abstracts 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.
Presentation

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.
Presentation

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 asympt