Next Meeting

Abstracts

Victoria Hoskins (Essen) - Relative non-reductive geometric invariant theory and applications

After motivating and introducing (non-reductive) geometric invariant theory, I will report on joint work on a relative approach to non-reductive geometric invariant theory with E. Hamilton and J. Jackson. I hope to explain some applications where this relative viewpoint is particular useful: notably for moduli of unstable objects and moduli of representations of a quiver with multiplicities. The latter of these applications is joint work with J. Jackson and T. Vernet, and the geometry of these new moduli spaces should hopefully relate to the representation theory of symmetrisable Kac-Moody Lie algebras, similar to the case for classical moduli spaces of quiver representations.

George Cooper (Pisa) - Intrinsic non-reductive geometric invariant theory

I will present a stack-theoretic generalisation of the existence of non-reductive GIT quotients. Namely, let $\mathcal{X}$ be an algebraic stack with a weak action of the monoid stack $\Theta = [\mathbb{A}^1/\mathbb{G}_m]$ for which the centre $\mathcal{Z}$ admits a good moduli space $\mathcal{Z} \to Z$ and for which the retraction to the centre has flat relative inertia. Then, for appropriate invertible sheaves $\mathcal{L}$ on $\mathcal{X}$, the $\Theta$-semistable locus $\mathcal{X}^{ss}(\mathcal{L})$ is open in $\mathcal{X}$ and admits a moduli space $X(\mathcal{L})$ which is projective over $Z$, with $\mathcal{X}^{ss}(\mathcal{L}) \to X(\mathcal{L})$ factoring as a gerbe followed by a good moduli space. By taking $\mathcal{X}$ to be the quotient stack associated to the action of an internally graded group, we recover both the existence of non-reductive GIT quotients and the non-reductive Hilbert-Mumford criterion, in a way which naturally extends to both the relative setting and to positive characteristic.

Time permitting, I will indicate how this result can be used to both simplify many existing constructions of moduli spaces of unstable objects in algebraic geometry, and also to construct new examples of quasi-projective coarse moduli spaces of principal bundles over curves with non-reductive structure groups.

This talk is based on joint work with L. Modin (Hanover).

Chunyi Li (Warwick) - Bridgeland stability conditions on projective varieties

Slope stability for vector bundles on curves was introduced by Mumford in the 1960s, providing a rigorous foundation for the construction of moduli spaces. In higher dimensions, the notion of stability splits into slope stability and Gieseker stability. While both retain many of the desirable properties as in the curve case, each also presents subtle technical limitations.

The Bridgeland stability condition can be viewed as a generalized slope stability on curves to higher-dimensional varieties in a more unified and robust way, combining advantages of both slope and Gieseker stability. A central question in the theory has been to determine which varieties admit Bridgeland stability conditions. In this talk, I will discuss recent progress on the existence of stability conditions on all projective varieties.

Gergely Bérczi (Aarhus) - The blow-up construction in non-reductive geometric invariant theory

Abstract forthcoming.

Frances Kirwan (Oxford) - Degenerate Morse theory and higher quivers

Abstract forthcoming.