Journées thématiques
de Cergy-Pontoise
11-12-13 December 2023: Moduli spaces of vector bundles and curves. This session will be dedicated to moduli spaces of vector bundles and curves, and their role in enumerative geometry, representation theory, and Hodge theory.
Schedule:
Monday 11 December
- 9h: Welcome coffee
- 10h-10h50: David Holmes (1/3)
- 11h00-11h50: Florent Schaffhauser (1/3)
Lunch - 14h00-14h50: Alexander Soibelman (1/3)
- 15h00-15h50: David Holmes (2/3)
Coffee break - 16h30-17h20: Florent Schaffhauser (2/3)
- 9h30-10h20: Alexander Soibelman (2/3)
- 10h30-11h20: David Holmes (3/3)
Lunch - 13h30-14h20: Sergej Monavari
- 14h30-15h20: Denis Nesterov
Coffee break - 16h00-16h50: Okke van Garderen
- 9h30-10h20: Alexander Soibelman (3/3)
- 10h30-11h20: Florent Schaffhauser (3/3)
Lunch - 13h30-14h20: Gabriele Rembado
- 14h30-15h20*: Veronica Fantini
Coffee break - 16h00-16h50*: Noah Arbesfeld
Lectures:
- David Holmes (Universiteit Leiden, the Netherlands).
Line bundles on logarithmic curves and the double ramification cycle
Abstract: We will begin by studying the basic structure of jacobians of degenerating families of curves, with a particular focus on the universal curve over the moduli space of stable curves. We will then start to think about the double ramification cycle (a fancy name for the locus where a line bundle on a curve is fibrewise-trivial), initially for smooth curves, then for "nice" (compact type) singular curves. We will then introduce the a little of the machinery of logarithmic curves and logarithmic line bundles, to extend this definition to all stable curves. Finally, we will discuss how to give a formula for the resulting tautological cycle, and a couple of ways to prove the formula.
Tentative outline:
Hour 1:
- Very quick reminder on (pre)stable curves;
- Structure of jacobians of prestable curves;
- Explain why the big jacobian of the universal curve over Mg,n is almost never separated, and the small one does not have extension property;
- Potentially an aside on Raynaud's construction of Néron models by taking the quotient by the closure of the unit section, and how to extend to higher-dimensional families.
Hour 2:
- The DR cycle for smooth curves;
- Extending DR to compact type by twisting;
- Log structures on curves (Deligne-Faltings approach);
Hour 3:
- LogDR as the locus where line bundle is log trivial;
- The log Chow ring;
- Statement of LogDR formula;
- Idea of two proofs of the formula, one via localisation, the other via GRR.
- Florent Schaffhauser (Universität Heidelberg, Germany).
Higgs bundles for non-constant groups and applications
Abstract: Nonabelian Hodge theory can be seen as a vast generalization of the theory of harmonic forms on smooth and projective complex analytic curves. It is due to Hitchin, Simpson, Donaldson and Corlette. The fundamental result of the theory is the existence of a correspondence between the “Betti cohomology” and the “Dolbeault cohomology” of the variety. In his 1992 paper ‘Higgs bundles and local systems’, Carlos Simpson observed that the Betti side generalises to non-constant coefficients, and asked about the corresponding Dolbeault space in that case. The purpose of this mini-course is to explain one possible answer to Simpson’s question.
- Alexander Soibelman (IHES, France).
Motivic invariants for moduli of Higgs bundles on a curve
Abstract: Motivic classes can be used to encode certain invariants in algebraic geometry in terms of elements of the Grothendieck ring of varieties or, more generally, of stacks. We begin with an introduction to motivic class computations from the point of view of counting rational points of an algebraic variety over a finite field. We will then familiarize ourselves with some of the tools used in performing these computations, such as Hall algebras, before moving on to their motivic versions. After giving a brief rundown of some known computations related to moduli of vector bundles on a curve, we will focus on motivic classes for moduli stacks of vector bundle and connection pairs, moduli stacks of semistable Higgs bundles, as well as their parabolic versions. As an application, we will look at how motivic class computations for parabolic Higgs bundles on the projective line relate to the Deligne-Simpson problem. We conclude with a discussion of motivic classes of moduli of irregular parabolic connections and moduli of irregular parabolic Higgs bundles.
- Noah Arbesfeld (University of Vienna, Austria).
Computing vertical Vafa-Witten invariants
Abstract: I'll present a computation in the algebraic approach to Vafa-Witten invariants of projective surfaces, as introduced by Tanaka-Thomas. The invariants are defined using moduli spaces of stable Higgs pairs on surfaces and are formed from contributions of components. The physical notion of S-duality translates to conjectural symmetries between these contributions. One component, the "vertical" component, is a nested Hilbert scheme on a surface. I'll explain work in preparation with M. Kool and T. Laarakker in which we express invariants of this component in terms of a quiver variety, the instanton moduli space of torsion-free framed sheaves on P^2. As a consequence, we deduce constraints on Vafa-Witten invariants, including a formula for the contribution of the vertical component to refined invariants in rank 2.
- Veronica Fantini (IHES, France).
Stokes invariants for local P2 and modularity
Abstract: The generating function of Gromov—Witten invariants for local P2 has well-known modularity properties, as regarded as function on the Kahler moduli. However, a different source of invariants can be defined from the study the so-called fermionic spectral traces for local P2.
Fermionic spectral traces are the coefficients in the expansion at the orbifold point (in the moduli space of the mirror curve) of a function related to the quantum mirror curve. Then, the Stokes invariants are defined by studying the asymptotics of the fermionic spectral traces in certain limits (the semiclassical and its dual). Indeed, by the TS/ST correspondence, these limits correspond to expansions at different points in the moduli space of local P2. In a joint work with C. Rella, we conjecture that the generating function of the Stokes invariants for local P2 has certain modular properties.
- Okke van Garderen (University of Luxembourg, Luxembourg). Gauge theory for sheaves in Calabi-Yau threefolds
On a Calabi-Yau threefold, holomorphic vector bundles can be described in two ways: algebraically as moduli spaces of semistable sheaves, or gauge-theoretically as the critical points of a holomorphic Chern-Simons functional. This dual perspective allowed Donaldson and Thomas to use enumerative geometry to define meaningful invariants for the infinite dimensional spaces appearing in the latter description. In this talk I explain how the gauge theoretic perspective extends to moduli of sheaves supported on a closed subvariety of a threefold by constructing a natural Chern-Simons functional. The construction depends only on a local choice of volume, and therefore applies also to non-compact varieties. I will moreover show how the functional reduces to a potential on a finite dimensional quiver moduli space, which can be used to refine and categorify curve-counting invariants.
- Sergej Monavari (EPFL Lausanne, Switzerland).
Tetrahedron instantons in Donaldson-Thomas theory
Abstract: Tetrahedron instantons were recently introduced by Pomoni-Yan-Zhang in string theory, as a way to describe systems of D0-D6 branes with defects. We propose a rigorous geometric interpretation of their work by the point of view of Donaldson-Thomas theory. We will explain how to naturally construct the moduli space of tetrahedron instantons as a Quot scheme, parametrizing quotients of a torsion sheaf over a certain singular threefold, and how to construct a virtual fundamental class in this setting using quiver representations and the recent machinery of Oh-Thomas (which is in principle designed for moduli spaces of sheaves on Calabi-Yau 4-folds). Furthermore, we will show how to formalize mathematically the invariants considered by Pomoni-Yan-Zhang (initially defined via supersymmetric localization in Physics) and how to rigorously compute them, solving some open conjectures. Joint work with Nadir Fasola.
- Denis Nesterov (University of Vienna, Austria).
Unramified Gromov-Witten and Gopakumar-Vafa invariants
Abstract: Kim, Kresch and Oh constructed compactified moduli spaces of unramified maps from curves to a smooth projective target. The associated invariants are called unramified Gromov-Witten invariants. In dimension 3, Pandharipande conjectured that unramified Gromov-Witten invariants are equal to Gopakumar-Vafa invariants (BPS invariants) for Fano and primitive Calabi-Yau classes. After an introduction to unramified Gromov-Witten theory, I will present a certain wall-crossing technique that can be used to prove this conjecture.
- Gabriele Rembado (Université de Montpellier, France).
Moduli spaces of wild connections: deformations and quantisations
Abstract: Moduli spaces of meromorphic connections on (principal bundles over) Riemann surfaces have a rich geometric structure. In the logarithmic case, they encompass the complex character varieties of pointed Riemann surfaces, which assemble into flat Poisson/symplectic fibre bundles upon deforming the surface; after geometric/deformation quantisation, this yields (projectively) flat vector bundles over the base space of deformations, famously including the Knizhnik--Zamolodchikov connection from 2d conformal field theory. In this talk we will aim at a review of part of this story, and then present recent work about extensions involving deformations and quantisations of moduli spaces of irregular singular (`wild') meromorphic connections. These are joint work with (P. Boalch, J. Douçot, M. Tamiozzo) and (G. Felder, R. Wentworth).
Location : Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95300 Cergy-Pontoise. The room of the seminar is E554 appart from the last talks (indicated with star) which will be in the E1 amphitheater.
Organizers: Bertrand Deroin, Louis Ioos, and Adrien Sauvaget.