9:30-10:30 Kirwan I
11:00-12:00 Szenes I
9:30-10:30 Kirwan II
11:00-12:00 Halpern-Leistner I
9:30-10:30 Halpern-Leistner II
11:00-12:00 Szenes II
Titles and Abstracts
Frances Kirwan (Oxford)
Talk 1: Applications of non-reductive geometric invariant theory
In general GIT for non-reductive linear algebraic group actions is much less well behaved than for reductive actions. However when the unipotent radical U of a linear algebraic group is graded, in the sense that a Levi subgroup has a central one-parameter subgroup which acts by conjugation on U with all weights strictly positive, then GIT for a linear action of the group on a projective scheme is almost as well behaved as in the reductive setting, provided that we are willing to multiply the linearisation by an appropriate rational character. This has potential applications for the construction of moduli spaces of 'unstable' objects of fixed type, such as sheaves of fixed Harder-Narasimhan type.
Talk 2: Moment maps for actions of non-reductive complex groups with graded unipotent radicals
When a complex reductive group acts linearly on a projective variety the GIT quotient can be identified with an appropriate symplectic quotient. The aim of this talk is to discuss an analogue of this description for GIT quotients by suitable non-reductive actions.
Daniel Halpern-Leistner (Columbia)
Beyond geometric invariant theory
Geometric invariant theory is an essential tool for constructing moduli spaces in algebraic geometry. Its advantage, that the construction is very concrete and direct, is also in some sense a drawback, because for a given moduli problem it is often intractable to explicitly describe GIT semistable objects in an intrinsic and simple way. I will discuss the theory of Theta-stratifications, a general framework for studying moduli problems directly, without the need to approximate a moduli problem as an orbit space for a reductive group on a quasi-projective scheme.
Over the last few years, we have been developing a program to understand many of the classical structures and results of geometric invariant theory in this broader context. I will discuss the notions of stability, of Harder-Narasimhan filtrations, $\Theta$-stratifications, and good moduli spaces for a general algebraic stack. I will also report on some new progress in this program: joint with Jarod Alper and Jochen Heinloth, we give a simple necessary and sufficient criterion for an algebraic stack to have a good moduli space. This leads to the construction of good moduli spaces in many new examples, such as the moduli of Bridgeland semistable objects in derived categories. Time permitting, I will discuss how this theorem can be used to understand the birational geometry of these moduli spaces.
András Szenes (Geneva)
Integrating over geometric subsets of Hilbert schemes
We present a rather general scheme for calculating the tautological Chern numbers of geometric subsets of Hilbert schemes. Our formula works in dimension 2, but modulo certain conjectures, it potentially holds in any dimension. (Joint work with Gergely Berczi.)
Jørgen Rennemo (Oxford)
Tautological integrals over geometric subsets of the Hilbert scheme
The Göttsche conjecture, proved by Tzeng and Kool-Shende-Thomas, states (among other things) that given a "generic" d-dimensional linear system |L| of curves on a surface S, the number of d-nodal curves in |L| is computed by a polynomial in the Chern number of the pair (S,L). Tzeng's proof of the conjecture reduces the problem to one on the Hilbert scheme of points on S, by first showing that the number of d-nodal curves equals the integral of a Chern class of a tautological bundle L^[3d] over a certain natural subset W in S^[3d]. The second, harder part of the proof is to show that this integral is given by a polynomial in Chern numbers of (S,L).
In generalising the Göttsche conjecture to counting curves with other types of singularities, one is led to consider Chern class integrals over other natural subsets of the Hilbert scheme, which we call "geometric". The second part of Tzeng's proof generalises to a statement that the integral of Chern classes of a tautological bundle over a geometric subset of the Hilbert scheme is given by a polynomial in the Chern numbers of (S,L). The aim for the talk is to explain what this means and why it's true.
Victoria Hoskins (FU Berlin)
Quotients of unstable GIT strata and moduli of sheaves of fixed Harder-Narasimhan type.
Many moduli spaces are constructed as a GIT quotient of a reductive group G acting on a projective variety X and the moduli problem inherits a notion of semistability from GIT. In this largely introductory talk, I will describe how on the GIT side, there is a stratification of the unstable locus in X by G-invariant subvarieties due to work of Hesselink, Kempf, Kirwan and Ness, and I will explain how constructing quotients of the unstable strata in X naturally leads to non-reductive GIT (we will see various instances of this in the talks of Kirwan and Jackson). In particular, I will focus on the example of moduli spaces of sheaves over a projective variety, whose moduli space of semistable sheaves is constructed as a GIT quotient of a Quot scheme. I will compare the GIT stratification of this Quot scheme with the stratification by Harder-Narasimhan (HN) types, and explain joint work in progress with G. Berczi, J. Jackson and F. Kirwan on constructing moduli spaces of sheaves of fixed HN type.
Joshua Jackson (Oxford)
Moduli Spaces of Unstable Curves
Instability in GIT is a structured phenomenon. More precisely, this means that along with a quotient of a suitable open semistable subset, GIT gives an invariant stratification of the complement, whose strata we can hope to perform quotients of using recent progress in non-reductive GIT (the U-hat theorem). This allows one to construct separate 'moduli spaces of unstable objects' which parameterise those objects left out of the usual one.
In this talk I will report on work in progress studying the stratification of the unstable locus arising from the GIT construction of the moduli space of stable curves of fixed genus, in analogy with the Harder-Narasimhan picture one sees in the case of sheaves.
In our context instability is 'caused' by singularities of the curve, and we study the interesting relationship that emerges between the measure of instability provided by the stratification and some existing measures of singularity, with a view to building moduli spaces of curves of fixed 'singularity type'.
Andrew Dancer (Oxford)
We review recent work on implosion constructions for symplectic
and hyperkahler manifolds, and disucss links with nonreductive
GIT and with gauge theory