Final Report Summary - MOTZETA (Motivic zeta functions and the monodromy conjecture)
The most exciting developments in mathematics occur when seemingly distinct fields of research start to interact in novel ways. In our project, we have discovered such interactions between number theory, geometry, and mirror symmetry, a mysterious geometric phenomenon that was first observed in the context of mathematical physics (string theory).
One of the most influential proposals to explain the geometric foundations of mirror symmetry is the Strominger-Yau-Zaslow conjecture, which predicts the existence of a special type of structure (now commonly called an SYZ fibration) on the geometric objects where mirror symmetry occurs. A groundbreaking idea of Kontsevich and Soibelman was that such SYZ fibrations should appear naturally in the context of non-archimedean geometry, a branch of geometry that was initially designed to solve problems in number theory. By building a bridge to yet another part of geometry (birational geometry and the minimal model programme) we have been able to prove the existence of non-archimedean SYZ fibrations and to verify some of their expected properties. This circle of ideas has then led us to a solution of a longstanding open problem on Igusa zeta functions (Veys’ conjecture from 1999), unveiling new connections between number theory and geometry.
Solutions of important problems often open new perspectives and unlock unexpected applications. The novel techniques we have developed to understand Igusa zeta functions have proven to be quite powerful in at least two other directions. We have first applied them to enumerative geometry, which deals with various types of counting problems for geometric objects. Our main result in this context is a proof of the Davison-Meinhardt conjecture from 2011 on motivic nearby fibers of weighted homogeneous polynomials. Our method has also yielded a conceptual short proof of the integral identity conjecture of Kontsevich and Soibelman from 2008, one of the technical cornerstones in motivic Donaldson-Thomas theory, which was originally proven by Lê Quy Thuong in 2015 by means of a related but less flexible method. In a second direction, we have used motivic techniques to answer a longstanding question of fundamental importance in algebraic geometry: specialization of stable rationality in families of algebraic varieties. This is a very active field of research, and we are confident that our novel methods will lead to further breakthroughs.
One of the most influential proposals to explain the geometric foundations of mirror symmetry is the Strominger-Yau-Zaslow conjecture, which predicts the existence of a special type of structure (now commonly called an SYZ fibration) on the geometric objects where mirror symmetry occurs. A groundbreaking idea of Kontsevich and Soibelman was that such SYZ fibrations should appear naturally in the context of non-archimedean geometry, a branch of geometry that was initially designed to solve problems in number theory. By building a bridge to yet another part of geometry (birational geometry and the minimal model programme) we have been able to prove the existence of non-archimedean SYZ fibrations and to verify some of their expected properties. This circle of ideas has then led us to a solution of a longstanding open problem on Igusa zeta functions (Veys’ conjecture from 1999), unveiling new connections between number theory and geometry.
Solutions of important problems often open new perspectives and unlock unexpected applications. The novel techniques we have developed to understand Igusa zeta functions have proven to be quite powerful in at least two other directions. We have first applied them to enumerative geometry, which deals with various types of counting problems for geometric objects. Our main result in this context is a proof of the Davison-Meinhardt conjecture from 2011 on motivic nearby fibers of weighted homogeneous polynomials. Our method has also yielded a conceptual short proof of the integral identity conjecture of Kontsevich and Soibelman from 2008, one of the technical cornerstones in motivic Donaldson-Thomas theory, which was originally proven by Lê Quy Thuong in 2015 by means of a related but less flexible method. In a second direction, we have used motivic techniques to answer a longstanding question of fundamental importance in algebraic geometry: specialization of stable rationality in families of algebraic varieties. This is a very active field of research, and we are confident that our novel methods will lead to further breakthroughs.