MPS Conference on Higher Dimensional Geometry, May 610, 2024
Meeting Summary
The final (sixth) inperson Higher Dimensional Geometry Conference supported by the Simons Foundation was held from Monday May 6th to Friday May 10th, 2024, at the Simons Center in Stony Brook. The conference had 22 speakers: Frederic Campana (Nancy), Adrien Dubouloz (Poitiers), Hélène Esnault (Berlin), Enrica Floris (Poitiers), Osamu Fujino (Kyoto), Paul Hacking (Seattle), Andreas Höring (Nice), Hiroshi Iritani (Kyoto), Elham Izadi (San Diego), Lena Ji (Ann Arbor), János Kollár (Princeton), Adrian Langer (Warsaw), Radu Laza (Stony Brook), Brian Lehmann (Boston), Konstantin Loginov (Moscow), Shigeru Mukai (Kyoto), Mihnea Popa (Boston), Christian Schnell (Stony Brook), Stefan Schreieder (Hannover), Sho Tanimoto (Nagoya), Yuri Tschinkel (New York) and Olivier Wittenberg (Paris). The research level of the conference was very high, and there was considerable interaction amongst the participants. During every coffee and lunch break, we had multiple discussions, which continued after the talks were over.
János Kollár opened the conference on Monday with an excellent talk about his resent results on the generalization of old problem of Hironaka about smoothing algebraic cycles. Recall that Hironaka proved that the Chow group of ddimensional cycles on a ndimensional smooth variety X is generated by smooth subvarieties if 2d
During the final day of the conference, Friday, Osamu Fujino spoke on weakly positive sheaves, and Mihnea Popa spoke on the topology and Hodge theory of singular varieties. In fact, Osamu presented an example (discovered by him with Sho Ejiri and Masataka Iwai) of an almost nef vector bundle that is not weakly positive, which gives a negative answer to a question posed by Demailly, Peternell and Schneider, and Mihnea described recent progress in understanding the filtered de Rham (or Du Bois) complex of a complex algebraic variety.
After the last talk, we learned a tragic news that Jim Simons passed away on this day. We were all deeply saddened by this. However, our series of conferences on Higher Dimensional Geometry was a great success — and this serves as a testament to Jim Simons’ vison of collaboration in advancing the frontiers of math and the physical sciences. All of our conferences were very well attended with many young researchers among participants and speakers. The final conference was not an exception. All coffee breaks and lunch breaks were combined with rich mathematical discussion, which has already resulted in many new collaboration projects.
Organizers:
Paolo Cascini, Imperial College
Ivan Cheltsov, University of Edinburgh
Christopher Hacon, University of Utah
Rob Lazarsfeld, Stony Brook University
James McKernan, UCSD
Burt Totaro, UCLA
Chenyang Xu, Princeton University
Meeting Goals:
The Simons Conference on Higher Dimensional Geometry (May 610, Stony Brook) will focus on recent developments in algebraic geometry. Topics include Kstability, varieties defined in positive and mixed characteristic, CalabiYau type varieties, foliations and topology of varieties.

Agenda
Monday, May 6^{th}
8:30 AM CHECKIN & BREAKFAST 9:30 AM János Kollár  Smoothing Algebraic Cycles Below the Middle Dimension 10:30 AM BREAK 11:00 AM Christian Schnell  A HodgeTheoretic Proof of Hwang's Theorem 12:00 PM LUNCH 1:15 PM Brian Lehmann  Restriction Theorems for Curves 2:15 PM BREAK 2:45 PM Lena Ji  Rationality Criteria for Conic Bundle Threefolds Over NonClosed Fields 3:45 PM BREAK 4:15 PM Paul Hacking  On The Morrison Cone Conjecture for CalabiYau 3Folds 5:15 PM DAY ONE CONCLUDES Tuesday, May 7^{th}
8:30 AM CHECKIN & BREAKFAST 9:30 AM Hélène Esnault  A NonAbelian Version of Deligne's Fix Part Theorem 10:30 AM BREAK 11:00 AM Hiroshi Iritani  Decomposition of Quantum Cohomology Under Blowups 12:00 PM LUNCH 1:15 PM Konstantin Loginov  Finite Abelian Subgroups in the Space Cremona Group 2:15 PM BREAK 2:45 PM Stefan Schreieder  Curves on Powers of Hyperelliptic Jacobians 3:45 PM BREAK 4:15 PM Olivier Wittenberg Levels 0f Function Fields of Real Varieties 5:15 PM DAY TWO CONCLUDES Wednesday, May 8^{th}
8:30 AM CHECKIN & BREAKFAST 9:30 AM Shigeru Mukai  Vinberg Surface of Discriminant 3 and Cubic 4Folds with Many Cusps 10:30 AM BREAK 11:00 AM Adrien Dubouloz  Additive Group Actions, Polar Cylinders and Rigidity of BrieskonPham Hypersurfaces 12:00 PM LUNCH 1:15 PM Adrian Langer  Projective Contact Log Varieties 2:15 PM BREAK 2:45 PM Enrica Floris  On Algebraic Subgroups of The Cremona Group 3:45 PM BREAK 4:15 PM Radu Laza  Remarks on CalabiYau Degenerations 5:15 PM DAY THREE CONCLUDES Thursday, May 9^{th}
8:30 AM CHECKIN & BREAKFAST 9:30 AM Frederic Campana  On a conjecture of Mihnea Popa 10:30 AM BREAK 11:00 AM Elham Izadi  Szego Kernels and the Scorza Map on Moduli Spaces of Spin Curves 12:00 PM LUNCH 1:15 PM Yuri Tschinkel  Equivariant Birational Geometry 2:15 PM BREAK 2:45 PM Andreas Horing  KLT Degenerations of Projective Spaces 3:45 PM BREAK 4:15 PM Sho Tanimoto  Campana Rationally Connectedness and Weak Approximation 5:15 PM DAY FOUR CONCLUDES Friday, May 10^{th}
8:30 AM CHECKIN & BREAKFAST 9:30 AM Osamu Fujino  Some Remarks on Weakly Positive Sheaves 10:30 AM BREAK 11:00 AM Mihnea Popa  On The Topology and Hodge Theory Oof Singular Varieties 12:00 PM LUNCH 1:00 PM MEETING CONCLUDES 
Abstracts
Frédéric Campana
University of LorraineOn a conjecture of Mihnea Popa
Popa conjectured that if $f:X\to Y$ is a submersive projective morphism with connected fibers between complex quasiprojective 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
CRNSAdditive Group Actions, Polar Cylinders and Rigidity of BrieskornPham Hypersurfaces
Adrien Dubouloz will give an overview of two recent and developing advances concerning applications of the correspondence between anticanonical polar cylinders in Fano varieties and homogeneous actions of the additive group on the spectra of their anticanonical rings: one concerns the study of automorphism groups of socalled BrieskornPham affine hypersurfaces via a reduction to wellformed 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 BerlinA NonAbelian Version of Deligne’s Fix Part Theorem
We prove a nonabelian 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) monodromyweight 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 PoitiersOn 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 UniversitySome 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 MassachusettsOn the Morrison Cone Conjecture for CalabiYau 3Folds
The Morrison cone conjecture asserts that the action of the birational automorphism group of a CalabiYau 3fold 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 CalabiYau 3fold X, then it holds for any CalabiYau 3fold deformation equivalent to X.
Andreas Höring
Université Côte d’AzurKLT 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 wellknown 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 UniversityDecomposition 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 Z1) copies of QH(Z). The proof relies on Fourier analysis of equivariant quantum cohomology.
Elham Izadi
UC San DiegoSzego 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 thetachracteristic 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 welldefined in all genera. We determine the limits of the Scorza correspondence at generic points of the vanishing thetanull 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 thetacharacteristic. We further show that limit of the Scorza quartic at abelian varieties with vanishing thetanulls is twice the quadric tangent cone to the theta divisor at the vanishing thetanull.
Lena Ji
University of MichiganRationality Criteria for Conic Bundle Threefolds Over NonClosed 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 wellunderstood 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 krationality 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 krationality 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 UniversitySmoothing 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 WarsawProjective 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 quasiprojective varieties.
Radu Laza
Stony Brook UniversityRemarks on CalabiYau 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 Kstability. The remaining case, that of Ktrivial 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 AlexeevEngel for K3 surfaces), Radu Laza will propose a canonical minimal compactification for the Ktrivial 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 CollegeRestriction 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 UniversityFinite 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 UniversityVinberg Surface of Discriminant 3 and Cubic 4Folds 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 10fold 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 UniversityOn 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 UniversityA HodgeTheoretic Proof of Hwang’s Theorem
Christian Schnell will explain a Hodgetheoretic 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 HannoverCurves 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 ColemanOort 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 UniversityEquivariant Birational Geometry
Slides (PDF)Yuri Tschinkel will discuss new results and constructions in equivariant birational geometry.
Sho Tanimoto
Nagoya UniversityCampana 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 NordLevels 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.

Contacts
Registration and Travel Assistance
Ovation Travel Group
[email protected]
(917) 4088384 (24Hours)
www.ovationtravel.comMeeting Questions and Assistance
Meghan Fazzi
Manager, Events and Administration, MPS, Simons Foundation
[email protected]
(212) 5246080