Positivity in varieties of maximal Albanese dimension
type: Item
instance of: scholarly article
title: Positivity in varieties of maximal Albanese dimension
zbMATH Open document ID: 1387.14033
author: Jungkai A. Chen
author: Zhi Jiang
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2018-03-07T00:00:00Z
review text: In this paper, the authors study some aspects of the birational geometry of smooth complex projective varieties of maximal Albanese dimension, i.\ e., varieties \(X\) admitting a generically finite morphism \(f:X \to A\) to an abelian variety \(A\). In the first part of the paper, given such a morphism \(f\), the authors prove a general decomposition theorem for \(f_*\omega_X\), showing that this sheaf is not far from being M-regular, in the sense introduced and studied by \textit{G. Pareschi} and \textit{M. Popa} [J. Am. Math. Soc. 16, No. 2, 285--302 (2003; Zbl 1022.14012)]. Roughly speaking this implies that \(f_*\omega_X\) is sufficiently positive on \(A\). From this result they obtain that if, in addition, \(X\) is of general type, then the bicanonical map of \(X\) is generically finite to its image. The same result for the tricanonical map has been proved by the second author et al. [Int. Math. Res. Not. 2013, No. 13, 2984--3005 (2013; Zbl 1312.14033)]. In the second part of the paper, the authors investigate the structure of \(X\) when \(X\) is of general type of maximal Albanese dimension and \(\chi(X,\omega_X)=0\). In general it is known that \(\chi(X,\omega_X) \geq 0\), and several studies seem to indicate that the case \(\chi(X,\omega_X)=0\) is the most difficult to deal with. Inspired by the results in their paper [J. Reine Angew. Math. 691, 203--227 (2014; Zbl 1307.14008)], which is the starting point of this investigation, they formulate a conjecture. A key role in the formulation is played by varieties of the above type that the authors call ``primitive of \(\chi=0\)'', since they can reduce to this case up to considering suitable fibrations whose general fiber is of this type again. \(X\) is conjectured to be primitive of \(\chi=0\) if and only if there exist simple abelian varieties \(K_1, \dots ,K_{2k+1}\), coming from double coverings \(F_i \to K_i\), where \(F_i\) is a normal projective variety of general type with involution \(\sigma_i\), and an…
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14K99
zbMATH DE Number: 6847722
zbMATH Keywords: irregular smooth complex projective variety of general type
zbMATH Keywords: maximal Albanese dimension
zbMATH Keywords: holomorphic Euler characteristic
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
full work available at URL: crelle-2015-0027
OpenAlex ID: W2562463473
cites work: Q4023939
cites work: Varieties with vanishing holomorphic Euler characteristic
cites work: Pluricanonical maps of varieties of maximal Albanese dimension
cites work: On the Irregularity of the Image of the Iitaka Fibration
cites work: On coverings of simple abelian varieties
cites work: Singularities of theta divisors and the birational geometry of irregular varieties
cites work: On Kähler fiber spaces over curves
cites work: Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville
cites work: Higher Obstructions to Deforming Cohomology Groups of Line Bundles
cites work: On the finiteness theorem for rational maps on a variety of general type
cites work: A derived category approach to generic vanishing
cites work: On the birational geometry of varieties of maximal Albanese dimension
cites work: Q5470192
cites work: On the Iitaka Fibration of Varieties of Maximal Albanese Dimension
cites work: Higher direct images of dualizing sheaves. I
cites work: Higher direct images of dualizing sheaves. II
cites work: Q4847164
cites work: Duality between D(X) and with its application to picard sheaves
cites work: Regularity on abelian varieties I
cites work: GV-sheaves, Fourier-Mukai transform, and generic vanishing
cites work: Regularity on abelian varieties III: relationship with Generic Vanishing and applications
cites work: Subspaces of moduli spaces of rank one local systems
DOI: 10.1515/CRELLE-2015-0027
description: scientific article; zbMATH DE number 6847722