} \end{matrix}] .

Similarly, the twistor line l_{v}(ψ^{(2)}(ξ)) intersects Z̃_{ξ^{−1}} at

F^{(2)}( [\begin{matrix}q^{(2)}(ξ) \\ p^{(2)}(ξ) \end{matrix}] , ψ^{(2)}(ξ))

where

(6.4.2)  [\begin{matrix}q^{(2)}(ξ) \\ p^{(2)}(ξ) \end{matrix}] := [\begin{matrix}ξq_{0} + ip̄_{0} \\ ξp_{0} − iq̄_{0} \end{matrix}] .

Conversely, the twistor line containing

F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(ζ)) ∈ Π−1 ; Z̃(𝔸_{1})

intersects Z̃_{0} at

F^{(1)}(\frac{1}{1 + ζζ̄} [\begin{matrix}p + iζq̄ \\ q − iζp̄ \end{matrix}] , ψ^{(1)}(0))

and the twistor line containing

F^{(2)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(2)}(ξ)) ∈ Π−1 ; Z̃(𝔸_{2})

intersects Z̃_{∞} at

F^{(2)}(\frac{1}{1 + ξξ̄} [\begin{matrix}ξ̄p + iq̄ \\ ξ̄q − ip̄ \end{matrix}] , ψ^{(2)}(0)).

6.5. The real structure on the twistor space. The antipodal map ℙ^{1} ⟶^{α} ℙ^{1} lifts to the twistor space Z̃ preserving the fibration and the twistor lines. In terms of the product fibration in the coordinate charts, the lift α̃ to Z̃ is given by:

F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(ζ)) \underleftrightarrow{α̃} F^{(2)}( [\begin{matrix}−iξ̄p̄ \\ iξ̄q̄ \end{matrix}] , ψ^{(2)}(−ξ̄))

where ξζ = 1 and

(6.5.1)  F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(0)) \underleftrightarrow{α̃} F^{(2)}( [\begin{matrix}−ip̄ \\ iq̄ \end{matrix}] , ψ^{(2)}(0)).

Lemma 6.1.2 and the antiholomorphicity of α imply that α̃ is an antiholomorphic involution of Z.

6.6. Symplectic geometry of the twistor space. The complexsymplectic structures defined in (5.2.3) fit together giving a holomorphic exterior 2-form on the complex manifold Ω on Z, taking values in the holomorphic line bundle Π∗ ; Z̃O(2), where ΠZ̃ denotes the twistor fibration.

Let u ∈ S^{2}. There is a family of complex-symplectic structures on V_{ℍ} which are holomorphic with respect to u, which are parametrized by a unit vector u_{1} ∈ T_{1}S^{2}. Namely (5.2.3) defines a complex-symplectic structure Ω_{(u_{1},u_{2})}, where u_{1} ∈ u^{⊥} and u_{2} = u × u_{1}. Let u be I_{ζ}^{(1)} or I_{ξ}^{(2)}. Define u_{1} (respectively u_{2}) as J_{ζ}^{(1)}, J_{ξ}^{(2)} (respectively K_{ζ}^{(1)}, K_{ξ}^{(2)}), where:

J_{ζ}^{(1)} := (1 + ζK)J(1 + ζK)^{−1}
K_{ζ}^{(1)} := (1 + ζK)K(1 + ζK)^{−1}
J_{ξ}^{(2)} := (ξ̄+ K)J(ξ̄+ K)^{−1}
(6.6.1)  K_{ξ}^{(2)} := (ξ̄+ K)K(ξ̄+ K)^{−1},

analogous to the definition (6.1.1) of I_{ζ}^{(1)} and I_{ξ}^{(2)}.

Evidently, (J_{0}^{(1)}, K_{0}^{(1)}) = (J, K) and (J_{0}^{(2)}, K_{0}^{(2)}) = (−J, K). For ζ, ξ ∈ ℂ, let

Ω_{ζ}^{(1)} := (1 + ζζ̄)Ω_{(J_{ζ}^{(1)},K_{ζ}^{(1)})}
Ω_{ξ}^{(2)} := (1 + ξξ̄)Ω_{(J_{ξ}^{(2)},K_{ξ}^{(2)})}.

Corresponding to ψ^{(1)}(0) ∈ 𝔸_{1} and ψ^{(2)}(0) ∈ 𝔸_{2} are

Ω_{0}^{(1)} = Ω_{(J,K)}, Ω_{0}^{(2)} = Ω_{(−J,K)},

respectively.

Lemma 6.6.1. Let α and β be tangent vectors and ζ, ξ ∈ ℂ. Then

Ω_{ζ}^{(1)}(α, β) = Ω_{(J,K)}((1 − ζK)α, (1 − ζK)β)
Ω_{ξ}^{(2)}(α, β) = Ω_{(J,K)}((ξ − K)α, (ξ − K)β)

Proof. First recall the metric g as defined in (5.2.1). For any α and β,

g((1 + ζK)α, β) = g(α, (1 − ζK)β)
g((ξ̄+ K)α, β) = g(α, (ξ − K)β)

since g(hα, β) = g(α, −hβ) for any purely imaginary quaternion h ∈ ℍ.

Ω_{ζ}^{(1)}(α, β) = (1 + ζζ̄) (g(J_{ζ}^{(1)}α, β) + ig(K_{ζ}^{(1)}α, β))
= (1 + ζζ̄) (g((1 + ζK)J(1 + ζK)^{−1}α, β)
+ ig((1 + ζK)K(1 + ζK)^{−1}α, β))

= g((1 + ζK)J(1 − ζK)α, β) + ig((1 + ζK)K(1 − ζK)α, β) = g(J(1 − ζK)α, (1 − ζK)β) + ig(K(1 − ζK)α, (1 − ζK)β)

= Ω_{(J,K)}((1 − ζK)α, (1 − ζK)β)

and similarly for Ω_{ξ}^{(2)}(α, β). □

Corollary 6.6.2. If ζξ = 1, then Ω_{ξ}^{(2)} = ξ^{2}Ω_{ζ}^{(1)}.

Proof. Since ξ − K = ξ(1 − ζK) and Ω_{(J,K)} is holomorphic with respect to I,

Ω_{ξ}^{(2)}(α, β) = Ω_{(J,K)}((ξ − K)α, (ξ − K)β)
= ξ^{2}Ω_{(J,K)}((1 − ζK)α, (1 − ζK)β)
= ξ^{2}Ω_{ζ}^{(1)}(α, β)

as desired. □

Thus {Ω^{(1)}, Ω^{(2)}} defines a Π∗ ; Z̃O(2)-valued exterior 2-form Ω on Z̃. Denoting δ the meromorphic section of O(−2) in (6.1.5), we see that Ω ⊗ Π_{Z}^{∗}δ is a (scalar-valued) exterior 2-form on Z̃.

Corollary 6.6.3. Ω is holomorphic.

Proof. Ω is holomorphic in the fiber directions since Ω_{ζ}^{(1)}(I_{ζ}^{(1)}(α), β) = iΩ_{ζ}^{(1)}(α, β) on ℍ^{k} × 𝔸_{1}. Now

Ω_{ζ}^{(1)}(α, β) = Ω_{(J,K)}((1 − ζK)α, (1 − ζK)β)
= Ω_{(J,K)}(α, β)
+ ζ (Ω_{(J,K)}(Kα, β) + Ω_{(J,K)}(α, Kβ))
+ ζ^{2} Ω_{(J,K)}(Kα, Kβ)
= Ω_{(J,K)}(α, β) − 2 ζ ω_{I}(α, β) + ζ^{2} Ω_{(−J,K)}(α, β).

That is,

(6.6.2)  Ω_{ζ}^{(1)} = Ω_{(J,K)} − 2ζω_{I} + ζ^{2}Ω_{(−J,K)};

therefore, Ω^{(1)} is holomorphic with respect to ζ.

Similarly, Ω_{ξ}^{(2)}(I_{ξ}^{(2)}(α), β) = iΩ_{ξ}^{(2)}(α, β) on ℍ^{k} × 𝔸_{2}, and

(6.6.3)  Ω_{ξ}^{(2)} = Ω_{(−J,K)} − 2ξω_{I} + ξ^{2}Ω_{(J,K)}.
□

(Formulas (6.6.2),(6.6.3) differ slightly from formula (3.87) of [HKLR], due to our differing conventions; see the discussion after (6.1.2).)

Lemma 6.6.4. Let α̃ be the real structure on Z̃ and Ω the O(2)- valued holomorphic exterior 2-form defined above. Then

(α̃)^{∗}Ω = −\overline{Ω}.

Proof. The action of α̃ on the coefficient bundle O(2) is defined in Lemma 6.1.3. Now

Ω_{0}^{(1)} = Ω_{(J,K)} = −\overline{Ω_{(−J,K)}} = −\overline{Ω_{0}^{(2)}}.

Let ξζ = 1. Applying (6.6.2),

Ω_{−ξ̄}^{(1)} = −ξ̄^{2}Ω_{ζ}^{(1)}.
□

6.7. The lattice quotient. We have described the essential structures on the twistor space Z̃ ⟶ ℙ^{1} of the quaternionic vector space V_{ℍ}. That is Z̃ ≅ ℍ^{k} × ℙ^{1} with the complex structure which restricts to the fiber over [v] ∈ ℙ^{1} by the corresponding complex structure defined by (6.1.2). The coordinate charts F^{(i)} defined in (6.3.2) trivialize the holomorphic vector bundle Z̃ over the affine patches 𝔸_{1}, 𝔸_{2} of ℙ^{1}.

The twistor space for the hyperkähler manifold V_{ℍ}/L is the quotient Z = Z̃/L with the fibration Z ⟶^{Π_{Z}} ℙ^{1} induced from ΠZ̃. Denote the fiber Π_{Z}^{−1}(ψ^{(1)}(ζ)) by Z_{ζ} and the fiber Π_{Z}^{−1}(ψ^{(2)}(0)) by Z_{∞}, similar to Z̃_{ζ} and Z̃_{∞}. The complex structure on Z̃, the complex Π∗ ; Z̃O(2)- valued exterior form Ω and the real structure α̃ are all invariant under translations in L, and therefore induce corresponding structures on the quotient Z = Z̃/L.

The collection of twistor lines l_{v} is invariant under the action of L. If γ ∈ L, then

l_{v+γ} = l_{v} + γ.

Thus the twistor lines in Z̃ define twistor lines in Z, namely holomorphic sections with normal bundle isomorphic to ℂ^{2k} ⊗ O(1).

In summary we obtain the theorem of Hitchin-Karlhede-Lindström-Roc̆ek [HKLR]:

Theorem 6.7.1. The twistor space Z is a complex manifold with holomorphic fibration Z ⟶^{Π_{Z}} ℙ^{1}. Furthermore there exists a holomorphic Π_{Z}^{∗}O(2)-valued exterior 2-form Ω on Z inducing a nondegenerate complex-symplectic structure on each fiber of Π_{Z}. There exists a family of holomorphic sections l_{v} of Π_{Z} each with normal bundle isomorphic to ℂ^{2k} ⊗O(1). There exists a real structure α̃ on Z covering the antipodal map ℙ^{1} ⟶^{α} ℙ^{1}, preserving each twistor line l_{v} and (α̃)^{∗}Ω = −\overline{Ω}.

In terms of the coordinate atlas, L acts on Z̃ as follows. On the fiber Z̃_{0}, the action of γ ∈ L is:

(6.7.1)  F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(0)) ⟼^{γ} F^{(1)}( [\begin{matrix}q + γ \\ p \end{matrix}] , ψ^{(1)}(0)).

Applying (6.5.1), the action of γ ∈ L on the opposite fiber Z̃_{∞} is:

F^{(2)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(2)}(0)) ⟼^{γ} F^{(2)}( [\begin{matrix}q \\ p − iγ̄ \end{matrix}] , ψ^{(2)}(0))

Applying (6.4.1) and (6.4.2), the action of γ ∈ L on the other fibers are:

F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(ζ)) ⟼^{γ} F^{(1)}( [\begin{matrix}q + γ \\ p − iζγ̄ \end{matrix}] , ψ^{(1)}(ζ))
F^{(2)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(2)}(ξ)) ⟼^{γ} F^{(2)}( [\begin{matrix}q + ξγ \\ p − iγ̄ \end{matrix}] , ψ^{(2)}(ξ))

6.8. Functions and flows. The holomorphic ℂ^{∗}-action on the fiber Z_{0} = T^{∗}Jac(X) in §3 extends to a holomorphic ℂ^{∗}-action on Z. This ℂ^{∗}-action covers a holomorphic ℂ^{∗}-action on ℙ^{1}, where the action of λ ∈ ℂ^{∗} is defined by:

ψ^{(1)}(ζ) ⟼ ψ^{(1)}(λζ)
ψ^{(2)}(ξ) ⟼ ψ^{(2)}(λ^{−1}ξ).

The ℂ^{∗}-action on Z preserves the foliation by twistor lines. Since the fiber Z_{0} is a cross-section for this foliation, (6.4.1) implies:

Proposition 6.8.1. The ℂ^{∗}-action is described as:

F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(ζ)) ⟼^{λ} F^{(1)}( [\begin{matrix}\frac{1+|λζ|^{2}}{1+|ζ|^{2}} q + iζ \frac{|λ|^{2}−1}{1+|ζ|^{2}} p̄ \\ λp \end{matrix} ] , ψ^{(1)}(λζ))

in 𝔸_{1} and

F^{(2)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(2)}(ξ)) ⟼^{λ} F^{(2)}( [\begin{matrix}λ^{−1} \frac{|λ|^{2}+|ξ|^{2}}{1+|ξ|^{2}} q + iλ^{−1}ξ \frac{|λ|^{2}−1}{1+|ξ|^{2}} p̄ \\ p \end{matrix} ] , ψ^{(2)}(λ^{−1}ξ))

in 𝔸_{2}. In particular, if |λ| = 1, then

F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(ζ)) ⟼^{λ} F^{(1)}( [\begin{matrix}q \\ λp \end{matrix}] , ψ^{(1)}(λζ)).

Proof. The point represented by

F^{(1)}( [\begin{matrix}q \\ p \end{matrix}] , ψ^{(1)}(ζ)) ∈ Π−1 ; Z̃(𝔸_{1})

lies on the same twistor line as

F^{(1)}(\frac{1}{1 + |ζ|^{2}} [\begin{matrix}q − iζp̄ \\ p + iζq̄ \end{matrix}] , ψ^{(1)}(0))

which maps under λ ∈ ℂ^{∗} to F^{(1)}(v_{λ}, ψ^{(1)}(0)), where

v_{λ} := [\begin{matrix}q_{λ} \\ p_{λ} \end{matrix}] := \frac{1}{1 + |ζ|^{2}} [\begin{matrix}q − iζp̄ \\ λ(p + iζq̄) \end{matrix}] .

The twistor line containing F^{(1)}(v_{λ}, ψ^{(1)}(0)) intersects the the fiber Z_{λζ} at

F^{(1)}( [\begin{matrix}q_{λ} + i(λζ)p̄_{λ} \\ p_{λ} − i(λζ)q̄_{λ} \end{matrix}] , ψ^{(1)}(λζ)).

The other claims follow similarly. □

As λ ⟶ 0, the orbit converges to a point in M which is fixed by the ℂ^{∗}-action and hence lies in the critical set of the Hitchin map (3.3.3).

The circle action inside the ℂ^{∗}-action is Hamiltonian for the symplectic structure ω_{I}. The Hamiltonian potential function is the energy function defined in (3.3.5):

ℍ^{k}/L ⟶^{ϕ} ℝ
[q + pJ] ⟼ \frac{1}{2}p · p̄.

Compare (3.3.6). In contrast:

Theorem 6.8.2. The function ϕ is a Kähler potential for the metric g and the complex structure

e^{Iθ}J = cos(θ)J + sin(θ)K

for each θ ∈ ℝ.

Let (M, g, J) be a Kähler manifold, that is, M is a manifold with complex structure J and Riemannian metric g which is Kählerian with respect to J. Recall that a Kähler potential is a function M ⟶^{f} ℝ such that

−\frac{1}{2}d(df ∘ J) = ω_{J}.

Here df denotes the exterior derivative and df ∘ J denotes the 1-form (essentially the Hodge ⋆-operator applied to df) obtained by composition

T M ⟶^{J} T M ⟶^{df} ℝ.

The notation ω_{J} was introduced in (5.2.2). The proof of Theorem 6.8.2 divides into several lemmas:

Lemma 6.8.3. For every u ∈ S^{2}, the function

ℍ^{k} ⟶^{ρ} ℝ
q + pJ ⟼ \frac{1}{2}(q · q̄ + p · p̄)

is a Kähler potential for the Riemannian metric

g = dq · dq̄ + dp · dp̄

on ℍ^{k}.

Proof. The rotation group SO(3) acts transitively on S^{2} and isometrically on (ℍ^{k}, g). Thus, by symmetry we may assume that u = I. Now

ρ = \frac{1}{2}(q · q̄ + p · p̄)
dρ = \frac{1}{2}(q · dq̄ + q̄ · dq + p · dp̄ + p̄ · dp)
dρ ∘ I = \frac{1}{2}(−iq · dq̄ + iq̄ · dq − ip · dp̄ + ip̄ · dp)
= \frac{i}{2}(−q · dq̄ + q̄ · dq − p · dp̄ + p̄ · dp)

and

d(dρ ∘ I) = \frac{i}{2}(−dq · dq̄ + dq̄ · dq − dp · dp̄ + dp̄ · dp)
= ω_{I}

as claimed. □

If u is a complex structure, then a function M ⟶^{ψ} ℝ is u-pluriharmonic if d(dψ ∘ u) = 0. The sum of a Kähler potential with a pluriharmonic function is again a Kähler potential. Theorem 6.8.2 follows from adding to ρ (the Kähler potential of Lemma 6.8.3) the pluriharmonic function −ψ defined in the following lemma:

Lemma 6.8.4. For each θ ∈ ℝ, the function

ℍ^{k}/L ⟶^{ψ} ℝ
q + pJ ⟼ q · q̄ − p · p̄

is e^{Iθ}J-pluriharmonic.

Proof. From the definition of J,

dp ∘ J = dq̄
dq ∘ J = −dp̄
dp̄ ∘ J = −dq
dq̄ ∘ J = dp.

Now

ψ = q · q̄ − p · p̄
dψ = q · dq̄ + q̄ · dq − p · dp̄ − p̄ · dp
dρ ∘ e^{Iθ} = e^{−iθ}q · dq̄ + e^{iθ}q̄ · dq − e^{−iθ}p · dp̄ − e^{iθ}p̄ · dp
dρ ∘ e^{Iθ} ∘ J = e^{−iθ}q · dp + e^{iθ}q̄ · (−dp̄) − e^{−iθ}p · (−dq) − e^{iθ}p̄ · dq̄
= e^{−iθ}q · dp − e^{iθ}q̄ · dp̄ + e^{−iθ}p · dq − e^{iθ}p̄ · dq̄
= d(e^{−iθ}p · q − e^{iθ}p̄ · q̄)

is closed, that is, d(dρ ∘ (e^{Iθ}J)) = 0, as claimed. □

The function ϕ is not a Kähler potential for any of the other complex structures in the hyperkähler family.

7. The moduli space and the Riemann period matrix

Here we explicitly compute these moduli spaces in terms of classical invariants. We return to the original point of view, involving the fundamental group and the Betti moduli space.

7.1. Coordinates for the Betti moduli space. Under the direct product decomposition of Betti moduli space

Hom(π, ℂ^{∗}) ⟶^{≅} Hom(π, U(1)) × Hom(π, ℝ^{+})
ρ ⟼ (ρ_{u}, ρ_{ℝ}),

the representation ρ decomposes into a unitary part ρ_{u} and a positive real part ρ_{ℝ} as:

ρ_{u}(γ) = \frac{ρ(γ)}{|ρ(γ)|}, ρ_{ℝ}(γ) = |ρ(γ)|.

By the real analytic isomorphism ℝ^{+} ⟶^{log} ℝ, we identify

Hom(π, ℝ^{+}) ⟶ Hom(π, ℝ) ⟶ H^{1}(X, ℝ) ≅ ℝ^{2k}

Let A_{1}, B_{1}, …, A_{k}, B_{k} ∈ π be generators for the presentation (2.1.1). In terms of these generators, a representation ρ ∈ Hom(π, ℂ^{∗}) is determined by an arbitrary 2k-tuple

(ρ(A_{1}), …, ρ(B_{k})) ∈ (ℂ^{∗})^{2k}.

The unitary part ρ_{u} is then given by

(\frac{ρ(A_{1})}{|ρ(A_{1})}, …, \frac{ρ(B_{k})}{|ρ(B_{k})|}) ∈ U(1)^{2k}

and the positive-real part by

(|ρ(A_{1})|, …, |ρ(B_{k})|) ∈ (ℝ^{+})^{2k}

which we identify with the real vector

( log |ρ(A_{1})|, …, log |ρ(B_{k})|) ∈ ℝ^{2k}.

With a conformal structure on X, ρ_{u} corresponds to a point in Jac(X) = V/L, where V ≅ ℂ^{k} is a complex vector space and L is the lattice in ℂ^{k} spanned by ℤ^{k} and the columns of the period matrix Π. The real part ρ_{ℝ} corresponds to a point in the real symplectic vector space H^{1}(X; ℝ) ≅ ℝ^{2k}. Thus, the conformal structure provides a linear ℂ^{∗}-action generated by a complex structure 𝕁_{Π} depending on Π.

7.2. Abelian differentials and their periods. The homology classes corresponding to the generators A_{1}, B_{1} …, A_{k}, B_{k} ∈ π determine a symplectic basis (also denoted by A_{1}, B_{1}, … A_{k}, B_{k}) of H_{1}(X; ℤ) satisfying:

A_{j} · A_{l} = A_{j} · A_{l} = 0
A_{j} · B_{l} = −B_{j} · A_{l} = δ_{j,l}

with respect to the intersection form on H_{1}(X; ℤ). If ϕ, ψ are 1-forms on X, then

(7.2.1)  ∫_{X} ϕ ∧ ψ = ∑_{j=1}^{k} ( ∫_{A_{j}} ϕ ∫_{B_{j}} ψ − ∫_{B_{j}} ϕ ∫_{A_{j}} ψ).

Recall that the space V̄ = ℋ^{1,0}(X) of abelian differentials is a complex vector space for which the Hermitian inner product defined by (3.3.1) is negative definite. Choose a basis {ω_{1}, …, ω_{k}} of abelian differentials. Call such a basis unitary if the corresponding basis {ω̄_{1}, …, ω̄_{k}} of V is orthonormal with respect to the positive definite Hermitian inner product on V induced by (3.3.1).

We record this basis as a column vector of 1-forms

ω = [ω_{1} ; ⋮ ; ω_{k}] .

If {ω′_{1}, …, ω′_{k}} is another basis, then the corresponding vectors ω and ω′ satisfy

ω′ = Uω,

where U ∈ GL(k, ℂ) is an invertible k × k matrix. If ω is a unitary basis, then ω′ is unitary if and only if U ∈ U(k) is a unitary matrix.

If α, β ∈ V̄ are abelian differentials, then ⋆α = iα and ⋆β̄ = −iβ, and (3.3.1) implies

⟨α, β⟩ = ∫_{X} α ∧ β̄

The period matrix for ω is defined as the k × 2k matrix [A B], where the k × k-matrices A, B are defined by:

(7.2.2)  A_{j,l} := ∫_{A_{l}} ω_{j},  B_{j,l} := ∫_{B_{l}} ω_{j}

respectively. Formula (7.2.1) implies that A and B are invertible. If ω′ is another basis related to ω by ω′ = Uω, then the period matrix [A′ B′] of ω′ relates to the period matrix of ω by:

(7.2.3)  [A′ B′] = U [A B] .

Since the (2, 0)-form ω_{j} ∧ ω_{l} is identically zero,

0 = ∫_{X} ω_{j} ∧ ω_{l}

and (7.2.1) implies:

(7.2.4)  0 = AB^{†} − BA^{†}.

Suppose that ω is a unitary basis. Then

−i δ_{j,l} = ∫_{X} ω_{j} ∧ ω̄_{l}

and (7.2.1) implies

(7.2.5)  −i𝕀 = AB̄^{†} − BĀ^{†}.

The normalized basis of abelian differentials {ω′_{1}, …, ω′_{k}} is the basis such that

∫_{A_{j}} ω′_{l} = δ_{j,l}.

Together with (7.2.3), this implies:

ω′ = A^{−1}ω,

where A is defined in (7.2.2). By (7.2.3), the corresponding period matrix is

[A′ B′] = [𝕀 Π]

where Π = A^{−1}B is the matrix

Π_{i,j} := ∫_{B_{j}} ω′_{i}.

(7.2.4) and (7.2.5) imply the Riemann bilinear relations:

Π = Π^{†}
Im(Π) = \frac{Π − Π̄}{2i} = \frac{1}{2}(A^{−1})\overline{(A^{−1})}^{†} > 0.

That is, Π is a k×k symmetric matrix whose imaginary part is positive definite.

For a symmetric k × k complex matrix Π with positive definite imaginary part, the matrix [A B] corresponding to a unitary basis can be recovered. Namely, let \sqrt{2 Im(Π)} denote the unique positive definite symmetric square-root of 2 Im(Π) and let U ∈ U(k) be an arbitrary unitary matrix. Then

A := U(\sqrt{2 Im(Π)})^{−1}
B := U(\sqrt{2 Im(Π)})^{−1} Π

define a period matrix [A B] which is unique up to the choice of U.

7.3. Flat connections. Consider a flat connection D on a trivial line bundle over X. Write

D = D_{0} + ψ + ϕ,

where D_{0} is the flat connection arising from the trivialization, ψ purely imaginary and ϕ real 1-forms. By a gauge transformation, one may write

ψ = Ψ − Ψ̄
ϕ = Φ + Φ̄,

where Ψ ∈ V and Φ ∈ V̄. The corresponding vectors q, p ∈ ℂ^{k} satisfy

Ψ = ∑_{j=1}^{k} q_{j}ω̄_{j} = ω̄^{†} · q, Φ = ∑_{j=1}^{k} p_{j}ω_{j} = ω^{†} · p,

where

q = [q_{1} ; ⋮ ; q_{k}] , p = [p_{1} ; ⋮ ; p_{k}] .

The Higgs pair consists of the holomorphic structure D′′ = D′′_{0}+q^{†}ω̄ and the Higgs field Φ = p^{†}ω. Corresponding to q, p ∈ ℂ^{k} is the connection D = D_{0} + η, where

η = (q^{†}ω̄ − q̄^{†}ω) + (ω̄^{†} · q − ω^{†} · q̄) + (ω^{†} · p + ω̄^{†} · p̄).

To simplify calculations, henceforth let ω be the normalized basis of abelian differentials such that A = 𝕀 and B = Π.

If D is a flat unitary connection, then D = D_{0} + ψ, where ψ is purely imaginary. By a gauge transformation, one may assume that ψ = Ψ − Ψ̄, where Ψ ∈ ℋ^{0,1}(X). The corresponding vector q ∈ ℂ^{k} is defined by

Ψ = ∑_{j=1}^{k} q_{j}ω̄_{j} = ω̄^{†} · q

and

ψ = ω̄^{†} · q − ω^{†} · q̄ = 2i Im (ω̄^{†} · q),

with corresponding periods

∫_{A_{l}} ψ = 2i Im(q_{l})

∫_{B_{l}} ψ = 2i Im ( ∑_{j=1}^{k} q_{j}Π̄_{j,l}).

In the notation

a := [∫_{A_{1}} ψ ; ⋮ ; ∫_{A_{k}} ψ] , b := [∫_{B_{1}} ψ ; ⋮ ; ∫_{B_{k}} ψ] ,

the periods are:

a = 2i Im(q)
b = 2i Im(Π̄ · q) = 2i( Re(Π) · Im(q) − Im(Π) · Re(q)).

The lattice L in ℂ^{k} consists of q such that the periods a, b ∈ 2πiℤ, that is,

Im(q) ∈ πℤ^{k}, Re(Π) Im(q) − Im(Π) Re(q) ∈ πℤ^{k}

which is equivalent to:

q ∈ π Im(Π)^{−1}{ℤ^{k} + Πℤ^{k}}.

Thus L is equivalent (by π Im(Π)^{−1} ∈ GL(k, ℂ)) to the lattice ℤ^{k}+Πℤ^{k} spanned by the 2k columns of [𝕀_{k} Π]. Therefore Jac(X) is the quotient of ℂ^{k} by ℤ^{k} + Πℤ^{k}. (See, for example, Gunning [Gu1, Gu3], §2.5.)

7.4. Higgs fields. The space Hom(π, ℝ^{+}) of positive real characters inherits structure from the conformal structure of X through the period matrix Π. For ρ ∈ Hom(π, ℂ^{∗}), write

α_{j} := \frac{1}{2} log |ρ(A_{j})|, β_{j} := \frac{1}{2} log |ρ(B_{j})|.

We determine the corresponding flat ℝ^{+}-connection with holonomy (α, β).

After applying a gauge transformation, the 1-form ϕ ∈ ℋ^{1}(X; ℝ) decomposes as ϕ = Φ+Φ̄, where Φ ∈ ℋ^{1,0}(X) is a holomorphic 1-form. The vector p ∈ ℂ^{k} corresponding to Φ is defined by

Φ = ∑_{j=1}^{k} p_{j}ω_{j} = ω^{†}p

which has periods

α_{j} = \frac{1}{2} ∫_{A_{j}} ϕ = Re ∫_{A_{j}} Φ = Re(p_{j})

β_{j} = \frac{1}{2} ∫_{B_{j}} ϕ = Re ∫_{B_{j}} Φ = Re ( ∑_{l=1}^{k} Π_{j,l}p_{l})).

Thus

α = Re(p)
β = Re(Π) Re(p) − Im(Π) Im(p)

with inverse mapping

Re(p) = α
Im(p) = Im(Π)^{−1}( Re(Π)α − β).

As matrices,

[\begin{matrix}Re(p) \\ Im(p) \end{matrix}] ⟼ [\begin{matrix}𝕀_{k} & 0 \\ Re(Π) & − Im(Π) \end{matrix}] [\begin{matrix}α \\ β \end{matrix}]

and

[\begin{matrix}α \\ β \end{matrix}] ⟼ [\begin{matrix}𝕀_{k} & 0 \\ Im(Π)^{−1} Re(Π) & − Im(Π)^{−1} \end{matrix}] [\begin{matrix}Re(p) \\ Im(p) \end{matrix}] .

7.5. The ℂ^{∗}-action in terms of the period matrix. We describe the ℂ^{∗}-action explicitly as follows. We break p into its real and imaginary parts:

p = Re(p) + i Im(p)

Scalar multiplication p ⟼ ip is:

[\begin{matrix}Re(p) \\ Im(p) \end{matrix}] ⟼ [\begin{matrix}Im(p) \\ − Re(p) \end{matrix}] = [\begin{matrix}0 & 𝕀_{k} \\ −𝕀_{k} & 0 \end{matrix}] [\begin{matrix}Re(p) \\ Im(p) \end{matrix}] .

Multiplying the Higgs field Φ by i transforms the periods α, β by:

[\begin{matrix}α \\ β \end{matrix}] ⟼ 𝕁_{Π} [\begin{matrix}α \\ β \end{matrix}]

where

𝕁_{Π} = [\begin{matrix}𝕀_{k} & 0 \\ Re(Π) & − Im(Π) \end{matrix}] [\begin{matrix}0 & 𝕀_{k} \\ −𝕀_{k} & 0 \end{matrix}] [\begin{matrix}𝕀_{k} & 0 \\ Im(Π)^{−1} Re(Π) & − Im(Π)^{−1} \end{matrix}]

= [\begin{matrix}− Im(Π)^{−1} Re(Π) & Im(Π)^{−1} \\ − Re(Π) Im(Π)^{−1} Re(Π) − Im(Π) & Re(Π) Im(Π)^{−1} \end{matrix}] . (7.5.1)

Then the ℂ^{∗}-action is given by

Re(λ)𝕀_{2k} + Im(λ)𝕁_{Π}

for λ ∈ ℂ^{∗}.

7.6. The ℂ^{∗}-action and the real points. The ℂ^{∗}-action on Hom(π, ℂ^{∗}) fixes the unitary component and acts as above on the Higgs field. That is, λ ∈ ℂ^{∗} maps

ρ = (ρ_{u}, ρ_{ℝ}) ⟼ (ρ_{u}, exp ((Re(λ)𝕀 + Im(λ)J_{Π}) log(ρ_{ℝ}))),

where 𝕁_{Π} is defined by (7.5.1).

The unitary projection Hom(π, ℂ^{∗}) ⟶ Hom(π, U(1)) associates to a Higgs line bundle the underlying holomorphic line bundle. It can be described purely in terms of the ℂ^{∗}-action as follows. Let ρ ∈ Hom(π, ℂ^{∗}). Then as λ ⟶ 0, the orbit λ · ρ approaches ρ_{u}. In particular this structure defines the foliation of Hom(π, ℂ^{∗}) by copies of Hom(π, ℝ^{+}) = H^{1}(X; ℝ).

Moreover the complex-symplectic structure (J, Ω) defined by (2.2.7) arises from the complex-orthogonal structure on ℂ^{∗}. By (2.2.9) (compare also (5.3.2)), the imaginary part of Ω is a real-symplectic structure −ω_{K} under which the decomposition

Hom(π, ℂ^{∗}) = Hom(π, U(1)) × Hom(π, ℝ^{+})

is a Lagrangian product (a real bipolarization). The ℂ^{∗}-action is the extra piece of structure to determine the Dolbeault moduli space (expressed in terms of the complex structure I). As Hom(π, U(1)) is ω_{K}- Lagrangian, ω_{K} identifies the tangent space with the normal space, which in turn identifies with H^{1}(X; ℝ). This real-symplectic space inherits a Hermitian structure from the complex structure 𝕁_{Π}, which defines a complex structure on Hom(π, U(1)). With this complex structure, Hom(π, U(1)) identifies with Jac(X). Thus the full structure of the Dolbeault moduli space of Higgs bundles arises from the dynamics of the ℂ^{∗}-action and the natural structures of the Betti moduli space which depend only on the fundamental group of Σ.

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