Draft:Nadel vanishing theorem

AFC comment (self): This theorem can potentially be merged into the multiplier ideal as a result related to multiplier ideal sheaves.

In mathematics, the Nadel vanishing theorem is a global vanishing theorem for multiplier ideals. This theorem is a generalization of the Kodaira vanishing theorem using singular metrics with (strictly) positive curvature, and also it can be seen as an analytical analogue of the Kawamata–Viehweg vanishing theorem.

Statement
Nadel vanishing theorem:  Let X be a smooth complex projective variety, D an effective $$\mathbb{Q}$$-divisor and L a line bundle on X, and $$\mathcal{J}(D)$$ is a multiplier ideal sheaves. Assume that $$L - D$$ is big and nef. Then

$$H^{i} \left(X, \mathcal{O}_{X}(K_X + L) \otimes \mathcal{J}(D) \right) = 0 \;\; \text{for} \;\; i > 0.$$

for analytic
Nadel vanishing theorem for analytic: Let $$(X, \omega)$$ be a Kähler manifold (X be a reduced complex space(Complex analytic variety) with a Kähler metric) such that weakly pseudoconvex, and let F be a holomorphic line bundle over X equipped with a singular hermitian metric of weight $$\varphi$$. Assume that $$\sqrt{-1} \cdot \theta(F) > \varepsilon \cdot \omega$$ for some continuous positive function $$\varepsilon$$ on X. Then

$$H^{i} \left(X, \mathcal{O}_{X}(K_X + F) \otimes \mathcal{J}(\varphi) \right) = 0 \;\; \text{for} \;\; i > 0.$$

Let arbitrary plurisubharmonic function $$\phi$$ on $$\Omega \subset X$$, then a multiplier ideal sheaf $$\mathcal{J}(\phi)$$ is a coherent on $$\Omega$$, and therefore its zero variety is an analytic set.

Footnote
Category:Theorems in algebraic geometry Category:Theorems in complex geometry