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On coverings of simple abelian varieties. (English) Zbl 1109.14017
Summary: To any finite covering $$f:Y\to X$$ of degree $$d$$ between smooth complex projective manifolds, one associates a vector bundle $$E_f$$ of rank $$d-1$$ on $$X$$ whose total space contains $$Y$$. It is known that $$E_f$$ is ample when $$X$$ is a projective space [R. Lazarsfeld, Math. Ann. 249, 153–162 (1980; Zbl 0434.32013)], a Grassmannian [L. Manivel, Invent. Math. 127, No.2, 401–416 (1997; Zbl 0906.14011)], or a Lagrangian Grassmannian M. Kim and L. Manivel, Topology 38, No. 5, 1141–1160 (1999; Zbl 0935.14008)]. We show an analogous result when $$X$$ is a simple abelian variety and $$f$$ does not factor through any nontrivial isogeny $$X'\to X$$. This result is obtained by showing that $$E_f$$ is $$M$$-regular in the sense of G. Pareschi and M. Popa [J. Am. Math. Soc. 16, No. 2, 285–302 (2003; Zbl 1022.14012)], and that any $$M$$-regular sheaf is ample.

##### MSC:
 14E20 Coverings in algebraic geometry 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14K02 Isogeny 14K05 Algebraic theory of abelian varieties 14K12 Subvarieties of abelian varieties
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