Higher Dimensional Geometry

Monday May 6th to Friday May 10th 2024

Organisers:

Paolo Cascini, Ivan Cheltsov, James McKernan and Chenyang Xu

Speakers




Schedule

Monday, May 6th

9:30 - 10:30 János Kollár

11:00 - 12:00 Christian Schnell

12:00 - 1:00 Lunch

1:15 - 2:15 Brian Lehmann

2:45 - 3:45 Lena Ji

4:15 - 5:15 Paul Hacking


Tuesday, May 7th

9:30 - 10:30 Hélène Esnault

11:00 - 12:00 Hiroshi Iritani

12:00 - 1:00 Lunch

1:15 - 2:15 Konstantin Loginov

2:45 - 3:45 Stefan Schreieder

4:15 - 5:15 Olivier Wittenberg


Wednesday, May 8th

9:30 - 10:30 Shigeru Mukai

11:00 - 12:00 Adrien Dubouloz

12:00 - 1:00 Lunch

1:15 - 2:15 Adrian Langer

2:45 - 3:45 Enrica Floris

4:15 - 5:15 Radu Laza


Thursday, May 9th

9:30 - 10:30 Frederic Campana

11:00 - 12:00 Elham Izadi

12:00 - 1:00 Lunch

1:15 - 2:15 Yuri Tschinkel

2:45 - 3:45 Andreas Horing

4:15 - 5:15 Sho Tanimoto


Friday, May 10th

9:30 - 10:30 Osamu Fujino

11:00 - 12:00 Mihnea Popa

12:00 - 1:00 Lunch


Titles and Abstracts

Frederic Campana

Title: On a conjecture of Mihnea Popa

Abstract: Popa conjectured that if $f:X\to Y$ is a submersive projective morphism with connected fibres between complex quasi-projective manifolds, the Logarithmic Kodaira dimensions of $X$ and $Y$ differ exactly by the Kodaira dimension of the fibres. Together with C. Schnell, he proved some important cases of this conjecture. We prove here, by a different approach, involving the `core map', that the conjecture holds when the fibres admit good minimal models.


Adrien Dubouloz

Title: Additive group actions, polar cylinders and rigidity of Brieskon-Pham hypersurfaces

Abstract: I 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

Title: Survey on properties of rigid local systems

Abstract: We’ll survey some properties of rigid local systems, based on our joint work with Michael Groechenig and in part with Johan de Jong.


Enrica Floris

Title: On algebraic subgroups of the Cremona group

Abstract: 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 regularisation theorem and the Minimal Model Program, to such a group we associate a rational Mori fibre space on which it acts regularly. In this talk, we will discuss the notion of maximal connected algebraic subgroups of the Cremona group, and its relation with the geometry of the associated Mori fibre spaces. This is a work in collaboration with A. Fanelli and S. Zimmermann.


Osamu Fujino

Title: Some remarks on weakly positive sheaves

Abstract: In this talk, I 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. In particular, we see that an almost nef vector bundle is not always weakly positive. This gives a negative answer to a question posed by Demaiily, Peternell, and Schneider. This is a joint work with Sho Ejiri and Masataka Iwai. If time permits, I would like to explain some recent weak positivity results coming from VMHS, and their applications. This part is a joint work with Taro Fujisawa.


Paul Hacking

Title: On the Morrison cone conjecture for Calabi--Yau 3-folds

Abstract: 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. I will present the following theorem of UMass 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 Hoering

Title: KLT degenerations of projective spaces

Abstract: Degenerations of projective spaces are a classical subject of complex algebraic geometry: if the central fibre 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 fibre. In this talk I will discuss what happens if we assume that the central fibre 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.


János Kollár

Title: Smoothing  algebraic cycles below the middle dimension

Abstract: 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.


Hiroshi Iritani

Title: Decomposition of quantum cohomology under blowups

Abstract: 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, I 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

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.


Lena Ji

Title: Rationality criteria for conic bundle threefolds over non-closed fields

Abstract: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–Tschinkel and Benoist–Wittenberg refined the Clemens–Griffiths rationality obstruction by introducing torsors over the intermediate Jacobian. Their results, together with work of Kuznetsov–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.


Adrian Langer

Title: Projective contact log varieties

Abstract: After recalling some known results on contact varieties, I will talk about contact structures on smooth complex projective log varieties. In particular, I 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 I also need to study more general contact structures on some special Lie algebroids. I will also show how such contact structures appear as natural generalizations of known contact structures on quasi-projective varieties.


Radu Laza

Title: Remarks on Calabi-Yau degenerations

Abstract: 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), I 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, Voisin [KLSV18] and respectively Green, Griffiths, and Robles [GGLR20].


Brian Lehmann

Title: Restriction theorems for curves

Abstract: 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 I will give several qualitative statements describing the behavior of these restrictions.  This is joint work with Eric Riedl and Sho Tanimoto.


Konstantin Loginov

Title: Finite abelian subgroups in the space Cremona group

Abstract: 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. In my talk I will consider this question in dimension three.


Shigeru Mukai

Title: Vinberg surface of discriminant 3 and cubic 4-folds with many cusps

Abstract: Vinberg(1983) studied two K3 surfaces of Picard number 20, and determined the structure of their (infinite) automorphism groups.  As a higher dimensional analogue I discuss the birational automorphism groups  Bir(X)  of holomorphic symplectic manifolds.   I 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 I will also discuss some interesting phenomena which we observe when taking mod 3 reduction of these algebraic varieties.


Mihnea Popa

Title: On the topology and Hodge theory of singular varieties

Abstract: I 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. I will also explain how one can use these developments to deduce basic results on the topology of singular varieties.


Christian Schnell

Title: A Hodge-theoretic proof of Hwang's theorem

Abstract: I 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

Title: Curves on powers of hyperelliptic Jacobians

Abstract:  For a curve of genus at least 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

Title: Equivariant birational geometry

Abstract: I will discuss new results and constructions in equivariant birational geometry.


Sho Tanimoto

Title: Campana rationally connectedness and weak approximation

Abstract: 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

Title: Levels of function fields of real varieties

Abstract: Let X be a smooth real algebraic variety of dimension d.  It has been knownsince 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.