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Canonical singularities, multiplier ideals and Kawamata-Vieweg vanishing theorem
Misha Verbitsky

Quarta-feira, 18 de julho de 2018, 15:30
Sala 228

Let $g$ be a singular metric on a trivial line bundle $L$ on a complex manifold. The multiplier ideal of $g$ is the ideal of $L^2$-integrable holomorphic sections of $L$. When $L$ is a nontrivial line bundle, its multiplier ideal is the ideal sheaf $I$ where $L\otimes I$ is the sheaf of $L^2$-integrable holomorphic sections of $L$. This complex-analytic notion can be used to define several classical concepts of algebraic geometry in more geometric fashion; also it allows to give a one-line proof of Kawamata-Viehweg vanishing theorem (called Kawamata-Viehweg-Nadel theorem in this version). I will define "singular metric" and "multiplier ideals" formally and state Nadel's theorems. Then I will define the canonical singularities via multiplier ideal sheaves and explain why all canonically embedded varieties have canonical singularities. The talk is supposed to be as elementary as possible; only basic knowledge of complex algebraic geometry (Kahler metrics, connection, curvature) is assumed.