A Characterization of Finite Quotients of Abelian Varieties. (24th December 2016)

• Record Type: Journal Article
• Authors: Lu, Steven; Taji, Behrouz
• Is Part Of: International mathematics research notices. Volume 2018:Number 1(2018)
• Page Start: 292
• Page End: 319
• Publication Date: 2016-12-24
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnw251
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We provide a characterization of quotients of Abelian varieties by finite groups actions that are free in codimension-one via vanishing conditions on the orbifold Chern classes. The characterization is given among a class of varieties with singularities that are more general than quotient singularities, namely among the class of klt varieties. Furthermore, for a semistable (respectively stable) reflexive ${\mathcal O}_{X}$ -module ${\mathscr{E}}$ with zero first and second orbifold Chern classes over such a variety $X$, we show that ${\mathscr{E}}|_{X_\mathrm{reg}^\mathrm{an}}$ is locally-free and flat, given by a linear (irreducible unitary) representation of $\pi_1(X_\mathrm{reg})$, and that it extends over a finite Galois cover $\widetilde X$ of $X$ étale over $X_\mathrm{reg}$ to a locally-free and flat sheaf given by an equivariant linear (irreducible unitary) representation of $\pi_1(\widetilde X)$ . These are generalizations to the singular setting that is more general than any orbifold strengthenings of the classical correspondences of Donaldson-Uhlenbeck-Yau-Simpson.