Pome72A62384B9ECLemma 13.13.48.72A62384B9EC Let $L$ be a holomorphic line bundle on $X$ with $\theta \in c_1(L)$. Assume that $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then there exists $k_0>0$ such that for each $k\geq k_0$, we have
Proof. By 13.13.47, there is $\psi \in \mathrm{PSH}(X,\theta )$ with $\theta _{\psi }$ a Kähler current and $\psi \leq \varphi $. Since $\mathcal{I}(k\psi )\subseteq \mathcal{I}(k\varphi )$, it suffices to prove the assertion with $\psi $ in place of $\varphi $. In this case, the result follows from Hörmander's $L^2$-estimate ([Hor66]), see [Theorem 13.21, Dem12]. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.