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.

Abstract. We establish a -and conjecture further -relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of the cohomology of twisted ideal sheaves. The main ingredient is the Generic Vanishing criterion established in [PP3], based on the Fourier-Mukai transform.

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincaré transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.

We note that all of these results hold in the general context of arbitrary integral functors defined by locally free kernels, characterizing the filtration of Coh(X) by GV k -sheaves. The definitions and statements are provided at the end of §5 for completeness.

We apply the theory of M -regularity developed by the authors [Regularity on abelian varieties, I, J. Amer. Math. Soc. 16 (2003), [285][286][287][288][289][290][291][292][293][294][295][296][297][298][299][300][301][302] to the study of linear series given by multiples of ample line bundles on abelian varieties. We define an invariant of a line bundle, called Mregularity index, which governs the higher order properties and (partly conjecturally) the defining equations of such embeddings. We prove a general result on the behavior of the defining equations and higher syzygies in embeddings given by multiples of ample bundles whose base locus has no fixed components, extending a conjecture of Lazarsfeld [proved in Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), [651][652][653][654][655][656][657][658][659][660][661][662][663][664]. This approach also unifies essentially all the previously known results in this area, and is based on Fourier-Mukai techniques rather than representations of theta groups.Theorem. Let A be an ample line bundle on an abelian variety X. The following hold:(1) A 2 is globally generated.(2) (Lefschetz Theorem) A 3 is very ample.(Ohbuchi's Theorem [Oh1]) If A has no base divisor, then A 2 is very ample. (4) (Bauer-Szemberg Theorem [BSz]) A k+2 is k-jet ample, and the same holds for A k+1 if A has no base divisor (extending (1), (2) and (3)). (5) (Koizumi's Theorem [Ko]) A 3 gives a projectively normal embedding. (6) (Ohbuchi's Theorem [Oh2]) A 2 gives a projectively normal embedding if and only if 0 X does not belong to a finite union of translates of the base locus of A (cf. §5 for the concrete statement). (7) (Mumford's Theorem [M2], [Ke1]) For k ≥ 4, the ideal of X in the embedding given by A k is generated by quadrics. In the embedding given by A 3 it is generated by quadrics and cubics. (8) (Lazarsfeld's Conjecture [Pa], extending results of Kempf [Ke2]) A p+3 satisfies property N p (extending (5) and (7)).( 9) (Khaled's Theorem [Kh]) If A is globally generated, then the ideal of X in the embedding given by A 2 is generated by quadrics and cubics.These results turn out to be-some quick while others non-trivial-consequences of the general global generation criterion in [PP], called the Mregularity criterion. Together with a more technical extension (the W.I.T. regularity criterion), described below, this approach yields new results and extensions as well. To introduce them, we first need some terminology.Let X be an abelian variety of dimension g over an algebraically closed field, with dual abelian varietyX, and let P be a suitably normalized Poincaré line bundle on X ×X. The Fourier-Mukai functor [Mu] is the derived functor associated to the functorŜ(F ) = pX * (p * X F ⊗ P) from Mod(X) to Mod(X). A sheaf F on X is said to satisfy the Weak Index Theorem (W.I.T.) with index i(F ) = k if R iŜ (F ) = 0 for all i = k, in which case R kŜ (F ) is simply denotedF. A weaker condition, introduced in [PP], is the following: F is called M -regular if codim(Supp R iŜ (F )) > i for all ...

We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.

Let A be an ample line bundle on an abelian variety X (over an algebraically closed field The purpose of this paper is to prove Lazarsfeld's conjecture. To put such matters into perspective, it is useful to review the case of projective curves. A classical theorem of Castelnuovo states that a curve X, embedded in projective space by a complete linear system |L|, is projectively normal as soon as deg L ≥ 2g(X) + 1, and a theorem of Mattuck, Fujita and Saint-Donat states that if deg L ≥ 2g(X) + 2, then the homogeneous ideal of X is generated by quadrics. Green ([G1]) unified, re-interpreted and generalized these results to a statement about syzygies. Specifically, given a (smooth) projective variety X and a very ample line bundle L on X, a minimal resolution of R L as a graded S L -module (notation as above) looks like) (where, since X is embedded by a complete linear system, a 0j ≥ 2 for any j), E 1 = j S L (−a 1j ) (where, since the image of X in P(H 0 (L) ∨ ) is not contained in any hyperplane, a 1j ≥ 2 for any j), and, in general, for p ≥ 1, E p = j S L (−a pj ) with a pj ≥ p + 1 for any j. Green introduced the following terminology: L is said to satisfy property N 0 if E 0 = S L . This means that the map S L → R L is surjective, i.e. that the embedded variety X is projectively normal. Moreover L is said to satisfy property N 1 if it satisfies N 0 and a 1j = 2 for any j, i.e. the homogeneous ideal of the embedded variety X is generated by quadrics. Inductively, one says that L satisfies condition N p if it satisfies condition N p−1 and a pj = p+1 for any j. So N 2 means that the relations between the quadrics defining X are generated by linear ones and, for arbitrary p ≥ 2, N p means that the first p − 1 maps of the resolution of the homogeneous ideal are matrices with linear entries. In a word N p means that, up to the p-th step, the resolution (1) is as "regular" as it could possibly be.

We prove and conjecture results which show that Castelnuovo theory in projective space has a close analogue for abelian varieties. This is related to the geometric Schottky problem: our main result is that a principally polarized abelian variety satisfies a precise version of the Castelnuovo Lemma if and only if it is a Jacobian. This result has a surprising connection to the Trisecant Conjecture. We also give a genus bound for curves in abelian varieties