30/08, UFF. Sala de seminários da pós-graduação, 7o. andar
Roi do Campo (UFF),
Jacobian discrepancies and rational singularities |
Resumo. In this talk I will introduce the notion of Jacobian discrepancy, an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. This invariant, very natural from the point of view of jet schemes and arc spaces, leads to a framework in which adjunction and inverstion of adjunction hold in full generality. Moreover, they allow us to give explicit formulas measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety, leading to characterizations of rational singularities in terms of discrepancies.
Filippo Viviani (Roma Tre),
Fourier-Mukai transform and autoduality for compactified Jacobians |
Resumo. Compactified Jacobians of singular curves with planar singularities appear as fibers of the Hitchin map for the moduli space of Higgs bundles. The classic limit of the geometric Langlands conjecture (as formulated by Donagi-Pantev) predicts a Fourier-Mukai autoequivalence of the fibers of the Hitchin map. For reduced curves, we are able to define a Poincar? sheaf and to show that the induced integral transform defines a Fourier-Mukai autoequivalence of any fine compactified Jacobian. This was previously establish for integral curves by D. Arinkin. As a corollary, we establish an autoduality result for compactified Jacobians, generalizing previous results of Arinkin, Esteves, Gagn e, Kleiman, Rocha. The major novelty for reducible curves is that there are many (possibly non-isomorphic) fine compactified Jacobians, all of which are birational Calabi-Yau projective varieties with mild singularities. More generally, we show that any two such compactified Jacobians are derived equivalent. This is
in line with Kawamata's conjecture that birational Calabi-Yau (smooth) varieties should be derived equivalent and it seems to suggest an extension of this conjecture to (mildly) singular Calabi-Yau varieties. This is a joint work with M. Melo and A. Rapagnetta.