The basepoint-freeness threshold, derived invariants of irregular varieties, and stability of syzygy bundles

My Ph.D. thesis consists of three distinct parts, concerning various aspects of abelian and irregular varieties.The first two are related each other by the common use of the generic vanishing theory as one of the principal tools, while the third one deals with the stability of special vector bundles on an abelian variety. The first chapter regards syzygies of polarized abelian varieties. The study of equations defining projective varieties comes out very naturally in algebraic geometry and it has received considerable attention over the years. In the 80s, Green realized that classical results of Castelnuovo, Mumford, Saint-Donat and Fujita for a smooth projective curve C endowed with a line bundle L, could be unified and generalized to a very satisfying statement about syzygies. After Green's result on curves appeared, several works focused on finding extensions of it to higher dimensional varieties. A natural candidate is the class of abelian varieties. Indeed, based on classical facts of Koizumi, Mumford and Kempf, and motivated by the result of Green on higher syzygies for curves, Lazarsfeld conjectured that, for an ample line bundle L on an abelian variety, L^m satisfies the property (N_p) if m is at least p+3. This was proved by Pareschi in 2000 with an argument working in characteristic zero. The property (N_p) means that the first p modules of syzygies of the section algebra of L are "linear", i.e. as simple as possible. Intuitively, these notions consist of an increasing sequence of "positivity" properties of L. We showed a general result that, in particular, provides at the same time a surprisingly quick proof of Lazarsfeld's conjecture, extending it to abelian varieties defined over a ground field of arbitrary characteristic, and a new proof of a criterion of Lazarsfeld-Pareschi-Popa, regarding syzygies of line bundles that are not necessarily multiples of other ones. Our main result is that the basepoint-freeness threshold, introduced by Jiang-Pareschi for the numerical class of an ample line bundle L, encodes information about the syzygies of L. Namely, if it is less than 1/(p+2) for some non-negative integer p, then the property (N_p) holds for L. The proof works over algebraically closed fields of arbitrary characteristic. In addition to syzygies, we prove that the basepoint-freeness threshold also gives information on the (local) positivity of the polarization L, indeed it controls the k-jet ampleness of L, and it is related to its Seshadri constant. The second chapter contains the results of a paper joint with G. Pareschi, published in the journal Algebraic Geometry . An extension of our main theorem to pushforward of pluricanonical bundles is also included. Given a smooth complex projective variety X, its derived category is a triangulated category naturally associated to X, whose objects are bounded complexes of coherent sheaves on X. Given another variety Y, we say that X is derived equivalent to Y if there exists an exact equivalence between their derived categories. It is very natural to ask which geometric information are preserved under derived equivalence. Over the past 20 years, a big deal of interest grew on this type of question, thanks especially to the works of Bondal, Orlov, Kawamata, Kontsevich, Popa, Schnell. The main conjecture in the field, often attributed to Kontsevich, predicts that derived equivalent varieties have the same Hodge numbers. However, up to now, the derived invariance of the Hodge numbers h^{0,j} is not even known. We prove a general result in this direction: the derived invariance of the cohomology ranks of the pushforward under the Albanese map of the canonical line bundle. In particular, in the case of varieties of maximal Albanese dimension, this settles in the affirmative conjectures of Popa and Lombardi-Popa, and it proves that the Hodge numbers h^{0,j} are derived invariants, for all j. The third chapter concerns a joint work (in progress) with Martí Lahoz. We settled a conjecture of Ein-Lazarsfeld-Mustopa, in the case of abelian varieties. Let (X, L) be a polarized smooth variety over an algebraically closed field. Ein-Lazarsfeld-Mustopa proved, in dimension 2, the slope stability of the kernel of the evaluation morphism of global sections, namely the syzygy bundle, of the line bundle L^d with respect to L, for d sufficiently large. They also conjectured that such result should hold for any smooth projective variety. The main result of the third chapter is basically a proof of this conjecture in the case of abelian varieties. Namely, let (A, L) be a polarized abelian variety defined over an algebraically closed field of arbitrary characteristic, and let d at least 2. Then the syzygy bundle M_{L^d} is Gieseker semistable with respect to L.