Pome

Lemma 11.22.225BD2AAB112A

Lemma 11.22.2.25BD2AAB112A Let $F:Y\to X$ be a holomorphic map between compact Kähler manifolds of dimensions $N\geq n$, with Kähler forms $\Omega $ and $\omega $. For $1\leq p\leq n$, there is a constant $C$ such that every smooth positive $(p,p)$-form $S$ on $X$ with $\mathrm{dd}^{\mathrm{c}}S=0$ satisfies

\[ \int _ Y F^*S\wedge \Omega ^{N-p} \leq C\int _ X S\wedge \omega ^{n-p}. \]

Proof. The positive closed current $\Gamma =F_*(\Omega ^{N-p})$ has bidegree $(n-p,n-p)$ on $X$. The classical $\mathrm{dd}^{\mathrm{c}}$-lemma for currents on compact Kähler manifolds gives a smooth real closed form $\gamma $ of this bidegree and a current $U$ with $\Gamma =\gamma +\mathrm{dd}^{\mathrm{c}}U$. For $p=n$, a closed current of degree zero is already a smooth locally constant function, and one may take $U=0$. Since $S$ is smooth, pairing it with $U$ is legitimate, and

\[ \int _ Y F^*S\wedge \Omega ^{N-p} =\langle \Gamma ,S\rangle =\int _ X\gamma \wedge S+\langle U,\mathrm{dd}^{\mathrm{c}}S\rangle =\int _ X\gamma \wedge S. \]

The pairing with the fixed smooth form $\gamma $ is bounded in absolute value by a constant times the trace mass $\int _ XS\wedge \omega ^{n-p}$ of the positive form $S$. This proves the estimate. $\square$

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