On coverings of simple abelian varieties

Abstract

Abstract. -To any finite covering f : Y → 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 ([9]), a Grassmannian ([11]), or a Lagrangian Grassmannian ([7]). We show an analogous result when X is a simple abelian variety and f does not factor through any nontrivial isogeny X → X. This result is obtained by showing that E f is M -regular in the sense of Pareschi-Popa, and that any M -regular sheaf is ample.Résumé (Sur les revêtements des variétés abéliennes simples). -On associeà tout revêtement fini f : Y → X de degré d entre variétés projectives lisses complexes un fibré vectoriel E f de rang d − 1 sur X dont l'espace total contient Y . On sait que E f est ample lorsque X est un espace projectif ([9]), une grassmannienne ([11]) ou une grassmannienne lagrangienne ([7]). Nous montrons un résultat analogue lorsque X est une variété abélienne simple et que f ne se factorise par aucune isogénie non triviale X → X. Ce résultat est obtenu en montrant que E f est M -régulier au sens de Pareschi-Popa, puis que tout faisceau M -régulier est ample.

Castelnuovo theory and the geometric Schottky problem

“…Another -this time homological -fact pointing towards such a parallel stems from previous work in which we have explored an abelian varieties analogue of for an overview of this circle of ideas). This yields results on the geometry of abelian varieties and their subvarieties which parallel classical facts of projective geometry ([PP1], [PP2], [PP3], [De3]). Although in this paper we will mainly use elementary geometric methods and Jacobian criteria based on the existence of trisecants to the Kummer, it is our hope that they will eventually naturally combine with homological methods.…”

Pushforwards of klt pairs under morphisms to abelian varieties

“…This theorem gives a detailed description about the positivity of the sheaf f * O X (D). It essentially says that the sheaf f * O X (D) is not just semipositive, but semiample, since M-regular sheaves are ample by [Deb06,Corollary 3.2].…”

Castelnuovo-Mumford Regularity and GV-Sheaves on Irregular Varieties

“…This gives a slightly coarser measure of positivity than CM-regularity; see Lemma 1.5 for a precise statement. The structure of cohomological support loci was addressed in [GL] and [Ha], and their connections with positivity were pursued in [De,PP1,PP4]. A key notion in these papers, as well as this note, is the following: F is said to be a GV-sheaf if codim(V i (F )) ≥ i for all i > 0.…”