Kahler manifolds (course 868K)

University of Maryland

Department of Mathematics

Autumn 2014






Time: Tuesdays, Thursdays at 12:30pm.
Room: 3114 EGR Building.

Teacher: Y.A. Rubinstein. Office hours: by appointment.


Course plan:
Kahler manifolds lie at the intersection of Differential Geometry, Complex Analysis, Algebraic Geometry, PDEs, Microlocal Analysis, Probability, and Topology. Introduced more than 80 years ago, the subject has proven an extremely fertile ground for interactions between the fields mentioned above. In this course we will attempt an introduction, but with PLENTY of examples to illustrate the theory. The course should be relevant to students interested in *one* (but not necessarily all!) of the following: PDE/Complex Analysis/Differential Geometry/Algebraic Geometry. Key elementary facts will be recalled as we go along, and the detailled proof of some of these facts will be given in the introductory course MATH742 (Geometric Analysis) that will be offered simultaneously in Autumn 2014. Thus, MATH868K might be interesting also for rather beginning students.

Webcast:
With the technical assistance of the UMD Global Classrooms Initiative within the Office of International Affairs, this course will be webcast live. If you are attending this course via satellite be sure to contact the instructor beforehand. To obtain all the relevant technical information as well as real-time troubleshooting please contact D'Mario Headen-Vance at dheaden@umd.edu, or 301-405-1634.

Textbook:
We will not be following a book. Here are, however, some good references for now:

Cannas da Silva, Symplectic Toric Manifolds
Fulton, Introduction to toric varieties, 1993.
Gauduchon, Calabi'ís extremal Kahler metrics: An elementary introduction (manuscript, 2014).
Griffiths, Harris, Principles of algebraic geometry, 1978.
Kobayashi, Nomizu, Foundations of differential geometry, Volumes 1-2, 1963..
Rubinstein, Geometric quantization and dynamical constructions on the space of Kahler metrics, 2008.
Rubinstein, Smooth and singular Kahler-Einstein metrics, 2014.
Tian, Canonical metrics in Kahler Geometry, 2000.

Requirements:
Occasional homework will be assigned in class. It will be beneficial for you to try to do all the homework on your own or with fellow students but you are not required to submit it, with one exception: if you are taking this course for credit you will be expected to type up solutions for one of the homeworks (the instructor will assign each homework to a different student). These solutions will then be posted for the benefit of the other students.

Additional references (as we go along):

Lecture notes:

Schedule:

  • Lecture 1
    Overview. Almost complex structures. Examples: S^2 and S^6.

  • Lecture 2
    The Schouten-van Dantzig-Kahler condition: Symplectic, Riemannian, complex, and holonomy viewpoints.

    HW1.

    HW1 solutions (please leave comments/questions in the Forum discussion "HW1 solutions").

  • Lecture 3
    The ddbar-lemma. Outlook to the space of Kahler metrics.

  • Lecture 4
    Calibrations, complex submanifolds, and the Kahler condition.

    HW2.

  • Lecture 5
    Complex projective geometry. Fubini-Study metrics. Hermitian line bundles and their curvature. Curvature of Kahler metrics. Example: the Fubini-Study metric is Kahler-Einstein.

  • Lecture 6
    Curvature of Kahler manifolds. Uniformization of Kahler manifolds of constant holomorphic sectional curvature.

    HW3.

  • Lecture 7
    Holomorphic vector fields: Cauchy-Riemann operators, Hermitian and Chern connections, characterziation of Kahler manifolds in terms of the Chern connection, definition of holomorphic vector fields. The Lie algebra of holomorophic vector fields. The Hodge decomposition applied to holomorphic vector fields. The holomorphy potential of a holomorphic vector field and its interpretation as a vector field on the space of all Kahler potentials representing a fixed Kahler class.

  • Lecture 8
    Rational maps, meromorphic functions, birational isomorphism, rational varieties. Projection from a point to a hyperplane and the blowup construction. Making friends among rational surfaces: the quadric surface, and the two point blowup of the projective plane.

  • Lecture 9
    A geometric description of the quadric surface and its lines. Invariants of birational maps.

    HW4.

  • Lecture 10
    Meromorphic/holomorphic Sections of line bundles, divisors, and line bundles, Picard group, linear equivalence, self-intersection of an exceptional divisor on the surface.

  • Lecture 11
    Divisors, linear series, and the canonical line bundle under blow-ups.

  • Lecture 12
    Chern class of a line bundle: metrics, connection, curvature of the connection, the Chern class of the line bundle associated to a divisor.

  • Lecture 13
    Proof of Castelnuovo-Enriques Criterion for blowing-down -1-curves on a surface.

  • Lecture 14
    Kodaira Embedding theorem.

  • Lecture 15
    Kahler identities. Kodaira Vanishing theorem.

    HW5.

  • Lecture 16
    Classification of del Pezzo surfaces.

  • Lecture 17
    Classification of rational surfaces, I: Noether's lemma characterizing rational surfaces in terms of linear series. Adjunction formula for a divisor. Rational ruled surfaces are projectivized rank-2 bundles. Grothendieck's lemma on decomposibility of vector bundles over P^1.

  • Lecture 18
    Classification of rational surfaces, II: understanding the geometry of Hirzebruch surfaces.

  • Lecture 19
    Classification of rational surfaces, III: completing the proof. Characterizing P^2 in terms of the canonical bundles and Betti numbers.

  • Lecture 20
    Higher-dimensional analogue of uniformization: the search for Kahler-Einstein metrics. Reduction of the KE equation to a complex Monge-Ampere equation. The importance of a priori estimates. The C^0 a priori estimate in the case of negative curvature.

  • Lecture 21
    The C^0 a priori estimate in the case of zero curvature.

  • Lecture 22
    Openness in the continuity method. The Laplacian a priori estimate for the Ricci continuity method.

  • Lecture 23
    Global C^3 estimates for the complex Monge-Ampere equation.

  • Lecture 24
    Local C^3 estimates for the complex Monge-Ampere equation.