Theorem 5.20.162985932B2B64

Theorem 5.20.16.2985932B2B64 Assume that $L$ is big and $T$ is a holomorphic line bundle on $X$. Fix a Hermitian metric $h_ T$ on $T$. Fix a Hermitian metric $h$ on $L$ and let $\theta =c_1(L,h)$. Take a Kähler form $\omega $ on $X$. Let $Y\subseteq X$ be a connected closed submanifold of dimension $m$. Suppose that $\varphi \in \mathrm{PSH}(X,\theta - \delta \omega )$ for some $\delta >0$ and $\varphi |_ Y\not\equiv -\infty $. Then there exists $k_0(\delta ,h_ T)>0$ such that for all $k \geq k_0$ and $s \in \mathrm{H}^0(Y, T\otimes L|_ Y^ k \otimes \mathcal{I}(k\varphi |_ Y))$⁶, there exists an extension $\tilde s \in \mathrm{H}^0(X, T\otimes L^ k \otimes \mathcal{I}(k\varphi ))$ such that

where $C>0$ is an absolute constant, independent of the data $(\varphi ,s,k)$.