Periodic Reporting for period 4 - MODSTABVAR (Moduli spaces of stable varieties and applications)
Okres sprawozdawczy: 2024-09-01 do 2025-08-31
Projective algebraic varieties are fundamental geometric objects of mathematics. By definition, they are common zero sets of multivariable polynomials, in the appropriate sense. As polynomials are the functions that can be assembled only using multiplication and addition, traces of projective varieties can be found already in the ancient greek's mathematics. For example pythagorean triples are points of a conic, a special type of projective varieties. The modern treatment of projective varieties is many times counted from Riemann's thesis, and ever since then, they have been central objets of mathematics. During this period of almost two centuries, a classification theory of these objects has crystallized as well. As a first approximation this classification theory has two steps. In the first step, called the Minimal Model Program, classification theory aims to decompose each projective variety to parts with purely "positive/zero/negative curvature". We call these parts the building blocks of the original projective variety. Here, the quotes mean that the definition of building blocks can be made really precise using curvature only when working over the complex numbers, otherwise one should take the curvature description only as an intuitive approximation of the actual definition. In the second step, called Higher dimensional moduli theory, classification aims to construct a moduli space for each type of building block. That is, it aims to construct a projective variety, the points of which parametrize all building blocks of given type.
The main object of the proposal is the above moduli space in the "negative curvature" case. In this case, the "negative curvature" varieties that the moduli space is supposed to parametrize are called stable varieties. They were originally introduced by Kollár and Shepherd-Barron, and they are also higher dimensional generalizations of the algebro-geometric notion of stable curves from many perspectives. Furthermore, over the complex numbers, stable varieties can be also defined surprisingly as the projective varieties admitting a negative curvature (singular) Kähler-Einstein metric by the work of Berman and Guenancia, or as the canonically polarized K-stable varieties by Odaka.
The fundamental objective of the project is to construct the coarse moduli space of stable surfaces with fixed volume over the integers (possibly excluding finitely many primes, not depending on the volume). In particular this involves showing the Minimal Model Program for varieties of dimension 3 the defining equations of which have integer coefficients (again possibly allowing denominators that are divisible only by primes from a fixed finite list). The main motivations are applications to general algebraic geometry and to the arithmetic of higher dimensional varieties. This main objective is then decomposed into 4 more specific objectives called birational objectives in the proposal. Besides the above main objective, the goal of the proposal is to further advance our knowledge around stable varieties, including connections to K-stable varieties. These latter goals of the proposal are labeled as additional/application objectives.
The main object of the proposal is the above moduli space in the "negative curvature" case. In this case, the "negative curvature" varieties that the moduli space is supposed to parametrize are called stable varieties. They were originally introduced by Kollár and Shepherd-Barron, and they are also higher dimensional generalizations of the algebro-geometric notion of stable curves from many perspectives. Furthermore, over the complex numbers, stable varieties can be also defined surprisingly as the projective varieties admitting a negative curvature (singular) Kähler-Einstein metric by the work of Berman and Guenancia, or as the canonically polarized K-stable varieties by Odaka.
The fundamental objective of the project is to construct the coarse moduli space of stable surfaces with fixed volume over the integers (possibly excluding finitely many primes, not depending on the volume). In particular this involves showing the Minimal Model Program for varieties of dimension 3 the defining equations of which have integer coefficients (again possibly allowing denominators that are divisible only by primes from a fixed finite list). The main motivations are applications to general algebraic geometry and to the arithmetic of higher dimensional varieties. This main objective is then decomposed into 4 more specific objectives called birational objectives in the proposal. Besides the above main objective, the goal of the proposal is to further advance our knowledge around stable varieties, including connections to K-stable varieties. These latter goals of the proposal are labeled as additional/application objectives.
The team has worked on all the 4 birational objective mentioned above, and also on many of the additional and application objectives. The team could solve 3 of the 4 birational objectives, most application and additional objectives, and even newer questions that were not in the original proposal. However, we were not able to get hold of the birational objective "locally stable reduction". This is a famous unsolved problem that many have tried to solve, but so far it has been eluding everyone, including the team.
The "MMP for arithmetic 3-folds" objective was resolved in the article “Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic" by Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi (team member), Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek. This article appeared in the Publications Mathématiques de l'IHÉS.
The objective "Gluing theory for surfaces" was showed in the articles by Quentin Posva (team member) titled "Gluing theory for slc surfaces and threefolds in positive characteristic", "Gluing for stable families of surfaces in mixed characteristic" and "Abundance for slc surfaces over arbitrary fields".
The objective "Local Kawamata Viehweg vanishing" was showed in the article "On the properness of the moduli space of stable surfaces over ℤ[1/30]" by Emelie Arvidsson, Fabio Bernasconi and Zsolt Patakfalvi (team member).
Work in the direction of the application/additional objectives were considering the following topics.
1.) Wall crossing property of the moduli space of stable varieties in the article "Wall crossing for moduli of stable log varieties" by Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro and Zsolt Patakfalvi (team member). This appeared in the Annals of Mathematics.
2.) Alternative construction of the moduli space of stable varieties in the article
"Moduli of Q-Gorenstein pairs and applications" by Stefano Filipazzi (team member) and Giovanni Inchiostro.
3.) Geometry of Calabi-Yau varieties by the team member Stefano Filipazzi with the articles "Baily--Borel compactifications of period images and the b-semiampleness conjecture" (with Benjamin Bakker, Mirko Mauri and Jacob Tsimerman) and "Boundedness of some fibered K-trivial varieties" (with Philip Engel, François Greer, Mirko Mauri and Roberto Svaldi).
4.) Positive characteristic geometry by the team members Jefferson Baudin, Zsolt Patakfalvi and Javier-Carvajal Rojas. Sample articles are "A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves" by Jefferson Baudin, and the article "The Demailly--Peternell--Schneider conjecture is true in positive characteristic" by Zsolt Patakfalvi and Ejiri Sho.
5.) Mixed characteristic geometry by the team member Zsolt Patakfalvi. For example the article "Test ideals in mixed characteristic: a unified theory up to perturbation" (with Bhargav Bhatt, Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek) gives a systematic treatment of the mixed characteristic test ideals (the mixed characteristic versions of the multiplier ideals).
The "MMP for arithmetic 3-folds" objective was resolved in the article “Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic" by Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi (team member), Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek. This article appeared in the Publications Mathématiques de l'IHÉS.
The objective "Gluing theory for surfaces" was showed in the articles by Quentin Posva (team member) titled "Gluing theory for slc surfaces and threefolds in positive characteristic", "Gluing for stable families of surfaces in mixed characteristic" and "Abundance for slc surfaces over arbitrary fields".
The objective "Local Kawamata Viehweg vanishing" was showed in the article "On the properness of the moduli space of stable surfaces over ℤ[1/30]" by Emelie Arvidsson, Fabio Bernasconi and Zsolt Patakfalvi (team member).
Work in the direction of the application/additional objectives were considering the following topics.
1.) Wall crossing property of the moduli space of stable varieties in the article "Wall crossing for moduli of stable log varieties" by Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro and Zsolt Patakfalvi (team member). This appeared in the Annals of Mathematics.
2.) Alternative construction of the moduli space of stable varieties in the article
"Moduli of Q-Gorenstein pairs and applications" by Stefano Filipazzi (team member) and Giovanni Inchiostro.
3.) Geometry of Calabi-Yau varieties by the team member Stefano Filipazzi with the articles "Baily--Borel compactifications of period images and the b-semiampleness conjecture" (with Benjamin Bakker, Mirko Mauri and Jacob Tsimerman) and "Boundedness of some fibered K-trivial varieties" (with Philip Engel, François Greer, Mirko Mauri and Roberto Svaldi).
4.) Positive characteristic geometry by the team members Jefferson Baudin, Zsolt Patakfalvi and Javier-Carvajal Rojas. Sample articles are "A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves" by Jefferson Baudin, and the article "The Demailly--Peternell--Schneider conjecture is true in positive characteristic" by Zsolt Patakfalvi and Ejiri Sho.
5.) Mixed characteristic geometry by the team member Zsolt Patakfalvi. For example the article "Test ideals in mixed characteristic: a unified theory up to perturbation" (with Bhargav Bhatt, Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek) gives a systematic treatment of the mixed characteristic test ideals (the mixed characteristic versions of the multiplier ideals).
All the work was beyond the state of the art. See the above entry box on "work performed...".
The project has ended, so no new results are expected until the end of the project.
The project has ended, so no new results are expected until the end of the project.