Modern Methods in Moduli

For an algebraic surface $S$ with canonical (resp. symplectic) singularities and for any positive integer $n$, I'll explain why the Hilbert scheme of $n$ points on $S$ also has canonical (resp. symplectic) singularities. In particular, these Hilbert schemes are normal varieties. The main results generalise well-known work of Fogarty (1968) and Beauville (1983) for non-singular surfaces. Our main tool is a generalisation of the Le Bruyn-Procesi theorem that describes the invariant algebra for the natural action of the product of general linear groups on the space of representations of a quiver for a given dimension vector. This is joint work with Ryo Yamagishi.

Quiver moduli spaces provide partial solutions to the problem of classifying isomorphism classes of quiver representations (or more generally, finite-dimensional modules over associative algebras). It appears to be an interesting question to try to classify these moduli spaces themselves. In the talk we present some results based mainly on classical invariant theory and combinatorial considerations that may serve as steps towards such a classification.

Intermediate Jacobians for smooth projective varieties play a very similar role as Jacobians play for smooth projective curves. While the relative Jacobian for families of curves is a well-studied concept, the relative intermediate Jacobian is a rather uncharted terrotory. In this talk, I will construct a sheaf of intermediate Jacobians for a family of cubic threefolds (not necessarily smooth). This sheaf will then be used to construct twists of certain intermediate Jacobian fibrations, a story parallel to twists of elliptic fibrations of K3 surfaces. This is a joint work with Mattei and Shinder.

It is a result by Belmans, Okawa, and Ricolfi that semiorthogonal decompositions of a derived category do have a very special algebraic structure in family: they form an algebraic space, which is étale over the base scheme. The aim of this talk is to present a conjecture on the behavior of bounded t-structures with noetherian heart, which should be analogous to the case of semiorthogonal decompositions. We will show how to prove part of this conjecture: in particular, a formal deformation statement for bounded t-structures with noetherian heart, which in itself has already several applications. This is joint work with Alexander Perry and Paolo Stellari, with the applications being part of a joint project together also with Chunyi Li and Xiaolei Zhao.

Nekrasov's 4D ADHM quiver is a generalisation of the 2D and 3D ADHM quivers. I will give a mathematical definition of the corresponding "gauge origami" partition function in terms of certain torus localized Oh-Thomas virtual cycles. I will relate the orientation problem of gauge origami to the one of Hilbert schemes of points on affine 4-space, which was previously analysed in physics by Nekrasov-Piazzalunga ("Magnificent Four") and in algebraic geometry in a joint work with J. Rennemo. I will provide a conjectural sheaf-theoretic interpretation of gauge origami in terms of certain moduli of 2-dimensional framed sheaves on $(\mathbb{P}^1)^4$. Joint work in progress with N. Arbesfeld and W. Lim.

A scheme $X$ is a splinter if for any finite surjective morphism $f\colon Y \to X$ the pullback map $\mathcal{O}_X \to f_* \mathcal{O}_Y$ splits as $\mathcal{O}_X$-modules. By the direct summand conjecture, now a theorem due to André, every regular Noetherian ring is a splinter. Whilst for affine schemes the splinter property can be viewed as a local measure of singularity, the splinter property imposes strong constraints on the global geometry of proper schemes over a field of positive characteristic. For instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology. I will report on joint work with Charles Vial concerning further obstructions on the global geometry of proper splinters in positive characteristic. Further, I will address the derived-invariance of the (derived-)splinter property.

I will present joint work with Fabian Reede and Ziyu Zhang. For an abelian surface $A$, we consider stable vector bundles on a generalized Kummer variety $\mathrm{K}_n(A)$ with $n>1$. We prove that the connected component of the moduli space which contains the tautological bundles associated to line bundles of degree 0 is isomorphic to the blowup of the dual abelian surface in one point. We believe that this is the first explicit example of a component of a moduli space of sheaves on a hyperkähler variety which is smooth with a non-trivial canonical bundle.

I will report on joint work with Yukinobu Toda (partially in progress) about the (categorical) Hall algebra of Higgs bundles on a curve. These categories have semiorthogonal decompositions in certain categories, called quasi-BPS, whose construction is inspired by the enumerative geometry of Calabi-Yau threefolds.

I will focus on two conjectural dualitities. The first is between the Hall algebra of semistable Higgs bundles of degree zero and a "limit" category. This equivalence aims to make precise the proposal of Donagi-Pantev of considering the classical limit of the de Rham Langlands equivalence. The second is a primitive (or BPS) version of the first, and it relates categories of sheaves on moduli of semistable Higgs bundles (for various degrees). This equivalence may be regarded as a version of the D-equivalence conjecture / SYZ mirror symmetry. We can prove (partial) versions of these conjectures for topological K-theory of these categories.

Kronecker moduli are algebraic varieties parametrizing linear algebra data up to base change. We consider generating series of their motives, and discuss results characterizing these series by functional and differential equations.

The geometry of Hilbert schemes of points is largely unknown, or known to be pathological in a precise sense. This should in principle make most Hilbert scheme invariants essentially inaccessible. We present a closed formula for the generating function of the motives (classes in the Grothendieck ring of varieties) of Hilbert schemes of points $\mathrm{Hilb}(\mathbb{C}^n,d)$, varying the ambient dimension $n$ and having fixed the number of points $d$. Joint work with M. Graffeo, S. Monavari and R. Moschetti.

The motivation for the presented results comes from the study of moduli spaces of Higgs bundles. Recent work of Hausel shows that the Hitchin fibration restricted to the minuscule upward flow is modelled on the spectrum of equivariant cohomology of a Grassmannian. This appears as a zero scheme of a vector field on the Grassmannian itself. In a joint work with Tamas Hausel, we show that it is a special case of a general phenomenon, where the spectrum of the equivariant cohomology ring shows up as a zero scheme associated with the group action. I will present those results, and not less interesting extensions such as those for spherical varieties. In that case the cohomology ring can be read off as a ring of functions on a non-affine zero scheme. We conjecture that equivariant K-theory can be computed from the fixed-point schemes in a similar way.

We define half spherical objects that induce (categorical) half twists. We give examples, in the context of the topological Fukaya category of a surface, where we show that the categorical half twist coincides with the geometric half twist, and in the context of divisors in smooth quasi-projective varieties. This is joint work with Alexander Polishchuk and Michel Van den Bergh.

Tannaka duality consists of a web of results that allow one to recover a geometric object from its category of sheaves. Although initially concerned with algebraic groups and their classifying spaces, in the last two decades Tannaka duality has found various applications to the geometry of algebraic stacks. In this talk I will give an introduction to this circle of ideas, and describe recent work which strengthens previous duality results.

The local Langlands correspondence (LLC) for covering groups is a conjectural parametrization of genuine smooth representations of (non-algebraic) covers of $p$-adic reductive groups by L-parameters. In this talk, I will explain how covering groups arise from transgressions of degree-4 characteristic classes, and why this perspective leads to a construction of the LLC whenever the covering degree is coprime to p. This is joint work with T. Feng, I. Gaisin, N. Imai, and T. Koshikawa.