Résumés moduli 2024

Let X be a toric hyperkaehler manifold, i.e. a complete hyperkaehler manifold of dimension 4n with T^n-symmetry. If T^n is the maximal torus of a compact Lie group and the Weyl group W of K acts on X, we can form the singular hyperkaehler manifold X/W and try to resolve the singularities of its twistor space Z/W to obtain a new complete hyperkaehler manifold. The ur-example is X=(R^3xS^1)^n with K=U(n) and W=S_n. The resolution of Z/W in this case gives the moduli space of charge n SU(2)-monopoles on R^3. In my talk I will explain how to resolve the twistor space of X/W for arbitrary X, and why, provided the volume growth of X is r^3n, the resolved hyperkaehler manifold should be described as a moduli space of solutions to Lie(K)-valued Nahm’s equations. This is joint work with Lorenzo Foscolo.

With Gonçalo Oliveira and Rosa Sena-Dias, we study the deformed Hermitian-Yang-Mills equation on the full flag manifold, both in rank one and in higher rank.

Almost all gravitational instantons that are hyperkahler (aka tesserons) can be realized as moduli spaces of monopoles. We establish this realization and use it to describe the parameter space of such spaces.

I will describe progress towards the construction of complete Calabi-Yau metrics on the complement of ample, simple normal crossings anti-canonical divisors. This talk will discuss joint works with Y. Li, and F. Tong and S.-T. Yau.

I will discuss the construction of large families of Yang-Mills instantons, i.e. solutions of the anti-self-duality equations of 4-dimensional gauge theory, on gravitational instantons with ALF (asymptotically locally flat) asymptotics. The latter are certain complete non-compact hyperkähler 4-manifolds which are prototypical examples of codimension-1 collapsing behaviour of special holonomy Ricci-flat manifolds. Our construction describes the behaviour of Yang-Mills instantons on ALF spaces in the collapsed limit where these 4-dimensional geometries are close to a 3-dimensional collapsed limit space. In this limit, the instantons are well approximated by superpositions of simple localised building blocks constructed out of monopoles on R^3. This is joint work with with Calum Ross and Jakob Stein.

I will report on a joint work with Olivier Biquard. We show that given a Fano manifold X and a smooth divisor D satisfying suitable conditions, one can construct positively curved Kähler-Einstein metrics with cone singularities along $D$ such that a suitable renormalization converges to the complete Ricci flat Tian-Yau metric on X\setminus D when the cone angle approaches its critical value.

Optimal degeneration problem aims to understand a canonical way of degenerating variety which does not admit canonical Kahler metric. The meaning of “canonical” depends on which invariant you would maximize, which are often related to some functional on Kahler metrics. There is an invariant of test configuration related Perelman entropy. I firstly explain how this is related to Perelman entropy. Then I explain a new formula which relates the invariant to Donaldson-Futaki invariant, using non-archimedean pluripotential theory.

Character varieties can be regarded in terms of flat connections on
oriented 2-manifolds, or in terms of representations of the fundamental group of a 2-manifold into a Lie group. They have a Poisson structure. The Poisson structure was originally defined by Bill Goldman (1984) or Atiyah-Bott (1983). I will outline the origin of the Poisson structure. I will also describe how to define an almost complex structure on the symplectic leaves, in some situations. symplectic leaves, in some situations. Some parts of the material presented are joint with Indranil Biswas, Jacques Hurtubise and Sean Lawton. Other parts are joint with Yukai Zhang.

Any holomorphic symplectic manifold contains entire curves as shown by Verbitsky using ergodicity, i.e., holomorphic symplectic manifolds are non-hyperbolic. More generally, together with S. Lu and Verbitsky (and later, with C. Lehn) we have established the Kobayashi conjectures in cases of Lagrangian fibrations. In this talk we shall explore generalizations of these results to primitive symplectic varieties. Together with C. Lehn we prove that if a primitive symplectic variety with second Betti number $b_2 \geq 5$ satisfies the rational SYZ conjecture, then it is hon-hyperbolic, and if $b_2 \geq 7$ then the Kobayashi pseudometric vanishes identically. In particular, this applies to all known examples of holomorphic symplectic manifolds. For Lagrangian fibrations with no multiple fibers in codimension one, we also have holomorphic dominability results with S. Lu, that imply the existence of a Zariski dense entire curve on a holomorphic symplectic manifold admitting such a Lagrangian fibration.

The known complete non-compact hyperKahler manifolds include several families of moduli spaces, including the moduli spaces of SU(2) monopoles on R^3 and the Hilbert schemes of points on C^2, among others. Beyond dimension 4, the asymptotic geometries of these spaces are not uniform, but exhibit singularities `at infinity’, presenting a challenge for geometric analysis. Indeed, the Hilbert schemes of dimension 4n for n > 1 are seminal examples of what Dominic Joyce called `quasi-asymptotically locally Euclidean’ (QALE) metrics, which generalize ALE metrics in dimension 4. I will report on a framework for geometric analysis for a broad class of `quasi-fibered boundary’ (QFB) metrics, which includes the class of QALE metrics as well as the putative `quasi-‘ generaliations (QALF, QALG, etc) of the non-exceptional classes of hyperKahler metrics in dimension 4. The point of view is to consider compactifications of these spaces as manifolds with corners, which can also be thought of as resolutions of certain stratified spaces. Through a pseudodifferential parametrix construction for the Hodge de Rham operator and an analysis relating weighted L2 cohomology with intersection cohomology, we prove a new case of Sen’s conjecture for the L2 cohomology of the charge 3 monopole moduli space, and of the Vafa-Witten conjecture for the L2 cohomology of Hilbert schemes in all cases. This is joint work with F. Rochon.

Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of noncollapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture. We will discuss how to describe the K ̈ahler potential at the C^0 level in the intermediate case for a large class of complete intersection examples.

TBA

Let X be a smooth connected projective K3 surface over the complex numbers and let Spl(r; c_1, c_2) be the moduli space of simple sheaves on X of fixed rank r and Chern classes c_1 and c_2. In 1984, Mukai proved that Spl(r; c1, c2) is a smooth algebraic space of dimension 2rc_2−(r − 1)c_1^2− 2r^2 + 2 with a natural symplectic sstricture, i.e., it has a non-degenerate closed holomorphic 2-form. In my talk, I will present a useful method to construct isotropic and Lagrangian subspaces of Spl(r; c_1, c_2). This is joint work with Barbara Fantechi.

The moduli space of surfaces homeomorphic to S endowed with a K=1 metric with conical ends is locally modelled on the moduli space of representations of the fundamental group of S in SU(2).
In this talk I will discuss certain topological property of this moduli space of representations and of its smooth locus.
This is a joint work with Dmitri Panov.

The Vafa-Witten equations are a higher-dimensional analogue of the Hitchin equations on compact Riemann surfaces for oriented four-manifolds. On a compact complex surface X, their solutions are polystable Higgs bundles (with Higgs fields \phi taking values in a holomorphic vector bundle E twisted by the canonical bundle of the surface); T-branes are solutions whose Higgs fields are non-abelian. In this talk, we consider a more general set of equations whose solutions are “wild” Vafa-Witten pairs (E,\phi: E \rightarrow E \otimes L), where L is any line bundle on X. In particular, we give necessary conditions for the existence of non-trivial solutions of the “wild” Vafa-Witten equations and describe some of their moduli.

Cristiano Spotti (Aarhus University)

Algebraic aspects of bubbling of Kähler-Einstein metrics

I will discuss the Nahm transform for multi-centered Taub-NUT spaces, with an emphasis on the invertibility of the transform and the induced isometry of moduli spaces. This is joint work with Sergey Cherkis and Andres Larrain-Hubach.

I will report on ongoing work with Sebastian Heller and Lothar Schiemanowski on solutions to Hitchin’s equation which are singular along a system of simple closed curves on the underlying Riemann surface. These are constructed by gluing methods. We use them to construct harmonic maps to the conformal 3-sphere with a specified behaviour when passing through the sphere at infinity. Earlier examples of such were obtained by Heller and Heller.

The moduli space of rank 2 Higgs bundles has a much studied very rich structure related to integrable systems, hyperkaehler reduction, mirror symmetry, and supersymmetric gauge theory. The space admits different notions of ideal points at infinity, arising from its various incarnations via the nonabelian Hodge theorem. In this talk, I will present results comparing this asymptotic behavior. One is a refinement of the Morgan-Shalen compactification of the Betti moduli space.

A second comes from the algebraic geometry of the C-star action on the moduli space. And third arises from analytic « limiting configurations » of solutions to the Hitchin equations. I will discuss how the nonabelian Hodge correspondence extends as a map between the latter two (partial) compactifications. Somewhat surprisingly, the extension is not continuous.

This lecture concerns the metric Riemannian geometry of Einstein manifolds. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics. My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. The metric geometry of Einstein manifolds is a central theme in modern differential geometry, and it is deeply connected to fundamental problems in algebraic geometry, analysis of nonlinear PDEs, and mathematical physics. We will introduce recent major progress and propose several open questions in the field.