On a Bogomolov type vanishing theorem

TL;DR

This paper generalizes Bogomolov's vanishing theorem to compact Kähler manifolds with pseudoeffective line bundles, establishing new cohomology vanishing results under curvature conditions.

Contribution

It extends classical vanishing theorems to broader settings involving pseudoeffective line bundles and Kähler manifolds, with new cohomology vanishing conditions.

Findings

Proves a new vanishing theorem for certain cohomology groups.

Generalizes Bogomolov's theorem to pseudoeffective line bundles.

Provides conditions under which specific cohomology groups vanish.

Abstract

Let $X$ be a compact K\"ahler manifold and $(L,h)\rightarrow X$ be a pseudoeffective line bundle, such that the curvature $i\Theta_{L,h}\geq 0$ in the sense of currents. The main result of the present paper is that $H^n(X,\mathcal{O}(\Omega^p_X\otimes L)\otimes \mathcal{I}(h))=0$ for $p\geq n-nd(L,h)+1$. This is a generalization of Bogomolov's vanishing theorem.