Pome

Proposition 16.6.1712DD9B82F232

Proposition 16.6.17.12DD9B82F232 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$.

  1. 7C6266FB792D

    For any big class $\alpha \in \mathrm{H}^{1,1}(X,\mathbb {R})$, $\pi ^*\alpha $ is big. Moreover, $\operatorname{vol}\pi ^*\alpha =\operatorname{vol}\alpha $.

  2. FDD4AF241C61

    For any big class $\beta \in \mathrm{H}^{1,1}(Y,\mathbb {R})$, $\pi _*\beta $ is big. Moreover, $\operatorname{vol}\pi _*\beta \geq \operatorname{vol}\beta $.

Proof. 1 Take a current $T_{\min }$ with minimal singularities in $\alpha $. Then $\pi ^*T_{\min }$ is a current with minimal singularities in $\pi ^*\alpha $. Our assertion follows from proposition 13.13.4.

2 Let $S_{\min }$ be a current with minimal singularities in $\beta $. Then $\pi _*S_{\min }$ is a positive current in the cohomology class $\pi _*\beta $. A current with minimal singularities in $\pi _*\beta $ is less singular than $\pi _*S_{\min }$, hence by the monotonicity theorem theorem 13.13.11

\[ \operatorname{vol}(\pi _*\beta )\geq \int _ X\left(\pi _*S_{\min }\right)^ n. \]

By proposition 13.13.4,

\[ \int _ X \left(\pi _*S_{\min }\right)^ n\geq \int _ Y S_{\min }^ n=\operatorname{vol}\beta . \]

We conclude $\operatorname{vol}\pi _*\beta \geq \operatorname{vol}\beta $. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub