Valery Alexeev (University of Georgia, Athens)

Title: KSBA compactifications of moduli spaces of K3 surfaces with a higher order automorphism

Abstract: By a result of Alexeev-Engel, the KSBA compactification of moduli spaces of K3 surfaces with a recognizable divisor is semitoroidal, corresponding to some semifan. The easiest example of a recognizable divisor is a component of the fixed locus of a nonsymplectic automorphism that has genus > 1. The case when the order of the automorphism is 2 was solved in joint work with Engel: there are 50 moduli spaces in that case, and for each of them we described the semifans and stable surfaces appearing on the boundary. In this talk I will present joint work with Deopurkar and Han which settles the order 3 case.

Ben Bakker (University of Illinois, Chicago)

Title: Some mixed characteristic hyperbolicity properties of moduli spaces

Abstract: Many varieties (including Calabi–Yau manifolds, curves, and most
hypersurfaces) satisfy a Torelli theorem, meaning their geometric
deformations are determined by the deformations of an appropriate
Hodge structure. Equivalently, the moduli spaces of such varieties
admit a quasifinite period map, and therefore inherit various
hyperbolicity properties from the negative curvature of the period
domain in Griffiths transverse directions. In this talk, I will begin
by surveying these properties over C, and then describe some recent
joint work with A. Shankar and J. Tsimerman proving some analogous
properties over Z_p. In particular, we will deduce a few important
arithmetic consequences for exceptional Shimura varieties.

Lukas Braun

Title: Some results on (non-)finitely generated quasiaffine algebras

Abstract: In this talk, I will report about work in progress on questions related to (non-)finite generation of quasiaffine algebras.

This in particular relates to certain Cox rings, reductive invariant rings of quasiaffine algebras, and singularities of such rings.

Results include abstract statements as well as computational methods.

Philip Engel (University of Illinois, Chicago)

Title: Boundedness theorems for abelian fibrations

Abstract: I will report on work-in-progress, with Filipazzi, Greer, Mauri, and Svaldi on boundedness results for abelian fibrations. We will outline a proof that irreducible Calabi-Yau varieties admitting an abelian fibration are birationally bounded in a fixed dimension; and similarly, that Lagrangian fibrations of symplectic varieties, in a fixed dimension, are analytically bounded.

Laure Flapan (Michigan State University)

Title: Cones of divisors on moduli spaces of K3 surfaces and hyperkähler manifolds

Abstract: We study the cones of pseudo-effective divisors and Noether-Lefschetz divisors on moduli spaces of K3 surfaces and hyperkähler manifolds. In particular, we establish the extremity in the pseudo-effective cone of certain extremal rays in the cone of Noether-Lefschetz divisors for these moduli spaces. This relies on joint work with Barros, Beri, and Williams as well as forthcoming work with Barros and Zuffetti.

Elham Izadi (University of California, San Diego)

Title: Szego kernels and the Scorza map on moduli spaces of spin curves

Abstract: The Scorza correspondence was first studied by Scorza. Starting with a spin curve of genus 3 (i.e., a curve of genus 3 with an even theta-chracteristic with no global sections), Scorza used his correspondence to construct a second plane quartic which gave a birational map from the moduli space of curves of genus 3 to the moduli space of spin curves of genus 3. Scorza’s results were further used by Mukai to construct the family of Fano threefolds of genus 12 and degree 22. Scroza’s correspondence is in fact well-defined in all genera. We determine the limits of the Scorza correspondence at generic points of the vanishing theta-null divisor and at generic points of boundary divisors. We further show that the Scorza quartic can be defined using Wirtinger duality which shows that it can, in a certain form, be defined for principally polarized abelian varieties with a theta-characteristic. We further show that limit of the Scorza quartic at abelian varieties with vanishing theta-nulls is twice the quadric tangent cone to the theta divisor at the vanishing theta-null.

Junpeng Jiao (Tsinghua University)

Title: On the boundedness of Calabi-Yau fibrations.

Abstract: A Calabi-Yau fibration is a fibration of projective varieties X->Z such that the canonical bundle K_X is numerically trivial over Z.
This class of varieties plays a significant role in algebraic geometry, appearing naturally in contexts such as good minimal models and elliptic
Calabi-Yau varieties. In this talk, I will present some results on the boundedness of Calabi-Yau fibrations under certain natural conditions,
based on joint work with Minzhe Zhu and Xiaowei Jiang.

Mirko Mauri (Institut de Mathématiques de Jussieu-Paris Rive Gauche)

Title: Decomposition theorem for logarithmic G-Hitchin systems

Abstract: I will report on an on going project with Mark de Cataldo, Andres
Fernandes Herrero and Roberto Fringuelli on the moduli space of
semistable logarithmic G-Higgs bundles over a smooth curve. For any
given degree in the algebraic fundamental group of the reductive group
G, we exhibit a uniform description of the decomposition theorem for the
Hitchin fibration.

Roberto Svaldi (Università degli Studi di Milano)

Title: Boundedness theorems for fibered CY.

Abstract: I will explain ideas and techniques behind recent results showing that fibered CY varieties are bounded, starting from the elliptic case and then moving to the case of higher relative dimension.

Valentino Tosatti (Courant)

Title: Pseudoeffective R-divisors with volume zero on K-trivial manifolds

Abstract:Pseudoeffective R-divisor classes with volume zero are limits of big R-divisors, and they satisfy a weak semipositivity condition (in analytic terms, they contain a closed positive current). While on general projective manifolds these semipositive objects can be rather badly behaved, I will argue that on K-trivial manifolds the situation should be substantially better for nef R-divisor classes. I will discuss in detail the case of K3 surfaces (joint with Filip) and of Calabi-Yau manifolds of Wehler type (joint with Filip and Lesieutre).

Lingyao Xie (University of California, San Diego)

Title: On the base-point-freeness theorem of foliations

Abstract: A Foliation is a saturated subsheaf of the tangent sheaf that is closed under Lie bracket. The study of foliations is a natural generalization of the study of the tangent bundle/differential sheaf, we expect the canonical divisor $K_F$ of the foliation to share some similar good properties as the $K_X$. In general a foliation could be very complicated and hard to handle, but when the foliation is algebraically integrable (i.e leaves are algebraic) then we are able to establish corresponding minimal model program to show that the base point free theorem holds for $K_F+A$, where $A$ is ample. This is a joint work with F. Meng and J. Liu.

Tony Yue Yu (Caltech)

Title: Decomposition of F-bundles and new birational invariants

Abstract: An F-bundle is a non-archimedean version of variation of nc-Hodge structures. I will discuss the spectral decomposition theorem for F-bundles, the resulting atomic decomposition of a smooth projective variety, and new birational invariants. Part is joint with Katzarkov, Kontsevich and Pantev, part with Hinault, Zhang and Zhang.

Student Talks

Zhiyuan Chen

Title: Stable degeneration of families of klt singularities with constant local volume

Abstract: We prove that for a locally stable family of klt singularities with constant local volume, the ideal sequences of the minimizing valuations for the normalized volume function form families of ideals with flat cosupport, which induce a degeneration to a locally stable family of K-semistable log Fano cone singularities.

Tai-Hsuan Chung

Title: Stable Reduction via the Log Canonical Model

Abstract: We will outline a proof of stable reduction for surfaces in large characteristic.

Zhiyuan Jiang

Title: Abundance for Kaehler Varieties via the Algebraic Reduction Map

Abstract: There has been a lot of recent exciting activity aimed at establishing the MMP for Kaehler varieties. By results of Campana, Das, Hacon, H\”oring, Paun, Peternell, and Ou, we know abundance and the MMP for threefolds and we have partial results for the MMP for fourfolds. We introduce a new approach to establish abundance for Kaehler varieties. The goal is to reduce abundance for Kaehler varieties to abundance for projective varieties, using the algebraic reduction map.

Hyunsuk Kim

Title: Canonical bundle formula and the equivalence of non-vanishing and Campana–Peternell conjectures

Abstract: Both the non-vanishing and the Campana–Peternell conjectures are special (and useful) cases of the abundance conjecture. While the Campana–Peternell conjecture is stronger than non-vanishing, Schnell showed that proving certain behavior on algebraic fibre spaces guarantees the equivalence of these two conjectures. He actually proves this behavior when the canonical bundle of the base is pseudo-effective. We see what the canonical bundle formula suggests in this case, and explain two possible ways to deal with the ‘problematic part’ appearing in the canonical bundle formula.

Junyao Peng

Title: G_m-equivariant degenerations of del Pezzo surfaces

Abstract: We study special G_m-equivariant degenerations of a smooth del Pezzo surface X induced by valuations that are log canonical places of (X,C) for a nodal anti-canonical curve C. We show that the space of special valuations in the dual complex of (X,C) is connected and admits a locally finite partition into sub-intervals, each associated to a G_m-equivariant degeneration of X. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement.

Shubham Saha

Title: Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves

Abstract: We will discuss some ongoing work on the subject, specifically in the rank 2, genus 2 case. The talk will start with a quick review of existing literature on $M_2$ and some of its étale covers, along with results and constructions involving moduli of rank 2 bundles. We will go over their generalizations to the universal setting and outline the usage of these tools for computing the Chow ring. Lastly, we shall go over some ideas to relate the generators to tautological classes.

Zhijia Zhang

Title: The volume preserving Cremona group

Abstract: We show that the group of birational automorphisms of P^n preserving the standard torus-invariant volume form is not generated by pseudo-regularizable maps, and is not simple, for n>2 and over various fields. This generalizes the same result about the Cremona group by Lin and Shinder. This is joint work with Konstantin Loginov.