Regularity on abelian varieties I

Abstract

We introduce the notion of Mukai regularity ( M M -regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

Generalized Clifford–Severi inequality and the volume of irregular varieties

“…• Continuous global generation of L outside of the ramification locus of the Albanese map of X, is implied by M -regularity of L (see [26]). …”

On coverings of simple abelian varieties

“…We refer to [12] and [13] for more details. For our purposes, the main result of [13] (Proposition 2.13) is that an M -regular coherent sheaf on an abelian variety is continuously globally generated.…”

“…The proof is a simple application of the results of [13] about global generation of sheaves on an abelian variety. More precisely, it is based on the remark that any M -regular sheaf ( § 3) on an abelian variety is ample (Corollary 3.2).…”

Generic vanishing index and the birationality of the bicanonical map of irregular varieties

“…Therefore F ⊗ I x itself is CGG outside Z (with respect to a). Since the same is true for L, it follows from [PP1,Prop 2.12] that for all α ∈ Pic 0 A, F ⊗ L ⊗ I x is globally generated outside Z. So the rational map associated to |F ⊗ L| is birational.…”

“…Hence by (6), (7) and (8), gv(a * (F ⊗ I x )) ≥ 1. By [PP1,Prop. 2.13], a * (F ⊗ I x ) is continuously globally generated (CGG, see [PP1]).…”