Regularity on abelian varieties III: relationship with Generic Vanishing and applications

Abstract

We describe the relationship between the notions of $M$-regular sheaf and $GV$-sheaf in the case of abelian varieties. The former is a natural strengthening of the latter, and we provide an algebraic criterion characterizing it among the larger class. Based on this we deduce new basic properties of both $M$-regular and $GV$-sheaves. In the second part we give a number of applications of generation criteria for $M$-regular sheaves to the study of Seshadri constants, Picard bundles, pluricanonical maps on irregular varieties, and semihomogeneous vector bundles. This second part of the paper is based on our earlier preprint math.AG/0306103, with some improved statements and shortened arguments.

25 pages; this replaces the older preprint math.AG/0306103 and roughly half of the content is new; prepared for the Proceedings of the Clay Institute Workshop on vector bundles, October 2006; updated references

References in corpus (5)

• Seshadri constants on algebraic surfaces
• Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality
• The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the jacobian
• Heisenberg invariant quartics and SU_C(2) for a curve of genus four
• Regularity on abelian varieties III: further applications

Cited by in corpus (23)

• Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis inequality
• Linear systems on irregular varieties
• Cohomological support loci for Abel-Prym curves
• The cohomological support locus of pluricaonical sheaves and the Iitaka fibration
• Singularities of pluri-theta divisors in Char $p>0$
• On varieties of maximal Albanese dimension
• Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories
• On the generic vanishing theorem of Cartier modules
• Cohomological rank functions on abelian varieties
• Automorphisms of surfaces of general type with q>=2 acting trivially in cohomology
• On the bicanonical map of irregular varieties
• Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic
• Theta-regularity and log-canonical threshold
• Irreducible Theta Divisors of PPAV's are Strongly F-regular
• Castelnuovo-Mumford Regularity and GV-Sheaves on Irregular Varieties
• Standard canonical support loci
• On the bicanonical maps of primitive varieties with $q(X)=dim(X)$: the degree and the Euler number
• Theta-regularity of curves and Brill-Noether loci
• Regularity of curves in abelian varieties
• The Equations of Singular Loci of Ample Divisors on (Subvarieties of) Abelian Varieties
• A note on the linear systems on the projective bundles over Abelian varieties
• Syzygies, Pluricanonical Maps, and the Birational Geometry of Varieties of Maximal Albanese Dimension
• On the vanishing of weight one Koszul cohomology of abelian varieties