2.10 Ampleness for \(\mathbb {R}\)-divisors
Suppose \(D\in \operatorname{\mathrm{Div}}_{\mathbb {R}} X\). Then \(D\) is ample if it can be expressed as a finite sum \(D = \sum c_i A_i\) with \(A_i\) integral ample divisors and \(c_i{\gt}0\).
Suppose \(A\) is an ample \(\mathbb {R}\)-divisor and \(V\) is a \(k\)-dimensional closed subvariety of \(X\). Then \(A^{k}\cdot V {\gt}0\).
Write \(A = \sum c_i A_i\) with \(A_i\) integral and expand out \(A^k\).
A \(\mathbb {Q}\)-divisor \(A\) is ample as an \(\mathbb {R}\)-divisor if and only if it is ample as a \(\mathbb {Q}\)-divisor.
Suppose first that \(A\) is ample as an \(\mathbb {R}\)-divisor. Then, by Lemma 2.30, \(A^k\cdot V{\gt}0\) for all \(k\)-dimensional subvarieties \(V\). So \(A\) is ample as a \(\mathbb {Q}\)-divisor. On the other hand, if \(A\) is ample as a \(\mathbb {Q}\)-divisor, then we can write \(A = \sum c_i A_i\) with \(c_i\) positive rational numbers. So \(A\) is clearly ample as an \(\mathbb {R}\)-divisor.