Global generation and very ampleness for adjoint linear series

TL;DR

This paper establishes new criteria for the global generation and very ampleness of adjoint linear series on smooth projective varieties, extending results to arbitrary characteristic and complex cases using Castelnuovo--Mumford regularity and vanishing theorems.

Contribution

It provides novel global generation and very ampleness results for adjoint line bundles and vector bundles, applicable in arbitrary characteristic and involving nef line bundles.

Findings

Global generation of $K_X \otimes L^{\otimes \dim X} \otimes A$

Very ampleness of $K_X \otimes L^{\otimes (\dim X+1)} \otimes A$

Global generation of pushforward sheaves in holomorphic submersions

Abstract

Let $X$ be a smooth projective variety over an algebraically closed field $\mathbb{K}$ with arbitrary characteristic. Suppose $L$ is an ample and globally generated line bundle. By Castelnuovo--Mumford regularity, we show that $K_X \otimes L^{\otimes \dim X} \otimes A$ is globally generated and $K_X \otimes L^{\otimes (\dim X+1)} \otimes A$ is very ample, provided the line bundle $A$ is nef but not numerically trivial. On complex projective varieties, by investigating Kawamata-Viehweg-Nadel type vanishing theorems for vector bundles, we also obtain the global generation for adjoint vector bundles. In particular, for a holomorphic submersion $f:X\longrightarrow Y$ with $L$ ample and globally generated, and $A$ nef but not numerically trivial, we prove the global generation of $ f_*(K_{X/Y})^{\otimes s}\otimes K_Y \otimes L^{\otimes \dim Y} \otimes A$ for any positive integer $s$.