Warwick Algebraic Geometry Seminar

Abstracts

Martijn Kool (Utrecht) - New directions in Vafa-Witten theory

In this talk I discuss (virtual) Euler characteristics of moduli spaces of stable sheaves on general type surfaces. These are part of so-called Vafa-Witten invariants, which were discovered in the context of supersymmetric Yang-Mills theory by Vafa-Witten in 1994 and mathematically defined by Tanaka-Thomas in 2017. The generating functions of these invariants have interesting modularity properties predicted by physics. Using virtual intersection theory, we can verify Vafa-Witten's original predictions in examples and generalize in several new directions. Joint work with Göttsche.

Weiwei Wu (University of Georgia) - Semi-toric spherical systems and symplectomorphism groups

We will explain a generalization of semi-toric systems. In dimension four, such systems can be easily obtained by generalizing the notion of "toric blow-up". As it turns out, this construction gains new understandings of the symplectic mapping class groups. We will explain its relation to a long-standing question between Lagrangian Dehn twists and symplectic mapping class groups of rational manifolds, and potential construction of exotic finite group actions. This talk is based on several on-going joint works with Liat Kessler, Jun Li and Tian-Jun Li.

Yoav Len (Georgia Tech) - Algebraic and tropical Prym varieties

My talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian varieties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concerning Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci. This is joint work with Martin Ulirsch.

Clélia Pech (Kent) - Quantum cohomology and mirror symmetry for odd symplectic Grassmannians

Odd symplectic Grassmannians are two-orbit algebraic varieties which share a lot of common features with homogeneous varieties, and are one of the nicest examples of a wider family of objects called horospherical varieties. In this talk, I will explain some old and new results about them concerning their quantum cohomology, their derived category, and their mirrors. Part of this work is joint with R. Gonzales, N. Perrin and A. Samokhin.

Travis Mandel (Edinburgh) - Tropical multiplicities from mirror polyvector fields and tropical quantum field theory

I will explain a formula for the multiplicity of a genus 0 tropical curve in terms of certain iterated brackets of polyvector fields. This formula follows from a more general (arbitrary genus) formula given in terms of a certain tropical quantum field theory, i.e., a symmetric monoidal functor whose domain is the category of "tropical cobordisms." This is joint work with H. Ruddat.

Christopher Lazda (Amsterdam) - A Neron-Ogg-Shafarevich criterion for K3 surfaces

The naive analogue of the Néron-Ogg-Shafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.