MPS Conference on Higher Dimensional Geometry, May 6-10, 2024

Frédéric Campana
University of Lorraine

On a conjecture of Mihnea Popa

Popa conjectured that if $f:X\to Y$ is a submersive projective morphism with connected fibers between complex quasi-projective manifolds, the logarithmic Kodaira dimensions of $X$ and $Y$ differ exactly by the Kodaira dimension of the fibers. Together with C. Schnell, Popa proved some important cases of this conjecture. Frédéric Campana will prove, by a different approach, involving the ‘core map,’ that the conjecture holds when the fibers admit good minimal models.

Adrien Dubouloz
CRNS

Additive Group Actions, Polar Cylinders and Rigidity of Brieskorn-Pham Hypersurfaces

Adrien Dubouloz will give an overview of two recent and developing advances concerning applications of the correspondence between anti-canonical polar cylinders in Fano varieties and  homogeneous actions of the additive group on the spectra of their anti-canonical rings: one concerns the study of automorphism groups of so-called Brieskorn-Pham affine hypersurfaces via a reduction to well-formed hypersurfaces and the other the construction of natural test configurations for Fano varieties possessing polar cylinders.

The talk is based on several joint works in progress with, among others, Michael Chitayat (Ottawa) and Takashi Kishimoto (Saitama).

Hélène Esnault
Freie Universität Berlin

A Non-Abelian Version of Deligne’s Fix Part Theorem

We prove a non-abelian version of Deligne’s Fix Part Theorem; a statement which is purely anchored in complex geometry. This motivates the consideration of a vaster program aimed at understanding (some parts of the) monodromy-weight conjecture in unequal characteristic by ’tilting it’ to a complex situation, for which we have the tools developed notably by Takuro Mochizuki; this lecture focuses on one small part of it.

Joint work with Moritz Kerz in progress.

Enrica Floris
Université de Poitiers

On Algebraic Subgroups of the Cremona Group

The study of connected algebraic subgroups of the Cremona group is a classical way of deepening the understanding of the Cremona group. Via the Weil regulariztion theorem and the minimal model program, to such a group we associate a rational Mori fiber space on which it acts regularly.

In this talk, Enrica Floris will discuss the notion of maximal connected algebraic subgroups of the Cremona group, and its relation with the geometry of the associated Mori fiber spaces. This is a work in collaboration with A. Fanelli and S. Zimmermann.

Osamu Fujino
Kyoto University

Some Remarks on Weakly Positive Sheaves

In this talk, Osamu Fujino will explain some remarks on weakly positive sheaves. The weak positivity was first introduced by Viehweg for the study of the Iitaka subadditivity conjecture. By constructing some explicit examples, we show that it is not necessarily preserved by extension. We see that an almost nef vector bundle is not always weakly positive. This gives a negative answer to a question posed by Demailly, Peternell and Schneider.

This is a joint work with Sho Ejiri and Masataka Iwai. If time permits, Fujino will describe some recent weak positivity results coming from VMHS, and their applications. This part is a joint work with Taro Fujisawa.

Paul Hacking
University of Massachusetts

On the Morrison Cone Conjecture for Calabi-Yau 3-Folds

The Morrison cone conjecture asserts that the action of the birational automorphism group of a Calabi-Yau 3-fold on its movable cone admits a rational polyhedral fundamental domain; in particular, there are finitely many orbits of faces of the cone. Paul Hacking will present the following theorem of University of Massachusetts postdoc Wendelin Lutz: If the Morrison cone conjecture holds for a Calabi-Yau 3-fold X, then it holds for any Calabi-Yau 3-fold deformation equivalent to X.

Andreas Höring
Université Côte d’Azur

KLT Degenerations of Projective Spaces

Degenerations of projective spaces are a classical subject of complex algebraic geometry: if the central fiber is smooth, it is isomorphic to the projective space by a well-known result of Siu. Similar results hold if we assume that the hyperplane class extends as an ample Cartier divisor to the central fiber.

In this talk, Andreas Höring will discuss what happens if we assume that the central fiber is a Fano variety with klt singularities. We will see that there are many possibilities and their geometry depends on the stability of the tangent sheaf. This is joint work with Thomas Peternell.

Hiroshi Iritani
Kyoto University

Decomposition of Quantum Cohomology Under Blowups

Quantum cohomology is a deformation of the cohomology ring defined by counting rational curves. We expect a close relationship between quantum cohomology and birational geometry. When the quantum parameter q approaches an “extremal ray,” the spectrum of the quantum cohomology ring clusters in a certain way (predicted by the corresponding extremal contraction), inducing a decomposition of the quantum cohomology.

In this talk, Hiroshi Iritani will discuss such a decomposition for blowups: quantum cohomology of the blowup of X along a smooth center Z will decompose into QH(X) and (codim Z-1) copies of QH(Z). The proof relies on Fourier analysis of equivariant quantum cohomology.

Elham Izadi
UC San Diego

Szego Kernels and the Scorza Map on Moduli Spaces of Spin Curves

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.

Lena Ji
University of Michigan

Rationality Criteria for Conic Bundle Threefolds Over Non-Closed Fields

An algebraic variety over a field k is said to be rational if it is birational to projective space. If a variety is rational over k, then it is geometrically rational, i.e., it becomes rational over the algebraic closure of k. However, in general, the converse need not hold. Rationality over k is well-understood when the dimension is at most 2, but the picture is less clear starting in dimension 3. In this talk, we study rationality obstructions for geometrically rational threefolds. Recently, Hassett and Tschinkel and Benoist and Wittenberg refined the rationality obstruction of Clemens and Griffiths by introducing torsors over the intermediate Jacobian. Their results, together with work of Kuznetsov and Prokhorov, showed that this refined obstruction can be used to characterize k-rationality for Fano threefolds of Picard rank 1. We study the rationality question for a family of threefolds that have Picard rank 2 and admit conic bundle structures. The intermediate Jacobian torsor obstruction does not always characterize k-rationality in this setting, and we explain how the Brauer group of k plays a role. This work is joint with S. Frei, S. Sankar, B. Viray and I. Vogt.

János Kollár
Princeton University

Smoothing Algebraic Cycles Below the Middle Dimension

Hironaka proved that the Chow groups $CH_d(X)$ are generated by smooth subvarieties if $2d<dim X$ and $d\leq 3$. Recently this was extended to all $2d<\dim X$ (with Voisin). The aim of this lecture is to explain the methods and sketch the proof.

Adrian Langer
University of Warsaw

Projective Contact Log Varieties

After recalling some known results on contact varieties, Adrian Langer will talk about contact structures on smooth complex projective log varieties. Langer will show how to study log contractions of contact log varieties using generalizations of some standard results on the loci of rational curves. To do so, Langer also needs to study more general contact structures on some special Lie algebroids. Langer will also show how such contact structures appear as natural generalizations of known contact structures on quasi-projective varieties.

Radu Laza
Stony Brook University

Remarks on Calabi-Yau Degenerations

It is a question of great interest to construct meaningful compactifications for the moduli of algebraic varieties of a specified type. For varieties of general type, and Fano type a fairly complete understanding of the compactification problem was obtained recently via the KSBA theory and respectively K-stability. The remaining case, that of K-trivial varieties turns out to be particularly challenging and the same time very interesting. After reviewing what we know in this case (especially new results, due to Alexeev-Engel for K3 surfaces), Radu Laza will propose a canonical minimal compactification for the K-trivial case and discuss some evidence towards it. (Versions of this conjecture previously occur in work of Ambro/Fujino/Shokurov, Odaka, and respectively GGLR.) The point of view taken here is that of Hodge theory. The talk is based on some joint work with R. Friedman. It is also closely related to joint work with Kollár, Saccà and Voisin [KLSV18] and respectively Green, Griffiths and Robles [GGLR20].

Brian Lehmann
Boston College

Restriction Theorems for Curves
Slides (PDF)

Let X be a smooth projective variety and let E be a vector bundle on X. A common way to analyze E is to fix a family of curves C on X and to study the restrictions of E to C.

In this talk, Brian Lehmann will give several qualitative statements describing the behavior of these restrictions. This is joint work with Eric Riedl and Sho Tanimoto.

Konstantin Loginov
HSE University

Finite Abelian Subgroups in the Space Cremona Group

Finite abelian groups are one of the simplest objects studied in algebra. In turn, rational varieties form a reasonably simple class of varieties considered in algebraic geometry. However, the question of which finite abelian groups can act on rational (or rationally connected) varieties, is far from being an easy question. In dimension 2 the answer to this question was given by A. Beauville and J. Blanc. Loginov will consider this question in dimension three.

Shigeru Mukai
Kyoto University

Vinberg Surface of Discriminant 3 and Cubic 4-Folds with Many Cusps

Vinberg (1983) studied two K3 surfaces of Picard number 20 and determined the structure of their (infinite) automorphism groups. As a higher dimensional analogue, Shigeru Mukai will discuss the birational automorphism groups Bir(X) of holomorphic symplectic manifolds. Mukai will explain how the group Bir(X) enlarges when X becoming from Vin3, one of two Vinberg surfaces, to its Hilbert square Vin3^[2] and to O’Grady type 10-fold Vin3^[OG]. If time permits, Mukai will also discuss some interesting phenomena observed when taking mod 3 reduction of these algebraic varieties.

Mihnea Popa
Harvard University

On the Topology and Hodge Theory of Singular Varieties

Mihnea Popa will describe recent progress in understanding the filtered de Rham (or Du Bois) complex of a complex algebraic variety, both in terms of general properties, and as a tool for the definition and study of refined classes of singularities. Popa will also explain how one can use these developments to deduce basic results on the topology of singular varieties.

Christian Schnell
Stony Brook University

A Hodge-Theoretic Proof of Hwang’s Theorem

Christian Schnell will explain a Hodge-theoretic proof for Hwang’s theorem, which says that if the base of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is smooth, then it must be projective space. The result is contained in a joint paper with Ben Bakker from last fall.

Stefan Schreieder
Leibniz Universität Hannover

Curves on Powers of Hyperelliptic Jacobians

For a curve of genus greater than or equal to four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a Jacobian of a curve. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties. The latter is closely related to the problem whether cubic threefolds are stably rational. This is joint work with Olivier de Gaay Fortman.

Yuri Tschinkel
Simons Foundation & New York University

Equivariant Birational Geometry
Slides (PDF)

Yuri Tschinkel will discuss new results and constructions in equivariant birational geometry.

Sho Tanimoto
Nagoya University

Campana Rationally Connectedness and Weak Approximation

Campana and Abramovich introduced the notion of Campana points which interpolate between rational points and integral points. Recently there are extensive activities on arithmetic geometry of Campana points and many conjectures have been proposed. In this talk, we discuss Campana curves/sections in the geometric setting. Campana conjectured that any klt Fano orbifold is Campana rationally connected. Assuming this conjecture, we prove that weak approximation at good places holds in the setting of Campana sections. This is a conjectural generalization of a theorem by Hassett and Tschinkel. Key tools to this theorem are log geometry and the notion of moduli stack of stable log maps. Finally, we verify our conjecture for certain classes of orbifolds. This is work in progress which is joint with Qile Chen and Brian Lehmann.

Olivier Wittenberg
Université Sorbonne Paris Nord

Levels of Function Fields of Real Varieties

Let X be a smooth real algebraic variety of dimension d. It has been known since Artin that -1 can be written as a sum of squares in the function field of X if and only if X has no real point. Under the hypothesis that X has no real point, what is then the minimum number of squares needed for this? We exhibit a link between this question and the geometry and cohomology of X, by showing that Pfister’s upper bound 2^d can be improved under various sets of assumptions on X. This is joint work with Olivier Benoist.

Registration and Travel Assistance
Ovation Travel Group
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Meeting Questions and Assistance
Meghan Fazzi
Manager, Events and Administration, MPS, Simons Foundation
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