# A Koll\'{a}r-type vanishing theorem

## TL;DR

This paper establishes a Kollár-type vanishing theorem for certain sheaves on complex manifolds, providing conditions under which higher direct images are reflexive, advancing the understanding of vanishing theorems in complex geometry.

## Contribution

It introduces a sufficient condition for the reflexivity of higher direct images of canonical bundles twisted by pseudo-effective line bundles, extending Kollár's vanishing theorem.

## Key findings

- Derived a Kollár-type vanishing theorem under new conditions
- Established reflexivity of higher direct images in complex geometry
- Provided a criterion for vanishing of certain cohomology sheaves

## Abstract

Let $f:X\rightarrow Y$ be a smooth fibration between two complex manifolds $X$ and $Y$, and let $L$ be a pseudo-effective line bundle on $X$. We obtain a sufficient condition for $R^{q}f_{\ast}(K_{X/Y}\otimes L)$ to be reflexive and hence derive a Koll\'{a}r-type vanishing theorem.