Mini-Courses
Yoshinori Namikawa (RIMS)
Lecture 1: Symplectic singularities and Kaledin's conjecture
Symplectic singularities play an important role in algebraic geometry and geometric representation theory. All known examples of such singularities show up with natural C^*-action. About 20 years ago, Kaledin conjectured that a symplectic singularity is always conical; more precisely, it admits a conical C^*-action where the symplectic form is homogeneous. Recently we proved Kaledin's conjecture conditionally, but in a substantially stronger form. The idea is to use Donaldson-Sun theory in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. This is a joint work with Y. Odaka.
Lecture 2: Towards a characterization of toric hyperkaehler varieties among symplectic singularities
Let (X, ω) be a conical symplectic variety of dimension 2n which has a projective symplectic resolution. Assume that X admits an effective Hamiltonian action of an n-dimensional algebraic torus T^n, compatible with the conical C^*-action. In this talk we prove that (X, ω) is isomorphic to a toric hyperkahler variety studied by Goto, Bielawski-Dancer, Hausel-Sturmfels, Konno, Proudfoot and others. This result can be regarded as a holomorphic symplectic analogue of Delzant's theorem on toric varieties.
Lewis Topley (University of Bath)
Representations of quantizations of symplectic singularities
Let X be a conic symplectic singularity. A quantization of X is a non-commutative algebra which deforms X in the sense that the first order approximation of the multiplication is given by the Poisson bracket. I will begin the lectures by explaining some basic connections between the representation theory of noncommutative algebras and the Poisson geometry of their degenerations. I will go on to discuss the quantizations of a conic symplectic singularity and illustrate the relationship between representation theory of Poisson geometry using examples from Lie theory: the nilpotent cones of simple Lie algebras, as well as their Hamiltonian reductions, the nilpotent Slodowy varieties. Time permitting, I will explain how the nice quantization theory of nilpotent Slodowy varieties can be used to prove concrete algebraic results regarding, for example, the relationship between finite W-algebras and Yangians.
Talks
Riccardo Carini (University of Bonn)
Symplectic quotients of Beauville-Mukai systems
I will explain how the Kaledin-Lehn-Sorger description of O’Grady’s singularity, phrased in terms of symplectic reductions modelled on nilpotent orbit closures, can be used to study quotients of singular Beauville-Mukai systems on symplectic surfaces. After taking the quotient by a suitable symplectic involution, some of these systems admit, at least locally, a symplectic resolution.
In the non-compact setting of a local curve, equivalently for Higgs bundles, this yields a smooth 18-dimensional quasi-projective symplectic manifold. In the compact K3 case, however, the involution acquires isolated fixed points, and the resulting quotient becomes a primitive symplectic variety with terminal singularities.
Wahei Hara (Kavli IPMU)
General K3 surfaces of degree 12 via flops and McKay correspondence
The aim of this talk is to revisit the derived categorical aspects of general K3 surfaces from the viewpoint of higher-dimensional birational geometry and the generalized McKay correspondence.
A celebrated result of Mukai shows that a general K3 surface of degree 12 can be realized as a linear section of a connected component of the orthogonal Grassmannian OG(5,10).
By Kuznetsov's homological projective duality, there exists the dual K3 surface that is derived equivalent to the original one.
Moreover, a general K3 surface of degree 12 and its dual are known not to be isomorphic in general.
According to results of Ito-Miura-Okawa-Ueda and Kapustka-Rampazzo, this duality can also be interpreted through the geometry of roofs in the sense of Kanemitsu, which are related to five- and six-dimensional quadric hypersurfaces and vector bundles over them.
In this talk, I will first review the geometry of such K3 surfaces, and then describe certain higher-dimensional flops associated with the roofs.
By applying the idea of the generalized McKay correspondence, we will show that these flops are derived equivalent, which in turn provides a new proof of the derived equivalence between the dual K3 surfaces.
Zelin Jia (Nagoya University)
Symplectic resolution of the moduli space of G-Higgs bundles over an elliptic curve
Until recently, the moduli space of semistable G-Higgs bundles (where G denotes a complex reductive Lie group) over an elliptic curve C had been studied. In the work of Franco, Prada and Newstead, these moduli spaces were shown to be related to the space spanned by the coroot lattice, quotiented by the associated Weyl group.
On the other hand, Groechenig established an isomorphism between the moduli space of parabolic GL-Higgs bundles over an elliptic curve C and the Hilbert scheme of points on the cotangent bundle of C. This result provides a symplectic resolution on both the Higgs side and the Hilbert side (via the Hilbert–Chow morphism).
In this talk, we will first discuss how to show that the symplectic structures on both sides coincide. Second, we will explain how to apply Groechenig's techniques to construct a symplectic resolution for the moduli space of semistable G-Higgs bundles. To obtain a tractable description, we will focus on the cases where G = SL or Sp.
Tasuki Kinjo (RIMS)
BPS cohomology for symplectic quotients
In this talk, we will introduce the BPS cohomology of symplectic varieties arising as Hamiltonian quotients. When the singularities can be resolved via variation of GIT, the BPS cohomology coincides with the cohomology of the corresponding symplectic resolution. I will explain that the BPS cohomology can be used in the formulation of the topological mirror symmetry conjecture for G-Hitchin system, extending the conjecture of Hausel and Thaddeus for type A groups. I will also discuss some speculations about the BPS cohomology, including counting problems of absolutely indecomposable G-bundles on smooth projective curves defined over finite fields. This talk is partially based on joint work with Chenjing Bu, Ben Davison, Andrés Ibáñez Núñez, and Tudor Pădurariu.
Ayako Kubota (Saitama University)
Examples and non-examples of invariant Hilbert scheme resolutions
For the Cox realization of an affine variety, one can associate an invariant Hilbert scheme that is birational to the original variety. We investigate whether this invariant Hilbert scheme provides a resolution of singularities, presenting both positive and negative examples arising from toric geometry. This talk is based on joint work with Yasunari Nagai.
Dmytro Matvieievskyi (UMass Amherst)
Counting leaves on symplectic dual varieties: nilpotent cones
Roughly speaking, symplectic duality is a statement that conical symplectic singularities often come in pairs of varieties that share or interchange certain geometric data and are expected to satisfy many interesting properties with regard to each other. One of such expectations is a relation between stratifications by symplectic leaves, sometimes stated as an expected bijection on the set of symplectic leaves. Looking at the main example of my talk (nilpotent cones for simple Lie group G and its Langlands dual G^\vee), we can easily see that the number of symplectic leaves is different. I will try to explain how to remedy this issue by considering certain covers of symplectic leaves and providing a bijection between them, which is a geometric interpretation of Achar duality and related to dualities of Sommers and Barbasch-Vogan-Lusztig-Spaltenstein. Time permitting, I will talk a bit on what to expect for more general symplectic dual pairs.
Raphaël Paegelow (University of Lille)
Gieseker spaces and Ariki-Koike algebras
We will present combinatorial correspondences between the irreducible components of the fixed points locus of the Gieseker space and the block theory of the Ariki-Koike algebra. First, we will describe the locus of fixed points in terms of Nakajima quiver varieties over the McKay quiver of type A. Then, we will present how to recover the combinatorics of cores of charged multipartitions, as defined by Fayers and developed by Jacon and Lecouvey, on the Gieseker side. In addition, we will present a new way to compute the multicharge associated with the core of a charged multipartition. Finally, if time permits, we will also explain how the notion of core blocks, discovered by Fayers, can be interpreted geometrically using a deep connection between quiver varieties and affine Lie algebras.
Benjamin Tighe (University of Oregon)
The LLV algebra for primitive symplectic varieties
The total Lie algebra associated to a smooth and projective complex variety $X$ is the amalgamation of the $sl_2$-algebras associated to the cohomology classes satisfying the hard Lefschetz theorem. When $X$ is a compact hyperkahler manifold, this endows the cohomology ring with the structure as an $so(V)$-representation, where V is the Mukai completion of the second cohomology group. This structure encapsulates the geometry of the underlying hyperkahler manifold and has been used in last decade to describe bounds on betti numbers and study compact analogues of the P=W conjecture for Lagrangian fibrations.
In this talk, I will describe an algebraic approach to this construction which extends to singular symplectic varieties by considering natural symmetries which arise in the intersection Hodge module and coming from the underlying holomorphic symplectic form. Time permitting, I will discuss how these methods give local vanishing results involving the Du Bois complex.