We will introduce a family of compactifications of the moduli space of quartic K3 surfaces from K-stability, focusing in detail on the locus of hyperelliptic K3s that arise as double covers of P1xP1 branched over a (4,4) curve. We will show that K stability provides a natural way to interpolate between the GIT moduli space and the Baily-Borel compactification and will relate this interpolation to VGIT wall crossings, verifying predictions of Laza and O’Grady. This is joint work with Kenny Ascher and Yuchen Liu.

Cluster varieties are log Calabi-Yau varieties which are unions of algebraic tori glued by birational "mutation" maps. They can be seen as a generalization of the toric varieties. In toric geometry, projective toric varieties can be described by polytopes. We will see how to generalize the polytope construction to cluster convexity which satisfies piecewise linear structure. As an application, we will see the non-integral vertex in the Newton Okounkov body of Grassmannian comes from broken line convexity. We will also see links to the symplectic geometry and application to mirror symmetry. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.

A key feature of stability theories in algebraic geometry is that unstable objects admit uniquely determined optimal destabilizations. This philosophy should hold for K-stability, which is a notion introduced by differential geometers to detect when Fano varieties admit Kahler-Einstein metrics. In this talk, I will discuss properties of certain optimal destabilization of K-unstable Fano varieties and uniqueness results. This is based on joint works with Yuchen Liu and Chuyu Zhou.

In general, the topology of the moduli space of semistable sheaves on an algebraic variety relies heavily on the choice of the Euler characteristic of the sheaves. We show a striking phenomenon that, for the moduli of 1-dimensional semistable sheaves on a toric del Pezzo surface (e.g. P^2), or the moduli of semistable Higgs bundles with respect to a divisor of degree > 2g-2 on a curve, the intersection cohomology (together with the perverse and the Hodge filtrations) of the moduli space is independent of the choice of the Euler characteristic. This confirms a conjecture of Bousseau for P^2, and proves a conjecture of Toda in the case of certain local Calabi-Yau 3-folds. In the proof, a generalized version of Ngô's support theorem plays a crucial role. Based on joint work with Davesh Maulik.

I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards cluster algebras. Let U be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in U uniquely determine a commutative associative algebra equipped with a compatible multilinear form. Although the statement of the theorem involves only elementary algebraic geometry, the proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. I will explain various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, our algebra generalizes, and gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK, as well as the positivity in the Laurent phenomenon, follow readily from the geometric description. This is joint work with S. Keel, arXiv:1908.09861. If time permits, I will mention another application towards the moduli space of KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs, joint with P. Hacking and S. Keel, arXiv: 2008.02299.

I will explain recent progress in the birational classification of algebraic foliations in low dimension inspired by the theory of the Minimal Model Program. This is joint work with Calum Spicer.

In positive characteristic, there exist fibrations between smooth varieties where every fiber is singular or even non-reduced. In the latter case, the generic fiber of the fibration is geometrically non-reduced. We study the failure of generic smoothness and obtain a structural result about geometrically non-reduced varieties, with applications to Fano varieties. This is joint work with Joe Waldron.

Given a smooth projective variety whose nonrationality is known, one can try to measure how far it is from being rational. I will explain recent progress in this direction for codimension two complete intersections.

Bogomolov-Sommese vanishing asserts that the $i$-th logarithmic reflexive differential form of a log canonical projective pair in characteristic zero does not contain a Weil divisorial sheaf of Iitaka dimension bigger than $i$. In positive characteristic, it is known that the vanishing theorem fails. Indeed, there is a simple normal crossing surface pair $(X, D)$ such that the first logarithmic differential form contains a big line bundle in each characteristic. On the other hand, in these counterexamples, we can see that $K_X+D$ is often big. In this talk, I will discuss Bogomolov-Sommese vanishing theorem for a log canonical surface pair $(X, D)$ such that $K_X+D$ is not pseudo-effective in positive characteristic. I also show an application to liftability of surface pairs.

It is a peculiar property of characteristic p>0 geometry that smooth varieties of non-negative Kodaira dimension could be uniruled. It has been widely believed that such behaviour could actually be mitigated by making appropriate cohomological assumptions. In the talk I will confirm this expectation in the case of varieties with trivial canonical class generalizing to arbitrary dimension the easier implication of Artin-Shioda conjecture describing the K3 surfaces story. The presentation will be based on the joint work with Zsolt Patakfalvi (EPFL)

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 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 Xiaolei Zhao.

I will explain how we obtain stronger versions of the Bogomolov-Gieseker inequality for stable vector bundles on certain varieties. As an application, we can construct Bridgeland stability conditions on some Calabi-Yau weighted hypersurfaces of dimension three. Tilt-stability conditions on the derived category play crucial roles in the arguments.

The inverse Hodge problem asks which possible Hodge diamonds can occur for smooth projective varieties. While this is a very hard problem in general, Paulsen and Schreieder recently showed that in characteristic 0 there are no restrictions on the modulo m Hodge numbers, besides the usual symmetries. In joint work with Matthias Paulsen, we extend this to positive characteristic, where the story is more intricate.