Periodic Reporting for period 1 - BTMG (Birational and Tropical Methods in Geometry)
Reporting period: 2018-09-03 to 2020-09-02
Inside mathematics, algebraic geometry is the study of algebraic varieties – these are spaces given by solutions to systems of polynomial equations in several variables. Classifying algebraic varieties is very hard and since the birth of this subject, extremely elaborate ideas have been developed to improve the understanding of algebraic varieties. One natural idea is to associate to a class of algebraic varieties a simpler object that captures enough geometric information. Elaborate examples of such are enumerative invariants or Chow rings.
One point of view this project took is to study algebraic varieties through their enumerative invariants which are given by counting 1-dimensional subobjects - curves - with fixed properties leading to Gromov-Witten invariants. Instead of studying the invariants directly, with various collaborators (Pierrick Bousseau, Andrea Brini, Jinwon Choi, Tom Graber, Sheldon Katz, Helge Ruddat, Nobuyoshi Takahashi) we found new relations that express potentially difficult to compute invariants though simpler ones, in particular by relating log and local Gromov-Witten invariants as well as log and local BPS invariants. Log Gromov–Witten theory is a very active domain of current research that is being developed by several groups, notably around Mark Gross (Cambridge), Dan Abramovich (Brown) and Bernd Siebert (UT Austin). This enumerative theory is central to the Gross–Siebert program for proving mirror symmetry, and it has wide-ranging and deep applications to algebraic geometry. In addition to this, we developed new tools to compute the invariants on either side of the correspondences.
The study of these new correspondences has lead to a flurry of research on it and and related questions as is evidenced by the growing number of citations that the papers written on this grant already have. There is now a small research community studying these and related questions. This work has consequences for string theory, which advances a unified theory of the inner workings of the physical world. The invariants that we study describe some of these physical systems and the new relations give new insight into string theory.
Another point of view I took in this project in collaboration with Christian Böhning and Hans-Christian Graf von Bothmer is to study varieties through their Chow rings. These are sophisticated invariants of algebraic varieties, but extremely difficult to understand. The first step of the method we propose is to take a limit of the variety to a simpler variety. This works by ''perturbing'' the equations of the variety to obtain a simpler one. Then, we study the prelog Chow ring of the limit and we relate this ring to the Chow ring of the original variety.
The prelog Chow ring R of the limit variety, which is simpler, can be computed explicitly and we introduced a computer implementable algorithm to do so. A key ingredient is that R remembers enough information from the Chow ring of the original variety so that properties of the original variety can be studied in R. In all examples we have calculated, we were even able to relate the entire Chow ring of the original variety to R and we are working towards being able to give a full solution in the general case. This is important for algebraic geometry due to the difficulty of calculations of Chow rings.
One point of view this project took is to study algebraic varieties through their enumerative invariants which are given by counting 1-dimensional subobjects - curves - with fixed properties leading to Gromov-Witten invariants. Instead of studying the invariants directly, with various collaborators (Pierrick Bousseau, Andrea Brini, Jinwon Choi, Tom Graber, Sheldon Katz, Helge Ruddat, Nobuyoshi Takahashi) we found new relations that express potentially difficult to compute invariants though simpler ones, in particular by relating log and local Gromov-Witten invariants as well as log and local BPS invariants. Log Gromov–Witten theory is a very active domain of current research that is being developed by several groups, notably around Mark Gross (Cambridge), Dan Abramovich (Brown) and Bernd Siebert (UT Austin). This enumerative theory is central to the Gross–Siebert program for proving mirror symmetry, and it has wide-ranging and deep applications to algebraic geometry. In addition to this, we developed new tools to compute the invariants on either side of the correspondences.
The study of these new correspondences has lead to a flurry of research on it and and related questions as is evidenced by the growing number of citations that the papers written on this grant already have. There is now a small research community studying these and related questions. This work has consequences for string theory, which advances a unified theory of the inner workings of the physical world. The invariants that we study describe some of these physical systems and the new relations give new insight into string theory.
Another point of view I took in this project in collaboration with Christian Böhning and Hans-Christian Graf von Bothmer is to study varieties through their Chow rings. These are sophisticated invariants of algebraic varieties, but extremely difficult to understand. The first step of the method we propose is to take a limit of the variety to a simpler variety. This works by ''perturbing'' the equations of the variety to obtain a simpler one. Then, we study the prelog Chow ring of the limit and we relate this ring to the Chow ring of the original variety.
The prelog Chow ring R of the limit variety, which is simpler, can be computed explicitly and we introduced a computer implementable algorithm to do so. A key ingredient is that R remembers enough information from the Chow ring of the original variety so that properties of the original variety can be studied in R. In all examples we have calculated, we were even able to relate the entire Chow ring of the original variety to R and we are working towards being able to give a full solution in the general case. This is important for algebraic geometry due to the difficulty of calculations of Chow rings.
The work falls into three categories which correspond to three sets of collaborations. The first is a collaboration with Jinwon Choi (Sookmyung Women's University Seoul), Sheldon Katz (University of Illinois Urbana-Champaign) and Nobuyoshi Takahashi (University of Hiroshima) and lead to three papers. The second collaboration is with Christian Böhning (University of Warwick) and Hans-Christian Graf von Bothmer (University of Hamburg) for which we wrote two articles. The third is a collaboration with Andrea Brini (University of Sheffield) and Pierrick Bousseau (University of Paris-Saclay) and led to three papers so far. An introduction to these articles is given in Part B of the report.
In the work 'Stable maps to Looijenga pairs' with Pierrick Bousseau and Andrea Brini, we vastly generalize the previous correspondences to also include invariants coming from counting punctured curves, sheaves and quiver representations. This shows that different enumerative theories of a priori rather distinct geometries are profoundly related. Besides translating structures from one theory to another, this gives new tools to compute counts of punctured curves (open Gromov-Witten invariants), which has long remained elusive, even though these invariants are of great importance to mirror symmetry. We expect this work to be fundamental to the understanding of enumerative geometry.
Based on our previous joint work, in ongoing work with Christian Böhning and Hans-Christian Graf von Bothmer, we combine our methods with the criterion of non-stable rationality of Colliot-Thélène and Voisin. Combining the two we are working towards showing non-stable rationality in a variety of new cases further advancing the state of the art of this rapidly advancing field of inquiry.
Based on our previous joint work, in ongoing work with Christian Böhning and Hans-Christian Graf von Bothmer, we combine our methods with the criterion of non-stable rationality of Colliot-Thélène and Voisin. Combining the two we are working towards showing non-stable rationality in a variety of new cases further advancing the state of the art of this rapidly advancing field of inquiry.