The cohomological support locus of pluri-canonical sheaves and the Iitaka fibration
type: Item
instance of: scholarly article
title: The cohomological support locus of pluri-canonical sheaves and the Iitaka fibration
zbMATH Open document ID: 1311.14016
author: Lei Zhang
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.3757
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 6355586
zbMATH Keywords: pluricanonical sheaves
zbMATH Keywords: Albanese maps
zbMATH Keywords: cohomological support
MaRDI profile type: Publication
OpenAlex ID: W2148073715
arXiv ID: 1212.3757
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DOI: 10.1112/JLMS/JDU042
published in: Journal of the London Mathematical Society. Second Series
review text: Let \(X\) be a smooth projective variety of non negative Kodaira dimension, \(A\) an abelian variety and \(F\) a sheaf on \(X\). The cohomological support of \(F\) with respect to a map \(a: X \to A\) is \(V^i(F,a)\), the set of divisors \(\alpha \in \mathrm{Pic}^0(A)\) for which the \(i\)-cohomology \(h^i(F \otimes a^*\alpha) \neq 0\). In the particular case in which \(F=\omega_X\) is the canonical sheaf and \(a\) the Albanese map the \(V^i(\omega_X, a)\)'s are of interest in relation with pluricanonical maps of \(X\) (in particular to prove results of birationality of these maps). The main result of the paper (see Theorem 1.2) is on the cohomological support of some pluricanonical sheaves \(\omega_X^m\). Under some particular assumptions (see AS(1,2)), the subgroup of \(\mathrm{Pic}^0(X)\) generated by the translates through the origin of the components of \(V^0(\omega_X^m,a)\) (\(a\) the Albanese map) is \(I^*\mathrm{Pic}^0(S)\) (\(I:X \to S\) the Iitaka fibration). Moreover \(q(S)\) can be computed from \(q(X)\) and dimension and Kodaira dimension of \(X\) and of the general fiber \(F\) of the Stein factorization of the Albanese map. Applications to pluricanonical maps (of \(F\)) and some considerations on the assumptions AS(1,2) are also provided.
reviewed by: Roberto Muñoz
description: scientific article; zbMATH DE number 6355586