Theorem 20.2.2871AECE2E70ED

Proof. Fix a Kähler form $\omega $ on $X$ and a Hermitian metric $h_ T$ on $T$. We may assume that $Y\neq X$ and that $\theta _{\varphi } \geq 3\delta \omega $ for some $\delta >0$. Let $(\varphi _ j)_ j$ be the decreasing quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\theta )$. We can assume that $\theta _{\varphi _ j} \geq 2\delta \omega $ for all $j\geq 1$. Also, there exists $\epsilon _0 >0$ such that $\theta _{(1+\epsilon )\varphi _ j} \geq \delta \omega $ for any $\epsilon \in (0,\epsilon _0)$. Take $k_0=k_0(\delta ,h_ T)$ as in theorem 5.20.16.

We fix $k \geq k_0$ and $s \in \mathrm{H}^0(Y, T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}(k\operatorname{Tr}_ Y^{\theta }(\varphi )))$. By theorem 5.16.21, there exists $\epsilon \in (0,\epsilon _0)$ such that $s \in \mathrm{H}^0(Y, T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}(k(1+\epsilon )\operatorname{Tr}_ Y^{\theta }(\varphi )))$.

Since $\operatorname{Tr}_ Y^{\theta }(\varphi ) \preceq _ P \varphi _ j|_ Y$, corollary 16.8.14 implies that $s \in \mathrm{H}^0(Y, T|_ Y\otimes L|_ Y^ k\otimes \mathcal{I}(k(1+\epsilon )\varphi _ j|_ Y))$. By theorem 5.20.16, there exists $\tilde s_ j \in \mathrm{H}^0(X, T\otimes L^ k \otimes \mathcal{I}(k(1+\epsilon )\varphi _ j))$ such that $\tilde s_ j|_ Y =s$, for all $j$.

But by definition of quasi-equisingular approximation, we obtain that for high enough $j$ the inclusion $\mathcal{I}(k(1+\epsilon )\varphi _ j) \subseteq \mathcal{I}(k\varphi )$ holds. As a result, $\tilde s_ j \in \mathrm{H}^0(X, T\otimes L^ k \otimes \mathcal{I}(k\varphi ))$ for high enough $j$, finishing the argument. $\square$