Global generation and very ampleness for adjoint linear series

Let $X$ be a smooth projective variety over an algebraically closed field $K$ with arbitrary characteristic. Suppose $L$ is an ample and globally generated line bundle. By Castelnuovo–Mumford regularity, we show that $K_X \otimes L^{\otimes \operatorname{dim} X} \otimes A$ is globally generated and $K_X \otimes L^{\otimes \operatorname{dim} X+1} \otimes A$ 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 $f : X \to Y$ with $L$ ample and globally generated, and $A$ nef but not numerically trivial, we prove the global generation of $f_{\ast} (K_{X/Y})^{\otimes s} \otimes KY \otimes L^{\otimes \operatorname{dim} Y} \otimes A$ for any positive integer $s$.

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This work was partially supported by China’s Recruitment Program of Global Experts and NSFC 11688101.

Received 1 November 2016

Accepted 9 April 2018

Published 30 December 2019