Tag - Hyperkähler manifolds

Johanna Bimmermann: From Magnetically Twisted to Hyperkähler

26th January, 2024

The tangent bundle of a Kähler manifold admits in a neighborhood of the zero section a hyperkähler structure. From a symplectic point of view, this means we have three symplectic structures all compatible with a single metric. Two of the three symplectic structures are easy to describe in terms of the canonic symplectic structure. The third one is harder to describe, but in the case of hermitian symmetric spaces, there is an explicit formula found by Biquard and Gauduchon. In this talk, I will construct a surprising diffeomorphism of the tangent bundle of a hermitian symmetric space that identifies this third symplectic structure with the magnetically twisted symplectic structure, where the twist is given by the Kähler form on the base.

Arend Bayer: Non-commutative abelian surfaces and generalized Kummer varieties

22nd November, 2023

Polarised abelian surfaces vary in 3-dimensional families. In contrast, the derived category of an abelian surface A has a 6-dimensional space of deformations; moreover, based on general principles, one should expect to get 'algebraic families' of their categories over 4-dimensional bases. Generalized Kummer varieties (GKV) are hyperkähler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have 4-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over 3-dimensional subvarieties.

I present a construction that addresses both issues. We construct 4-dimensional families of categories that are deformations of Db(A) over an algebraic space. Moreover, each category admits a Bridgeland stability condition, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from ℤ/2-actions on derived categories of K3 surfaces.

Stevell Muller: On symplectic transformations of OG10-type hyperkähler manifolds via cubic fourfolds

15th September, 2023

We know thanks to the work of L. Giovenzana, A. Grossi, C. Onorati and D. Veniani that OG10-type hyperkähler manifolds do not admit any non-trivial symplectic automorphisms. What about non-regular symplectic birational transformations? Given a cubic fourfold V, one can construct a hyperkähler manifold XV of OG10-type following a construction of R. Laza, G. Saccà, C. Voisin. Such manifolds are known as LSV manifolds. It can be shown that any symplectic automorphism on V induces a symplectic birational transformation on XV. In a couple of works with L. Marquand, we study and classify all possible cohomological actions on the OG10-lattice which can be realised as symplectic birational transformations. By investigating further the induced action on cohomology, we exhibit a criterion to decide which of these actions can be realised as induced from a cubic fourfold on an associate LSV manifold.

Sasha Viktorova: The defect of a cubic threefold

13th September, 2023

In this talk, we relate the defect σ(X) := b4(X) − b2(X) of a singular cubic threefold X to various geometric properties of X. The question is motivated by the construction of the exceptional example of a Hyperkähler manifold of type O'Grady 10 from a cubic fourfold by Laza, Saccà and Voisin. By a result of Brosnan, the defect of hyperplane sections of the cubic fourfold is an obstruction for the LSV construction to work. The talk is based on a joint work in progress with Lisa Marquand.

Giulia Saccà: Moduli spaces on K3 categories are Irreducible Symplectic Varieties

7th May, 2022

Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. I will rely on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

Claire Voisin: A topological characterization of hyper-Kähler fourfolds of Hilb2(K3) type

7th May, 2022

There are two known deformations types of hyper-Kähler (HK) fourfolds, namely Hilb2(K3) (Beauville, Fujiki) and the generalized Kummer variety K2(A) (Beauville). It is however still unknown whether there are other topological types or deformation types of HK fourfolds. Some strong restrictions on the Betti numbers of HK fourfolds are known by work of Beauville, S. Salamon, Verbitsky and Guan. In this talk, I will sketch the proof of the following:

Theorem. A hyper-Kähler fourfold X is a deformation of Hilb2(K3) if and only if it has two integral degree 2 cohomology classes satisfying the conditions l4=0, m4=0, l2m2=2. In particular, a HK fourfold which is homeomorphic to Hilb2(K3) is a deformation of Hilb2(K3).

Aleksander Doan: Holomorphic Floer theory and the Fueter equation

25th April, 2022

I will discuss an idea of constructing a category associated with a pair of holomorphic Lagrangians in a hyperkähler manifold, or, more generally, a manifold equipped with a triple of almost complex structures I, J, K satisfying the quaternionic relation IJ = -JI = K. This category can be seen as an infinite-dimensional version of the Fukaya-Seidel category associated with a Lefschetz fibration. While many analytic aspects of this proposal remain unexplored, I will argue that in the case of the cotangent bundle of a Lefschetz fibration, our construction recovers the Fukaya-Seidel category.

Yann Rollin: Lagrangians, symplectomorphisms and zeroes of moment maps

8th April, 2022

I will present two constructions of Kähler manifolds, endowed with Hamiltonian torus actions of infinite dimension. In the first example, zeroes of the moment map are related to isotropic maps from a surfaces in ℝ2n. In the second example, which is actually a hyperkähler moment map, the zeroes are related to symplectic maps of the torus T4. The corresponding modified moment map flows have short time existence. Polyhedral analogues of these constructions can be used to investigate piecewise linear symplectic geometry.