# Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

## TL;DR

This paper extends vanishing theorems for nef vector bundles on projective manifolds, establishing new conditions under which certain cohomology groups vanish, especially for indecomposable nef rank 2 bundles.

## Contribution

It proves a general vanishing result for nef vector bundles depending on specific cohomology, and shows vanishing for indecomposable nef rank 2 bundles on certain manifolds.

## Key findings

- Vanishing of $H^{n,n}(X, E)$ for specific nef vector bundles.
- Vanishing holds for indecomposable nef rank 2 bundles on certain projective manifolds.
- Vanishing also applies to nef and big rank 2 bundles on varieties with trivial canonical bundle.

## Abstract

In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.