Algebraic Geometry and Combinatorics Seminar: Geometry and topology of universal compactified Jacobians

Abstract: For a family of smooth curves, the relative Jacobian parametrizing degree 0 line bundles is a fundamental example of a family of principally polarized abelian varieties. Over the moduli space of stable curves, the relative Jacobian admits several different smooth toroidal compactifications. The study of such abelian fibrations appears in the study of Higgs bundles, hyperkahler manifolds, enumerative geometry and logarithmic geometry.

In this talk, I will focus on the ring structure on the rational cohomology/Chow group of fine compactified Jacobians. First, I will show that the ring structure on the rational cohomology group depends on the compactification. To address this, we degenerate the ring structure using the perverse filtration, and prove that this "intrinsic cohomology ring" is independent of the choice of stability conditions. Finally, I will present intersection computations of divisors in the (intrinsic) cohomology ring. These results draw on a combination of Fourier transforms, Decomposition theorem, logarithmic geometry and Gromov-Witten theory. These are joint works with D. Maulik, J. Shen, and Q. Yin; and with S. Molcho, and A. Pixton.

Host: Carl Lian