Positivity in varieties of maximal Albanese dimension
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 isogeny \(K_1 \times \dots \times K_{2k+1} \to A_X\) (where \(A_X\) is the Albanese variety of \(X\)) such that, up to the base change, \(X\) is birational to the quotient \((F_1 \times \dots \times F_{2k+1})/\langle \sigma_1 \times \dots \times \sigma_{2k+1}\rangle \). Any \(X\) constructed in this way is a primitive variety of \(\chi=0\). The authors succeed to prove the converse when \(A_X\) has exactly three simple factors.
irregular smooth complex projective variety of general type
scientific article; zbMATH DE number 6847722
scientific article