Martin Ulirsch - Vector bundles on Riemann surfaces and metric graphs

Oberseminar: June 2, 2022

Prof. Dr. Martin Ulirsch (Frankfurt) will speak on:

Vector bundles on Riemann surfaces and metric graphs

Abstract: The geometry of various moduli spaces of bundles on a compact Riemann surface is a pillar of modern geometry, since they are at the sweet spot of being at same time very intricate but also accessible to a manifold of different techniques. Recently the perspective that compact metric graphs are, in a way, a natural combinatorial, or "tropical", analogue of compact Riemann surfaces has gained significant traction. This is, in particular, due to its numerous applications in the context of enumerative geometry, the cohomology of moduli spaces, and Brill-Noether theory.

A tropical analogue of line bundles on metric graphs is, by now, well-understood and reflects the various compactifications of the Jacobian over semistable degenerations of compact Riemann surfaces. The goal of this talk is to propose an up-to-now still missing analogue of vector bundles of higher rank on metric graphs. After defining these objects I will, in particular, talk about tropical analogues of the Weil-Riemann-Roch-Theorem and of the Narasimhan-Seshadri correspondence. Then I will outline a tropicalization procedure that lets us connect this a priori only combinatorial theory with the classical story. As it turns out, this will work best in the case of the Tate curve.

Given time, I might indulge in some speculations concerning a new approach to degenerations of vector bundles using methods from logarithmic geometry that incorporates and expands on both the algebro-geometric and the tropical story.

The non-speculative aspects of this talk are based on joint work with Margarida Melo, Sam Molcho, and Filippo Viviani and joint work in progress with Andreas Gross and Dmitry Zakharov.