MPS Conference on Higher Dimensional Geometry, May 8-12, 2023

Hamid Abban
University of Nottingham

On K-stability and K-moduli of Fano hypersurfaces

It is accepted knowledge that K-stability provides the right framework for the compact moduli of Fano varieties. While it is conjectured that all smooth Fano hypersurfaces are K-polystable, very little is known about expectations on the structure of their K-moduli except in low degrees. I will survey known results on both K-stability of hypersurfaces and their moduli, with a focus on recent progress on quartic 3-folds in joint work with Cheltsov, Kasprzyk, Liu, and Petracci.
 

Nathan Chen
Columbia University

Higher index Fano varieties in positive characteristic with Bir(X) = 1

For a smooth complex hypersurface X of index 1 in projective space, the Noether-Fano method was used by many authors (Fano-Segre-Iskovskikh-Manin-Pukhlikov-Cort-Cheltsov-De Fernex-Ein-Mustaţă-Zhuang) to prove birational super-rigidity; in particular, the group Bir(X) of birational automorphisms is finite. Pukhlikov proved a similar result for most index 2 hypersurfaces. To our knowledge, there are no known results in index 3 and higher. In this talk, we will revisit a construction of Kollár which produces forms on certain p-cyclic covers in characteristic p, and use it to give examples of Fano varieties of arbitrarily large index with no nontrivial birational automorphisms. The tradeoff is that we must pass to positive characteristic and allow mild singularities. Our main observation is that there is a natural Bir(X)-equivariant ample line bundle on these varieties.
 

Ruadhai Dervan
University of Cambridge

Valuations and stability of polarised varieties

The valuative approach to the theory of K-stability of Fano varieties has led to many recent advances, for example to the construction of moduli spaces of Fano varieties. I will survey how valuations can be used to study K-stability of general polarised varieties, namely projective varieties endowed with an ample line bundle. I will begin by describing work of myself and Legendre defining a stability condition (“valuative stability”) using divisorial valuations, and will then describe work of Boucksom-Jonsson defining a stability condition (“divisorial stability”) using convex combinations of divisorial valuations. While valuative stability should only be expected to capture K-stability in the Fano setting (and should be strictly weaker in general), divisorial stability is conjectured to be equivalent to (uniform) K-stability for general polarised varieties. I will describe some applications of this new valuative approach: openness of the divisorially stable locus in the ample cone (Boucksom-Jonsson) and a description of the behaviour of divisorial stability under finite covers (joint work with Papazachariou).
 

Laure Flapan
Michigan State University

Kodaira dimension of moduli spaces of hyperkähler manifolds

We study the geometry of some moduli spaces of polarized hyperkähler manifolds. We use techniques of Gritsenko-Hulek-Sankaran involving the Borcherds modular form to determine a bound on the degree of the polarization after which these moduli spaces are always of general type. This is joint work with I. Barros, P. Beri, and E. Brakkee.
 

Giovanni Inchiostro
University of Washington

Moduli of Fano varieties with complements

I will discuss a new approach to build a moduli space of pairs (X,D) where X is a Q-Fano variety and D is a Q-divisor such that K_X + D is Q-rationally equivalent to 0. In the case of X=P^2, our approach gives a projective moduli space, which interpolates between KSBA-stability and K-stability. This is joint work with K. Ascher, D. Bejleri, H. Blum, K. DeVleming, Y. Liu, X. Wang.
 

Lena Ji
University of Michigan

Fano hypersurfaces with no finite order birational automorphisms

The birational automorphism group is a natural birational invariant associated to an algebraic variety. In this talk, we study the specialization homomorphism for the birational automorphism group. As an application, building on work of Kollár and work of Chen–Stapleton, we show that a very general n-dimensional complex hypersurface X of degree ≥ 5⌈(n+3)/6⌉ has no finite order birational automorphisms. This work is joint with Nathan Chen and David Stapleton.
 

Sándor Kovács
University of Washington

KSB stability is automatic in codimension 3

In this talk I will first review KSB/A stability, especially their local version and then discuss new results, joint with János Kollár, showing that it is enough to check these conditions, including flatness, up to codimension 2. This implies that we have a very good understanding of this stability condition in general, because local KSB-stability is trivial at codimension 1 points, and quite well understood at codimension 2 points, since we have a complete classification of 2-dimensional slc singularities.
 

Justin Lacini
University of Kansas

Syzygies of adjoint linear series on projective varieties

Syzygies of algebraic varieties have long been a topic of intense interest among algebraists and geometers alike. Starting with the pioneering work of Mark Green on curves, numerous attempts have been made to extend these results to higher dimensions. Ein and Lazarsfeld proved that if A is a very ample line bundle, then K_X + mA satisfies property N_p for any m>=n+1+p. It has ever since been an open question if the same holds true for A ample and basepoint free. In joint work with Purnaprajna Bangere we give a positive answer to this question.
 

Jihao Liu
Northwestern University

Optimal Bounds for Algebraic Invariants of Surfaces

In this talk, I will present several results on the optimal bounds for algebraic invariants of surfaces. Specifically, I will discuss our findings of the 1-gap of R-complementary thresholds, the smallest volume of ample log surfaces with reduced boundary, and the smallest minimal log discrepancy of klt Calabi-Yau surfaces. These results answer questions posed by V. Alexeev and W. Liu, and J. Kollár, and also reprove a recent result by L. Esser, B. Totaro, and C. Wang. As an application, I will also discuss our work on finding and classifying all exceptional Fano surfaces (Fano surfaces with Tian’s alpha invariant strictly greater than 1) that are not 1/11-klt. We have identified 25 such surfaces up to isomorphism. This is an ongoing joint work with V. V. Shokurov.
 

Enrica Mazzon
Imperial College London

(Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces

The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent result by Li relates explicitly the non-archimedean approach to the classical SYZ conjecture.

In this talk, I will give an overview of this subject and I will focus on families of Calabi-Yau hypersurfaces in P^n. For this class, in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey, we solve a non-archimedean conjecture proposed by Li and deduce that classical SYZ fibrations exist on a large open region of CY hypersurfaces.
 

Takumi Murayama
Purdue University

The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero

In 2010, Birkar, Cascini, Hacon, and McKernan proved a relative version of the minimal model program for projective morphisms of complex quasi-projective varieties, called the relative minimal model program with scaling. Their result is now fundamental to our understanding of the birational classification of quasi-projective varieties and has numerous applications.

In this talk, I will discuss recent joint work with Shiji Lyu that establishes the relative minimal model program with scaling for excellent schemes, excellent algebraic spaces, and analytic spaces simultaneously in equal characteristic zero. This not only recovers previous results for complex varieties, complex algebraic spaces, and complex analytic spaces, but also greatly extends the scope of the relative minimal model program with scaling to a broader class of geometric spaces, including formal schemes, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. Our results for (non-algebraic) schemes and rigid analytic spaces were previously only known in dimensions ≤3 and ≤2, respectively, and our results for formal schemes and Berkovich spaces are completely new.
 

Keiji Oguiso
University of Tokyo

Relative automorphisms of an abelian fibered Calabi-Yau threefold of positive entropy and an application to real forms

We show that there is exactly one abelian fibered Calabi-Yau threefold with a positive entropy automorphism preserving the fibration up to isomorphisms as fibered varieties. If time allows, as an application, I would like to discuss the finiteness problem of real forms of an abelian fibered Calabi-Yau threefold, which is one of my motivations for the above mentioned result.
 

Alex Perry
University of Michigan

The period-index problem

In the theory of Brauer groups, the longstanding period-index problem asks for a bound on one measure of complexity of a central simple algebra (its index) in terms of another (its period). I will discuss some recent progress on this problem which relies on a mixture of ideas from Hodge theory, noncommutative/derived algebraic geometry, and enumerative geometry. This is based on joint works with James Hotchkiss and with Johan de Jong.
 

Alexander Petrov
Max Planck institute for Mathematics

De Rham complex in positive characteristic

Deligne and Illusie proved that, remarkably, de Rham cohomology of a smooth proper variety over F_p admits an analog of Hodge decomposition, provided that the variety lifts to Z/p^2 and has dimension <=p. I will discuss what can be said about de Rham cohomology of liftable varieties of arbitrary dimension. It turns out that Deligne-Illusie’s theorem continues to hold for some classes of varieties, such as F-split and quasi-F-split ones. But in general it fails — de Rham cohomology of a smooth proper liftable variety might have smaller dimension than its Hodge cohomology.
 

Junliang Shen
Yale University

Cohomology of certain moduli spaces of sheaves

Cohomology of the moduli space of vector bundles on a curve has been studied for decades. In this talk, I will discuss some closely related moduli spaces, including moduli of Higgs bundles on a curve and moduli of sheaves on a Fano surface. Considerations from non-abelian Hodge theory and modern enumerative geometry of Calabi-Yau 3-folds predict surprising structures for the cohomology of these classical moduli spaces. I will discuss these predictions and some recent progress, without assuming any previous knowledge of non-abelian Hodge theory or enumerative geometry.
 

Fumiaki Suzuki
UCLA

Two coniveau filtrations and algebraic equivalence over finite fields

Over the complex numbers, the integral cohomology of a smooth projective variety is endowed with the coniveau and strong coniveau filtrations. The two filtrations differ in general as recently shown by Benoist and Ottem, and this result may be exnteded to the l-adic setting over any algebraically closed field of characteristic not 2. In this talk, I would like to discuss some consequences of the equality of the two filtrations for algebraic equivalence for codimension 2 cycles over finite fields. As an application, we show the vanishing of the third unramified cohomology for a large class of rationally chain connected threefolds over finite fields, confirming a conjecture of Colliot-Thelene and Kahn. This is a joint work with Federico Scavia.
 

Joe Waldron
Michigan State University

The log minimal model program for excellent threefolds

The log minimal model program has recently been completed for klt threefolds over regular excellent base schemes of residue characteristic $p>5$. In this talk I will survey the known results, together with some motivations and applications for working in this more general setup.
 

Shou Yoshikawa
Riken
and
Tokyo Institute of Technology

Varieties in positive characteristic with numerically flat tangent bundle

The positivity condition imposed on the tangent bundle of a smooth projective variety is known to restrict the geometric structure of the variety. Demailly, Peternell and Schneider established a decomposition theorem for a smooth projective complex variety over with nef tangent bundle. The theorem states that, up to an etale cover, such a variety has a smooth fibration admits a smooth algebraic fiber space over an abelian variety whose fibers are Fano varieties, so one can say that such a variety decomposes into the positive” part and the flat” part. A positive characteristic analog of the above decomposition theorem was proved by Kanemitsu and Watanabe. The “flat” part of their theorem is a smooth projective variety with numerically flat tangent bundle. In this talk, I will introduce the result that every ordinary variety with numerically flat tangent bundle is an etale quotient of an ordinary Abelian variety. In particular, we obtain the decomposition theorem for Frobenius splitting varieties with nef tangent bundle.
 

Ziquan Zhuang
Princeton University

Stability and boundedness of klt singularities

Donaldson and Sun showed that singularities on Gromov-Hausdorff limits of Kähler-Einstein Fano manifolds have algebraic metric tangent cones, and they conjectured that these metric tangent cones only depend on the algebraic structure of the singularities. In the study of this conjecture, Li and Xu proposed the Stable Degeneration Conjecture, predicting that every klt singularity has a canonical “stable degeneration” induced by the valuation with smallest normalized volume. In my talks, I’ll introduce some related backgrounds, survey the recent solution of these conjectures, and discuss some implications on boundedness of singularities.