ABELIAN VARIETIES, MODULI OF VECTOR BUNDLES AND MODULI OF COURVES

Harvard University, Cambridge MA

Investigators

Abstract

In work with G. Pareschi, Popa has introduced a notion of Mukai regularity for coherent sheaves on abelian varieties, defined using the Fourier-Mukai transform, and developed its theory. Popa and Pareschi are now interested in using the concept of Mukai regularity in the study of irregular varieties, especially to give effective results on adjoint linear series and pluiricanonical maps. Also, Popa and Pareschi intend to use invariants arising naturally from this theory in order to differentiate between Jacobians and other abelian varieties, and thus approach the Schottky problem in a new way. Using some of their results that carry through in the more general context of an arbitrary Fourier transform, Popa and Pareschi intend to obtain constraints on the existence of equivalences of derived categories, which is a main concern in this line of approach to mirror symmetry. In other work, Popa has established effective results for linear series on moduli spaces of vector bundles on curves. Popa is interested in approaching the Strange Duality by refining his methods based on the Fourier-Mukai transform of Verlinde bundles on Jacobians and combining it with representation theory techniques. Popa also intends to approach some optimal conjectures on effective base point freeness on these moduli spaces for general curves, using degeneration to stable curves. In work with G. Farkas, Popa has obtained Brill-Noether type non-existence results for rank 2 vector bundles on general curves, via generalized limit linear series and stability of pairs on reducible curves. Popa and Farkas intend to further refine these techniques in order to prove a conjecture of Bertram-Feinberger-Mukai on Brill-Noether theory for rank 2 vector bundles with canonical determinant, and to find non-existence results in higher ranks. This will allow them to define new divisors in moduli spaces of stable curves for appropriate genera. The ultimate goal of this project is to compute the class of these divisors and determine whether they provide counterexamples to the Harris-Morrison Slope Conjecture as expected. Algebraic curves and abelian varieties are ubiquitous objects in mathematics. Apart from their realizations in algebraic geometry, they appear in complex analysis (as Riemann surfaces or complex tori) or algebra (as field extensions or group schemes), and play a fundamental role in recent progress in mathematical physics. One of the most important problems in algebraic geometry is to classify algebraic varieties. For one-dimensional varieties (that is, for algebraic curves) this problem is approached by understanding the geometry of a space M(g) parametrizing all curves of given genus. In the case of abelian varieties, one way to approach this is to understand the totality of a specific kind of algebraic objects (called coherent sheaves) which can be associated to them. The investigator is pursuing these directions in research related to this proposal.