# Algebraic Geometry Seminar 23/24 Term 2

The algebraic geometry seminar in Term 2 2023/2024 will usually meet on Wednesdays at 3pm in B3.02, though we may sometimes change to allow speakers from other time zones.

See the talks from the previous term here.

**Wednesday 10 January 2024, 3pm. Speaker: -**

Title: -

**Wednesday 17 January 2024, 3pm. Speaker: -**

Title: -

**Wednesday 24 January 2024, 3pm. Speaker: TBA**

Title: TBA

**Wednesday 31 January 2024, 3pm. Speaker: Chunyi Li (Warwick)**

Title: Counting Stable Spherical Bundles on a K3 Surface

Abstract:

For a K3 surface S with higher Picard number, Huybretchs posts a question in his book asking if every spherical bundle is semistable with respect to some polarization and if there is a way to `counti' spherical bundles with a given character. Unfortunately, both problems fail in a naive way. More precisely,

1. there exists an example of a spherical vector bundle that is never semistable;

2. there exists an example of K3 surface and infinitely many spherical vector bundles with the same spherical Mukai vector v. Moreover, each of the vector bundles is stable with respect to some polarization.

However, we may put some assumptions on S so that the `counting theory' can make sense. In particular, when Nef(S) is rational polyhedral, we can do the actual counting. In the most interesting case that S is a generic elliptic K3 surface, we conjecture that on average there are ~(ln R)^2 many spherical bundles with a given character with rank r. The conjecture can be reduced to a question in classical analytic number theory.

I will discuss more details of spherical bundles and relevant questions in the talk. This is a joint work with Shengxuan Liu.

**Wednesday 7 February 2024, 3pm. Speaker: Ivan Cheltsov (Edinburgh)**

Title: Equivariant geometry of singular cubic threefolds

Abstract: I will report on a joint work with Yuri Tschinkel (Simons Foundation) and Zhijia Zhang (New York University) on linearizability of actions of finite groups on singular cubic threefolds.

**Wednesday 14 February 2024, 3pm. Speaker: Augustinas Jacovskis (University of Luxembourg)**

**Title:** Categorical Torelli for double covers

**Abstract:** Consider a threefold double cover X of (weighted) projective space, ramified in a canonically polarised surface Z. In this talk I'll describe a semiorthogonal decomposition of the mu_2-equivariant Kuznetsov component of X, and show that it contains a copy of Db(Z). This gives a relationship between the K-theory of the equivariant Kuznetsov component, and the primitive cohomology of Z. Using this relationship and classical Torelli theorems for hypersurfaces in (weighted) projective space, I'll show that for certain classes of prime Fano threefolds, an equivalence of Kuznetsov components implies that they're isomorphic. This is joint work with Hannah Dell and Franco Rota.

**Wednesday 21 February 2024, 3pm. Speaker: Chenjing Bu (Oxford)**

**Title:** Donaldson–Thomas invariants in type B/C/D

**Abstract:** We discuss the problem of constructing enumerative invariants for structure groups of type B/C/D, i.e. the orthogonal and symplectic groups. This includes counting principal bundles on a variety, or counting a certain version of quiver representations. In particular, we define Donaldson–Thomas invariants in type B/C/D for Calabi–Yau threefolds and quivers with potential, and discuss their wall-crossing behaviour under a change of stability conditions.

**Wednesday 28 February 2024, 3pm. Speaker: Daniel Loughran (Bath)**

Title: Arithmetic and geometry and cubic threefolds

Abstract: A famous theorem of Clemens and Griffiths states that any smooth cubic threefold over the complex numbers is non-rational. The key tool required in the proof was the intermediate Jacobian, which is a higher-dimensional version of the Jacobian of a curve. In this talk I will report on various applications with Javanpeykar on the intermediate Jacobian over non-algebraically closed fields, as well forthcoming work of my PhD student Tudor Ciurca on the rationality of cubic threefolds over fields of characteristic 2.

**Wednesday 6 March 2024, 3pm. Speaker: Xenia de la Ossa (Oxford)**

Title: On the arithmetic of families Calabi-Yau manifolds

Abstract: In this seminar I will discuss what I know (and donâ€™t know) about the arithmetic of Calabi-Yau 3-folds. The main goal is to explore whether there are questions of common interest in this context to physicists, number theorists and geometers. The main quantities of interest in the arithmetic context are the numbers of points of the manifold considered as a variety over a finite field. We are interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise for a physicist is that the numbers of points over a finite field are also given by expression that involve the periods of a manifold. The number of points are encoded in the local zeta function, about which much is known in virtue of the Weil conjectures. I will discuss interesting topics related to the zeta function and the appearance of modularity for one parameter families of Calabi-Yau manifolds. I will report on an example for which the quartic numerator of the zeta function factorises into two quadrics at special values of the parameter which satisfy an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. We note that these factorisations are due to a splitting of the Hodge structure and that these special values of the parameter are rank two attractor points in the sense of black hole solutions of type IIB supergravity. Modular groups and modular forms arise in relation to these attractor points. To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provided the first explicit examples of such points for Calabi-Yau manifolds of full SU(3) holonomy. The work presented is based on joint research with Philip Candelas, Mohamed Elmi and Duco van Straten and, time permitting, further work with Philip Candelas, Pyry Kuusela and Joseph McGovern. I will not be assuming familiarity with type II string theory.

**Wednesday 13 March 2024, 3pm. Speaker: Thomas Gauthier (UniversitĂ© Paris-Saclay)**

Title: Sparsity of Postcritically finite maps in higher dimension

Abstract: In this talk, we focus on connections between arithmetic and holomorphic dynamics. The first goal of the talks is to present several problems in arithmetic dynamics of endomorphisms of projective spaces, all inspired from classical problems in arithmetic geometry. The second goal is to explain how these problems are related to the notions of bifurcation currents and measures in complex dynamics. I will start with several motivations for the problem we study. If time allows, I will sketch a proof strategy to solve two problems at the same time. This is a joint work with Johan Taflin and Gabriel Vigny.