Mirror symmetry, Berkovich spaces and the Minimal Model Programme

点击翻译按钮获取中文摘要

Algebraic geometry studies the shapes of geometric objects that can be defined by means of polynomials equations, leading to profound interactions between number theory, algebra and geometry. Since the 1990s, physics has exerted a major and surprising influence on algebraic geometry through the theory of mirror symmetry, which grew out of string theory, a proposal to describe fundamental particles mathematically as tiny vibrating strings. Models of the universe in string theory take as input a specific type of object in algebraic geometry: a so-called Calabi-Yau variety, named after the mathematicians Eugenio Calabi and Shing-Tung Yau. These objects attracted the attention of physicists because of their special symmetry properties. The Calabi-Yau variety is responsible for the 6 "hidden dimensions" that are postulated in string theory to explain the fundamental properties of particles. Physicists soon realized that the Calabi-Yau variety is not uniquely determined by the physical model: rather, Calabi-Yau varieties seemed to come in "mirror pairs" giving rise to equivalent theories. Algebraic geometers were forced to take this idea very seriously after some spectacular applications to enumerative geometry (counting special types of curves living on Calabi-Yau varieties) around 1990. The main challenge for algebraic geometers was to provide mathematical foundations for these ideas, that is, give an exact definition of what it means to be a mirror pair, and devise techniques to construct such pairs. This is still an ongoing story, but much progress has been made. This project is concerned with one of the mathematical approaches to mirror symmetry, developed by Kontsevich and Soibelman: the non-archimedean approach to the Strominger-Yau-Zaslow (SYZ) conjecture. The SYZ conjecture is an ambitious attempt to give a geometric explanation of mirror symmetry, and it has been very influential in mathematics. Around 2000, Kontsevich and Soibelman had the groundbreaking insight that one should be able to find the structures predicted by the SYZ conjecture in a seemingly unrelated field: non-archimedean geometry, a branch of geometry and analysis that was originally designed to solve problems in number theory. In the last few years, I have realized an important part of Kontsevich and Soibelman's proposal, by introducing a new ingredient into the picture: the minimal model programme (MMP) in birational geometry. This programme is one of the most successful developments in algebraic geometry in the last 40 years; in 2018, the Cambridge mathematician Caucher Birkar received the Fields medal (the most prestigious award in mathematics) for his contributions to the MMP. The aim of the MMP is to classify all the geometric objects that arise in algebraic geometry. I have discovered that one can use non-archimedean geometry as a dictionary to transfer questions and results back and forth between the field of mirror symmetry and the MMP, thus providing new tools to study both fields simultaneously. The goal of this project is to further exploit these interactions between mirror symmetry, non-archimedean geometry, and birational geometry. In this way, I aim to prove some of the central conjectures in the non-archimedean approach to mirror symmetry, and to develop new tools to understand the MMP.