Periodic Reporting for period 3 - MODSTABVAR (Moduli spaces of stable varieties and applications)
Berichtszeitraum: 2023-03-01 bis 2024-08-31
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. So far the team could solve 3 of the 4 birational objectives, and for 2 of these the articles are publicly available by now: “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, as well as 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".
Additionally the team also worked on plenty of application and additional objectives. For example the article "Wall crossing for moduli of stable log varieties" by Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro and Zsolt Patakfalvi (team member) clarifies how the stable moduli spaces change as we vary their parameters. This phenomenon is generally called wall-crossing, and it was first noted and described for moduli spaces describing physics phenomena. Another examples is the article "Moduli of Q-Gorenstein pairs and applications" by Stefano Filipazzi (team member) and Giovanni Inchiostro, which gives an alternative construction of the moduli space of stable varieties in characteristic zero. Besides the above mentioned articles, the team completed about 15 other ones, which contain either research on the additional and application objectives, or research that we hope will eventually help resolving the remaining birational objective.
Additionally the team also worked on plenty of application and additional objectives. For example the article "Wall crossing for moduli of stable log varieties" by Kenneth Ascher, Dori Bejleri, Giovanni Inchiostro and Zsolt Patakfalvi (team member) clarifies how the stable moduli spaces change as we vary their parameters. This phenomenon is generally called wall-crossing, and it was first noted and described for moduli spaces describing physics phenomena. Another examples is the article "Moduli of Q-Gorenstein pairs and applications" by Stefano Filipazzi (team member) and Giovanni Inchiostro, which gives an alternative construction of the moduli space of stable varieties in characteristic zero. Besides the above mentioned articles, the team completed about 15 other ones, which contain either research on the additional and application objectives, or research that we hope will eventually help resolving the remaining birational objective.
Progress beyond the state of the art: in general many mathematics articles contain significant new ideas, and the same is true for the articles under the present project. So, in some sense all articles written under the current project go beyond state of the are. Listing all these here would be impossible. Hence, we mention only one particular example that goes well beyond the state of the art. 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 contains completely novel methodology of showing statements about the classification theory of algebraic varieties of mixed characteristic. Fundamentally, the article is based on the theory of perfectoid rings and prismatic cohomology developed by Peter Scholze and Bhargav Bhatt. The new methods introduced by the article have been already applied by multiple other articles, and we hope that in the long run they will find many more applications.
Expected results until the end of the project: the team will keep on working on putting out the article on the 3rd solved birational objective, as well as solving the 4th one. The latter one is a rather difficult problem, so it is hard to predict the outcome, nevertheless the team will put in serious effort to resolve it. The team will also investigate further additional and application objectives.
Expected results until the end of the project: the team will keep on working on putting out the article on the 3rd solved birational objective, as well as solving the 4th one. The latter one is a rather difficult problem, so it is hard to predict the outcome, nevertheless the team will put in serious effort to resolve it. The team will also investigate further additional and application objectives.