ICTS

Bridgeland stability conditions give a general notion of stability on the derived category of coherent sheaves. In complex geometry, a general philosophy states that stability should correspond to being able to solve a geometric PDE, determining a choice of metric or connection (with the Hitchin-Kobayashi correspondence being the classical example). It is thus natural to seek such a correspondence in this setting.

I will discuss the theory of deformed Hermitian Yang-Mills connections, and more generally Z-critical connections, which roughly play the role of analytic counterparts to stability in the sense of Bridgeland, but in the simplified setting of holomorphic vector bundles. The theory of deformed Hermitian Yang-Mills connections predates Bridgeland stability conditions, and was introduced with largely the same motivation, whereas the theory of Z-critical connections is more recent. The emphasis of the lectures will be on links with algebro-geometric stability, and applications to moduli.

This theory was largely developed in joint work with Lars Sektnan and John McCarthy.

When the study of Higgs bundles on a Riemann surface began 40 years ago, what is now known as the Hitchin integrable system was a sideshow – an observation. The talk will follow the development of this over the years in a variety of contexts and in particular describe the more central role it now plays in the geometry of the moduli space.

Bridgeland stability conditions give a general notion of stability on the derived category of coherent sheaves. In complex geometry, a general philosophy states that stability should correspond to being able to solve a geometric PDE, determining a choice of metric or connection (with the Hitchin-Kobayashi correspondence being the classical example). It is thus natural to seek such a correspondence in this setting.

I will discuss the theory of deformed Hermitian Yang-Mills connections, and more generally Z-critical connections, which roughly play the role of analytic counterparts to stability in the sense of Bridgeland, but in the simplified setting of holomorphic vector bundles. The theory of deformed Hermitian Yang-Mills connections predates Bridgeland stability conditions, and was introduced with largely the same motivation, whereas the theory of Z-critical connections is more recent. The emphasis of the lectures will be on links with algebro-geometric stability, and applications to moduli.

This theory was largely developed in joint work with Lars Sektnan and John McCarthy.

A multiplicative Higgs bundle is a pair formed by a principal bundle on a compact Riemann surface and a meromorphic section of the bundle of groups associated with the adjoint action of the structure group on itself. The moduli stack of these objects is equipped with an analogue of the Hitchin fibration, with similar properties. Multiplicative Higgs bundles also appear naturally in the theory of singular monopoles on three-manifolds.

In this talk, we will introduce multiplicative Higgs bundles from different perspectives, and explain several ways in which one can approach the construction of a moduli space classifying them. In particular, we will define stability conditions for them and formulate Hitchin-Kobayashi correspondences associated with these. This is part of an ongoing project in collaboration with Jacques Hurtubise and Oscar García-Prada.

The moduli space of Higgs bundles has a natural hyperkähler metric which depends on the complex structure of the Riemann surface. On the other hand it may also be identified with a character variety of the fundamental group which just depends on the topology. These two aspects generate a differential-geometric structure on the universal bundle over Teichmüller space which is the subject of this talk.

In joint work with B. Collier and O. Garcia-Prada we needed a notion of Harder-Narasminahn stratifications for variants of moduli spaces of Higgs bundles that admit a variation of stability conditions. In this talk I'd like to explain how these can be obtained easily from classical arguments of Kempf and Behrend once these are rephrased in terms of Theta-stability.

Very stable and wobbly Higgs bundles were introduced by Hausel and Hitchin, motivated by the study of mirror symmetry phenomena in moduli spaces of Higgs bundles over smooth projective complex curves. First, we will recall these notions and explain how very stable points can be classified whenever the Higgs field is generically regular by performing suitable Hecke transformations of certain sections of the Hitchin map. Then, motivated by similar mirror symmetry aspects that appear in moduli spaces of strongly parabolic G-Higgs bundles, we will explain how the aforementioned techniques can be extended to those moduli spaces, resulting in a classification of the wobbly points in terms of the combinatorics of the affine flag variety for G and the Bruhat order on its extended affine Weyl group.

The definition of the integrable system involves the invariant symmetric polynomials on the Lie algebra of a simple group, but there is an analogous construction using invariant alternating forms. This leads to information about the Hochschild cohomology of the moduli space of stable bundles. The talk will review this construction and consider it in the special case of the intersection of two quadrics in any dimension, where explicit formulas have recently been revealed.

It is well-known that the geometry of hyper-elliptic curves and that of quadrics are closely related. In particular, the Jacobian and the moduli of vector bundles of rank 2 have nice interpretation in terms of quadric geometry. Recently Sarbeshwar Pal and I have been trying to get such an interpretation for related spaces including Higgs moduli. I will discuss this ongoing work, with special focus on genus 2 curves.

The gravitating vortex (GV) equations are a dimensional reduction of the Kahler-Yang-Mills-Higgs equations to a Riemann surface. The Abelian GV equations contain as a special case, the Einstein-Bogomol'yni equations modelling (the as of now, hypothetical) cosmic strings in the Bogomolo'yni phase in the early universe. The non-abelian GV equations are expected to play a similar role for Electroweak cosmic strings. Moreover, these equations might have interesting moduli spaces and arise out of a moment map interpretation. I shall describe our existence, uniqueness, and obstruction (including a Donaldson-Uhlenbeck-Yau-Kobayashi-Hitchin correspondence) results (joint with L. Alvarez-Consul, M. Garcia-Fernandez, O. Garcia-Prada, and C. Yao) for Abelian GV equations, and an existence result for the non-abelian case.

Classical Geometric Invariant Theory, developed by David Mumford, provides powerful tools for constructing quotients by reductive group actions. Many moduli problems in geometry, however, involve non-reductive groups, notably the reparametrisation groups arising in jet spaces and singularity theory.
In joint work with Frances Kirwan and collaborators, we developed a general framework for non-reductive GIT for graded groups. After modifying the linearisation by a central character, we prove finite generation of invariants and construct projective quotients with a well-behaved stability theory extending the Hilbert–Mumford criterion.

Central problems in complex and algebraic geometry are the existence of special Kähler metrics and the construction of moduli spaces parameterizing complex projective varieties.
While both problems are open in full generality, the Yau–Tian–Donaldson conjecture predicts that one needs to impose some sort of algebro-geometric or analytic stability conditions on the varieties.
In this talk, we discuss this problem from the point of view of families of polarized varieties over the punctured disk, where the minimization of the degree of the Chow–Mumford (CM) line bundle can be seen as a sufficient numerical criterion for the separatednes of moduli spaces of manifolds admitting special Kähler metrics.
Ultimately, we present a result on equivariant CM minimization for extremal manifolds, generalizing a result by Hattori.

Classical Geometric Invariant Theory, developed by David Mumford, provides powerful tools for constructing quotients by reductive group actions. Many moduli problems in geometry, however, involve non-reductive groups, notably the reparametrisation groups arising in jet spaces and singularity theory.
In joint work with Frances Kirwan and collaborators, we developed a general framework for non-reductive GIT for graded groups. After modifying the linearisation by a central character, we prove finite generation of invariants and construct projective quotients with a well-behaved stability theory extending the Hilbert–Mumford criterion.

In the first part of this talk, we examine cluster-like structures that emerge from the moduli space of rank-two vector bundles on a smooth projective curve with fixed determinant. These structures are constructed by analyzing toric degenerations of the moduli spaces, specifically through the degeneration of the underlying smooth curve to a maximal nodal curve. By performing a change of variables and considering appropriate limits, these cluster-like structures reproduce the Plücker relations for the Grassmannian Gr(2, n). In the second part, we will discuss a combinatorial analog of the Torelli theorem for these limiting toric varieties.

This is a joint work with Pieter Belmans and Sergey Galkin.

Let X be a smooth projective curve genus at least 3, over an algebraically closed field k of arbitrary characteristics. Let M denote the moduli stack MX (H ) of H -torsors on X, when H quasi-split absolutely simple, simply connected connected group scheme. Using the theory of parahoric torsors and Hecke correspondences, we describe the cohomology groups Hi (M,TM ),i = 0, 1, 2 and Hi (M,ΩM ),i = 0, 1, 2 in terms of the curve X. The classical results of Narasimhan and Ramanan are derived as a consequence. The talk will outline in some detail the Hecke correspondences which is a basic tool in these computations. This is joint work with Yashonidhi Pandey.

Classical Geometric Invariant Theory, developed by David Mumford, provides powerful tools for constructing quotients by reductive group actions. Many moduli problems in geometry, however, involve non-reductive groups, notably the reparametrisation groups arising in jet spaces and singularity theory.
In joint work with Frances Kirwan and collaborators, we developed a general framework for non-reductive GIT for graded groups. After modifying the linearisation by a central character, we prove finite generation of invariants and construct projective quotients with a well-behaved stability theory extending the Hilbert–Mumford criterion.

Since the seminal work of Atiyah-Bott, Kähler reduction has proven to be an extremely fruitful framework for the construction and study of moduli spaces in complex geometry. In this talk, I will explain a symplectic reduction approach to the moduli space of Calabi-Yau metrics on a compact 2n-manifold M, that is, Riemannian metrics with holonomy SU(n). By Yau's solution of the Calabi Conjecture, this moduli space can be regarded as naturally fibering over the moduli of complex structures on M, and is endowed with a holomorphic structure after "complexifying" the fibres via B-fields. The construction of the moduli space metric and complex structure will naturally lead us to unexplored territory, involving pseudo-Kähler reduction by stages and Lie 2-algebra symmetries.

arXiv:2402.11310 (joint work with Sorin Dumitrescu).