Abstract:

The problem of determining the birational nature of the moduli space of curves of genus g has received constant attention in the last century and inspired a lot of development in moduli theory. I will discuss progress achieved in the last 12 months. On the one hand, making essential use of tropical methods it has been showed that both moduli spaces of curves of genus 22 and 23 are of general type (joint with D. Jensen and S. Payne). On the other hand I will discuss a proof (joint with A. Verra) of the uniruledness of the moduli space of curves of genus 16.

Abstract:

Given a smooth plane curve C defined over an arbitrary field k, we say that C is transverse-free if it has no transverse lines defined over k. If k is an infinite field, then Bertini's theorem guarantees the existence of a transverse line defined over k, and so the transverse-free condition is interesting only in the case when k is a finite field F_q. After fixing a finite field F_q, we can ask the following question: For each degree d, what is the fraction of degree d transverse-free curves among all the degree d curves? In this talk, we will investigate an asymptotic answer to the question as d tends to infinity. This is joint work with Brian Freidin.

Abstract:

In this talk we study certain moduli spaces of semistable objects in the Kuznetsov component of a cubic fourfold. We show that they admit a symplectic resolution \tilde{M} which is a smooth projective hyperkaehler manifold deformation equivalent to the 10-dimensional example constructed by O’Grady. As a first application, we construct a birational model of \tilde{M} which is a compactification of the twisted intermediate Jacobian fiberation of the cubic fourfold. Secondly, we show that \tilde{M} is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in the cubic fourfold, as conjectured by Castravet. This is a joint work with Chunyi Li and Laura Pertusi.

Abstract:

In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a more geometric/topological approach in the case of semisimple local systems adapting de Cataldo-Migliorini. As a byproduct, we can recover a weak form of Saito's decomposition theorem for variations of Hodge structures. Joint work in progress with Chuanhao Wei.

Abstract:

We will report on a recent joint work with Junyan Cao, cf. arXiv:2012.05063. The main topics we will discuss are revolving around the extension of pluricanonical forms defined on the central fiber of a family of Kaehler manifolds. For our results to hold we need the divisor of zeros of the said forms to be suficiently "nice", in a sense that will become clear during the talk.

Abstract:

We introduce a derived version of Lie algebras in characteric p and describe two recent applications: first, we use them to classify infinitesimal deformations, generalising the Lurie-Pridham theorem in characteristic zero; second, we prove a Galois correspondence for purely inseparable field extension, extending work of Jacobson at height one. This talk is based on joint works with Mathew and Waldron.

Abstract:

In this talk I will discuss properties of certain vector bundles on the moduli space of stable n-pointed curves which arise from admissible modules over certain vertex operator algebras. I will in particular describe conditions on the vertex algebra that guarantee that the factorization property holds for these vector bundles and discuss its consequences and open problems if time permits. This is based on a joint work with A. Gibney and N. Tarasca.

Abstract:

This describes joint work in progress with M. Mustata, in which we study the filtration on local cohomology sheaves induced by their natural mixed Hodge module structure. Special properties of this filtration, for instance strictness, lead to a number of different applications, including an injectivity theorem for dualizing complexes, local vanishing for forms with log poles, and especially a characterization of the local cohomological dimension of a closed subscheme in terms of data arising from a log resolution of singularities.

Abstract:

Algebraic geometry has had a huge influence on representation theory for (nearly) the last century. I'll talk about some historical instances of this relationship and recent progress in translating our algebraic understanding of representations of p-adic groups to geometric contexts. Parts of this talk are based on joint work with A. Ivanov and joint work with M. Oi.

Abstract:

I will work on a Calabi-Yau 3-fold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda for weak stability conditions, such as the quintic threefold. I will explain how wall-crossing with respect to weak stability conditions gives an expression of Joyce’s generalised Donaldson-Thomas invariants counting Gieseker semistable sheaves of any rank greater than or equal to one on X in terms of those counting sheaves of rank 0 and pure dimension 2. This is joint work with Richard Thomas.

Abstract:

Abstract:

K-stability is an algebraic condition that characterizes the existence of K\"ahler-Einstein metrics on Fano varieties. Recently there has been a lot of work on the construction of the K-moduli space, i.e. a good moduli space parametrizing K-polystable Fano varieties. Motivated by results in differential geometry, it is conjectured that this K-moduli space is proper and projective. In this talk, I'll discuss some recent progress in birational geometry that leads to a full solution of this conjecture. Based on joint work with Yuchen Liu and Chenyang Xu.

Abstract:

We introduce refined unramified cohomology and prove some general comparison theorems to cycle groups. Our approach has several applications. For instance, it allows to construct the first example of a smooth complex projective variety whose Griffiths group has infinite torsion subgroup.