| DCat2014-15 | Main | Venue | Program | Abstracts | Registration |
The talk will have three parts; we will discuss 1. Gromov's question concerning uniform rationality of smooth rational varieties, 2. birational isotriviality of fiber spaces all of whose fibers are birational to each other, and 3. dynamical spectra as a possible approach to irrationality of very general cubic fourfolds. Though these may at first sight appear as disparate topics, they seem to be linked in the context of the structure theory of Cremona groups, and cubic hypersurfaces figure prominently in all three of them.
The moduli space of k G instantons on C^2, where G is a classical gauge group, has a well known HyperKahler quotient formulation known as the ADHM construction. The extension to exceptional groups is an open problem.
In string theory this is realized using a system of branes, and the moduli space of instantons is identified with the Higgs branch of a particular supersymmetric gauge theory with 8 supercharges. A less known, and less studied aspect of moduli spaces of instantons is that they can be realized as the Coulomb branch of a supersymmetric gauge theory in 2+1 dimensions.
Recent developments on the understanding of the Coulomb branch gives us a nice solution to the problem where G is an exceptional group, thus allowing a systematic study of these moduli spaces. I will discuss these developments, and present the corresponding quivers, and the Coulomb branch Hilbert Series - the main tool which lead to the recent progress.
This is a joint work in progress with Álvaro Nolla de Celis and Kazushi Ueda. We consider group actions on dimer models. Our aim is to construct a consistent dimer model which is acted on by the group of symmetries of the characteristic polygon. This produces examples of non-commutative crepant resolutions of non-toric, non-quotient singularities.
There are several generalizations of two-dimensional McKay correspondence. I would like to talk about two projects in dimension three: non-abelian Reid's recipe and a generalization of special McKay correspondence.
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A vital component of the McKay correspondence is the moduli space construction of crepant resolutions as G-Hilbert schemes. Indeed, G-Hilbert schemes provide minimal resolutions for all quotient surface singularities, and they can always be calculated by quiver GIT from the McKay quiver.
I will discuss how quiver GIT can be used to construct minimal resolutions for all rational surface singularities by considering quivers more general than the McKay quiver. This generalises the G-Hilbert scheme moduli space construction which can only be defined for quotient singularities.
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The deformation of the Hilbert scheme of points on the projective plane is first constructed in the paper of Nevins and Stafford by an approach of non-commutative geometry. I will introduce the construction through an example of Hilb^2 P^2, and talk about the result on the minimal model program of the deformation of Hilb^n P^2 via wall-crossing on the Bridgeland stability space.
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Consider quiver varieties of type ADE and their Betti numbers. There are various approaches to compute them. From recent results on representation theory of current algebras (Chari-Ion and others), it follows that they are given by the specialization of Macdonald polynomials at $t=0$. I will explain a geometric approach to this result, which possibly be generalized to quiver varieties of other types.
Hilbert schemes of points on K3 surfaces are an important class of irreducible symplectic manifolds. They are highly symmetric, and this is reflected in their group of autoequivalences. In this talk, we present a new type of derived symmetries. It is a joint work with Pawel Sosna.
We prove a formula in the Grothendieck ring of varieties relating the Fano variety of lines on the cubic Y to the symmetric square of Y. Using this formula we compute various invariants of the Fano variety of lines and deduce a conditional criterion for irrationality of the cubic. Joint work with Sergey Galkin.
Bridgeland, King, and Reid realized a version of the McKay correspondence as a derived equivalence between two categories naturally associated to a finite subgroup G of SL(3): G-equivariant coherent sheaves on C^3 and coherent sheaves on a certain crepant resolution of C^3/G. One natural problem is to compute the images of special sheaves on either side under this equivalence. In my talk, I will discuss the related problem of how to identify the category of coherent sheaves on the crepant resolution as a full subcategory of the derived category of G-equivariant sheaves on C^3.
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In this talk I will explain how to use certain semi-orthogonal decompositions to construct autoequivalences of Hilbert schemes of points on Enriques surfaces and of Calabi-Yau varieties which cover them. This is joint work with Andreas Krug.
Auslander's version of the classical McKay correspondence is that there is a 1-1 correspondence between indecomposable non-free CM modules on the base singularity and exceptional curves its minimal resolution. The benefit of this approach is that representations are absent, so it lends itself to generalizations away from the quotient singularity case. In the talk, I will describe one such three-dimensional generalization, namely to (possibly singular) minimal models of cDV singularities. There are now two such correspondences. First there is a 1-1 correspondence between certain CM modules and minimal models, and then for each such pair there is a further correspondence, along the lines of the classical Auslander--McKay correspondence. This approach sounds algebraic, but the proofs are entirely geometric, and rely on realizing the Bridgeland--Chen flop functor as an appropriate mutation of a quiver with relations, together with an understanding of noncommutative deformations of curves. If there is time, I will discuss some other applications.