BTMG · Birational and Tropical Methods in Geometry

Birational and Tropical Methods in Geometry

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.

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