Global generation and very ampleness for adjoint linear series

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 $f:X⟶Y$ with $L$ ample and globally generated, and $A$ nef but not numerically trivial, we prove the global generation of $f_{\ast }(K_{X/Y}{)}^{\otimes s}\otimes K_Y\otimes L^{\otimes \dim Y}\otimes A$ for any positive integer $s$.

To appear in Comm. Anal. Geom

References in corpus (4)

• Injectivity theorem for pseudo-effective line bundles and its applications
• The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles
• On Fujita's freeness conjecture in dimension 5
• A vanishing theorem of Koll'ar-Ohsawa type