# Geometry of coherent sheaves: From Derived categories to Higgs bundles

Starting April 2022, there will be a series of one-day workshops at Goethe Universität, Frankfurt on varying themes in the realm of moduli spaces of vector bundles and coherent sheaves, representation varieties and non-abelian Hodge theory. The workshops will be in hybrid form: in person at Frankfurt University and streamed via Zoom. If you would like to receive updates and Zoom data about the upcoming workshops, please contact Inder, Johannes or Barbara. All times are Frankfurt time. You can use the following link to check what the stated time corresponds to in your city:

https://www.timeanddate.com/worldclock/converter.html

This workshop series is organized and funded as part of GAUS.

Upcoming Workshops

## Recent developments in GIT April 24th 2023

2:00 - 3:00 p.m : Victoria Hoskins (Nijmegen, speaking remotely)

An introduction to geometric invariant theory

The aim of this survey talk is to give an introduction to geometric invariant theory in order to prepare the audience for the subsequent talks as requested by the organisers. I will start by explaining how group actions often appear in moduli problems and we will see how constructing algebra-geometric quotients is related to 19th century invariant theory. I will explain why the theory is simplest for non-reductive group actions and, in this case, I will explain how Mumford constructs quotients (of certain open 'semistable' subsets) using geometric invariant theory, as well as giving combinatorial and numerical criteria for semistability. If there is time, I will briefly mention some recent developments to extend GIT to certain non-reductive group actions.

3:20 - 4:20 p.m : Joshua Jackson (Sheffield, speaking on-site)

Advances in Non-reductive GIT and applications

Following from the previous talk on reductive GIT, I will survey recent developments in extending this theory to non-reductive groups, with a particular focus on applications to moduli theory. Time permitting, I will then indicate how non-reductive GIT can be used in the study of sheaves, Higgs bundles, hypersurfaces, and singular curves.

4:40 - 5:40 p.m : Dario Weissmann (Essen, speaking on-site)

A stacky approach to identify the semi-stable locus of vector bundles

I report on recent joint work with Xucheng Zhang focusing on our Theorem A for vector bundles in characteristic 0: The semi-stable locus in the stack of bundles over a smooth projective curve is the maximal open locus admitting a schematic good moduli space. This gives an intrinsic motivation for semi-stability of vector bundles. Historically, semi-stability appeared in the quest for a moduli space of bundles and the classical construction of this moduli space uses a non-canonical GIT-construction. Theorem A also provides us with natural examples of good moduli spaces which are only algebraic spaces and not schemes.

Past Workshops

## Hodge and Poincaré polynomials of moduli spaces of bundles April 29th 2022

2pm-3pm: Vicente Muñoz (Málaga)(speaking remotely)

E-Polynomials of the SL(2,C)-character varieties of orientable surfaces

Abstract: Let X be an orientable surface of genus g, and G a complex Lie group. We fix a point p in X and a conjugacy class [D] of G. The representation variety is defined as the space parametrizing homomorphisms of the fundamental group $\pi_1(X-p)$ into G such that a loop around p goes to [D]. The associated moduli space is the GIT quotient by the conjugation action of G, and parametrizes flat G-bundles over X with prescribed monodromy around p. This is one of the incarnations of non-abelian Hodge theory, being homeomorphic to the moduli space of parabolic Higgs bundles in the case when [D] is diagonalizable.

In this talk, we compute the Hodge-Deligne polynomials (or E-polynomials) of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2,C), for any possible monodromy round the puncture. We shall explain the geometric technique introduced by Logares, Muñoz and Newstead based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations. The results are joint work with J. Martínez.

3:15pm-4:15pm: Dimitry Wyss (Lausanne)(speaking remotely)

Intersection cohomology from non-archimedean integrals

Abstract: Let $M(\beta,\chi)$ be the moduli space of one-dimensional semi-stable sheaves on a del Pezzo surface S, supported on an ample curve class $\beta$ and with Euler-characteristic $\chi$. Maulik and Shen recently showed that the intersection cohomology of $M(\beta,\chi)$ is independent of $\chi$ confirming a conjecture of Toda. Working over a non-archimedean local field F, we define a natural measure on the F-points of $M(\beta,\chi)$ and prove that the integral of a certain gerbe on $M(\beta,\chi)$ with respect to this measure is also independent of $\chi$. As a consequence we get a non-archimedean interpretation of the intersection cohomology of $M(\beta,\chi)$. Our results also apply to K3 surfaces and (meromorphic) Higgs bundles. This is joint work with Francesca Carocci and Giulio Orecchia.

## Branes in Higgs bundle moduli spaces May 20th 2022

2pm - 3pm: Emilio Franco (Lisboa) (speaking on-site)

Branes in the hyperKähler framework

Abstract: We will present a brief introduction to BBB- BAA- ABA- and AAB-branes arising in the work of Kapustin-Witten on Geometric Langlands Program.

3:15pm - 4:15pm: Ana Peón-Nieto (Birmingham) (speaking remotely)

Some branes over the singular locus of the Hitchin base

Abstract: In this talk I will speak about two families of BBB (and their dual BAA branes) that appear on the singular locus of the Hitchin system. The first one plays a key role in topological mirror symmetry, and is given by branes of fixed points by the n-torsion elements of the Jacobian variety. The second type of family is given by singular points of the moduli space, consisting of Higgs bundles with a reduction of the structure group to a maximal rank reductive subgroup. After explaining their description and relation, I will report on ongoing work with Franco, Gothen and Oliveira about their closure along the nilpotent cone.

4:30pm - 5:30pm: Sebastian Heller (Hannover) (speaking on-site)

Branes through finite group actions

Abstract: Higgs bundle moduli spaces are equipped with natural hyper-Kaehler structures. Since the work of Hausel and Thaddeus, and Kapustin and Witten, Higgs bundle moduli spaces have been related to mirror symmetry and geometric Langlands duality. Of particular interest are specific subspaces (branes), namely complex submanifolds called B-branes, and Lagrangians called A-branes, with respect to I, J and/or K. In this talk, I will report on natural constructions of mid-dimensional (B,B,B)-branes, e.g. hyper-Kaehler submanifolds, and their expected dual (B,A,A)-branes. This talk is based on joint work with L.Schaposnik.

## P=W Conjecture July 1st 2022

2p.m -3 p.m : Jochen Heinloth (Duisburg-Essen) (speaking on-site)

Introduction to the P=W conjecture

The organizers asked me to try to give an introduction to the P=W conjecture. This is a conjecture on the cohomology of a complex manifold that admits two different structures as algebraic variety, both coming form a moduli problem.

Hausel and Rodriguez-Villegas experimentally found a strange symmetry on the cohomology of one of these spaces, coming from a non-trivial weight filtration on the cohomology. This symmetry was reminiscent of the structures found in intersection cohomology of projective varieties, a structure that one can find in the other algebraic variety. De Cataldo-Hausel-Migliorini proved that for rank 2 bundles these two structures indeed coincide and conjectured that this should hold in general.

In the talk, I'll try to introduce this conjecture, give some background on the ingredients needed to formulate it and say a few words on progress that has been made.

3:20 - 4:20 p.m : Junliang Shen (Yale) (speaking remotely)

Symmetries of cohomology of Hitchin moduli spaces and the P=W conjecture

Nonabelian Hodge theory relates topological and algebro-geometric objects associated to a Riemann surface. Specifically, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces for the Riemann surface: the character variety and the Hitchin moduli space.

In 2010, de Cataldo, Hausel, and Migliorini proposed a conjectural relation — now called the P=W conjecture — between these two spaces. This conjecture gives a precise link between the topology of the Hitchin system and the Hodge theory of the character variety, imposing surprising symmetries for the cohomology of the Hitchin moduli space. I will start with a review of the P=W conjecture and certain symmetries of the cohomology of the Hitchin moduli space predicted by this conjecture. Then I will explain the idea of proving these symmetries directly on the Hitchin side using geometry in characteristic p. Based on joint work with Mark de Cataldo, Davesh Maulik, and Siqing Zhang.

4:40 - 5:40 p.m: Naoki Koseki (Edinburgh) (speaking remotely)

Cohomological chi-independence for Higgs bundles and Donaldson-Thomas invariants

P=W conjecture is an influential conjecture relating topology of the Dolbeaut moduli space and algebraic geometry of the Betti moduli space. One of the mysterious consequences of the P=W conjecture is the so-called chi-independence phenomenon on the Dolbeaut moduli spaces.

Using cohomological Donaldson-Thomas theory for Calabi-Yau threefolds, we were able to prove the above chi-independence and its stacky generalization, giving an evidence for the P=W conjecture. This is a joint work with Tasuki Kinjo (Tokyo).

## Non-abelian Hodge theorem for real Lie groups July 22nd 2022

2-3 p.m : Oscar Garcia-Prada (Madrid) (online)

Non-abelian Hodge correspondence for real groups and higher Teichmüller spaces

The non-abelian Hodge theorem on a compact Riemann surface X for a real semisimple Lie group G establishes a homeomorphism between the moduli space of G-Higgs bundles over X and the G-character variety of the fundamental group of X . After briefly recalling this correspondence, I will describe a general construction of a Higgs bundle parameterization of some special components of the G-character variety consisting entirely of discrete and faithful representations when G admits a positive structure. This is based on joint work with Bradlow, Collier, Gothen and Oliveira.

3:20 - 4:20 p.m : Florent Schaffhauser (Bogota) (on site)

Higher Teichmüller spaces for orbifolds

The Teichmüller space of a compact 2-orbifold X can be defined as the space of faithful and discrete representations of the fundamental group of X into PGL(2,R). It is a contractible space.

When X is a closed orientable surface, higher analogues of the Teichmüller space are, by definition, (unions of) connected components of representation varieties of \pi_1(X) that consist entirely of discrete and faithful representations. Historically, the most widely studied examples of such spaces are Hitchin components and maximal components.

In joint work with Daniele Alessandrini and Gye-Seon Lee, we show that the natural generalisation of Hitchin components to the orbifold case yield new examples of Higher Teichmüller spaces: Hitchin representations of orbifold fundamental groups are discrete and faithful, and share many other properties of Hitchin representations of surface groups. However, we also uncover new phenomena, which are specific to the orbifold case.

The goal of the talk is to give an overview of what is known about Hitchin components of orbifold groups.

4:40 - 5:40 p.m : Brian Collier (Riverside) (online)

Holomorphic curves and cyclic G2-Higgs bundles

The 6-pseudosphere is the space of norm -1 vectors in R^{4,3}, and is a pseudo-Riemann analogue of the 6-sphere. Like the 6-sphere, the 6-pseudosphere has a non-integrable almost complex structure which arises from the split octonions; the split real form of G2 is the automorphism group of this structure. In this talk, we will describe a class of J-holomorphic maps from the upper half plane to the 6-pseudosphere which are equivariant for representations of the fundamental group of a closed surface into the split real form of G2. We will describe a moduli space of such objects which fibers over the Teichmuller space of the surface with fibers given by a certain G2 Higgs bundles fixed by a Z/6 action. It turns out the components of the moduli space are labeled by an integer which lies in a fixed interval, and at one extreme is the space of G2-Hitchin representations. Other than this extremal case, the moduli spaces do not define connected components of the character variety for the split real form of G2. This is joint work with Jeremy Toulisse.

## Invariants and curve counting February 06th 2023

3pm - 4pm: Luca Battistella (Frankfurt), on site

Logarithmic and orbifold Gromov-Witten invariants

Logarithmic Gromov-Witten theory can be thought of as the study of curves in open manifolds, or, in other words, curves with tangency conditions to a boundary divisor. When the divisor is smooth, several techniques have been developed to compute the invariants, most notably orbifold stable maps. When the divisor is normal crossings, on the other hand, the logarithmic theory remains hardly accessible. The strategy of rank reduction, i.e. looking at the components of the boundary one at a time, is more directly applicable to other theories than the logarithmic one (as shown in Nabijou-Ranganathan and B.-Nabijou-Tseng-You) due to tropical obstructions. Inspired by one of the distinguishing features of the logarithmic theory - namely, birational invariance [Abramovich-Wise] - in joint work with Nabijou and Ranganathan we show that, when the genus is zero, tropical obstructions can be disposed of by blowing up the target sufficiently. The slogan is that the logarithmic theory is the limit orbifold theory under birational modifications along the boundary divisor. If time permits I will discuss work in progress towards understanding negative contact.

4:20pm-5:20pm: Georg Oberdieck (Stockholm), on site

Pandharipande-Thomas theory of elliptic threefolds and Jacobi forms

Pandharipande-Thomas theory is the study of the intersection theory of the moduli space of stable pairs of a threefold. The intersection numbers, called Pandharipande-Thomas invariants, may be viewed as counting curves on the threefold subject to given incidence conditions. In this talk we explore the properties of the generating series of Pandharipande-Thomas invariants of elliptically fibered threefolds. There will be two main conjectures: Quasi-Jacobi Property and Holomorphic Anomaly Equations. Together these essentially determine the modular properties of the generating series. The conjectures are motivated by the case of Calabi-Yau threefolds where by mirror symmetry computations Huang-Katz-Klemm conjectured that the series of PT invariants are Jacobi forms. I discuss several examples, in particular the equivariant geometry of K3xA^1. Here the conjectures lead to explicit new formulas for the invariants. Based on joint work with Maximilian Schimpf.