Cours spécialisé (GA, TA, Lie)
Contact : mauri à imj-prg.fr
Notes de cours : https://sites.google.com/view/mirko-mauri
Lagrangian fibrations, or integrable systems, are degenerate versions of fibrations into abelian varieties. They play a crucial role in physics, classical mechanics, representation theory, algebraic geometry and in the geometric Langland program. The decomposition theorem for Lagrangian fibrations attains a particularly elegant and structured form, explored in particular by Ngô and Arinkin, and which will be a central topic of the course. We will also study the compatibility of the decomposition with cup product in cohomology, which is at the core of the third proof of the famous P=W conjecture (Maulik--Shen--Yin, 2023). The coarse will be the opportunity to explore cutting-edge research, but at the same times it serves as a perfect motivation to introduce several general tools and objects in algebraic geometry, e.g., abelian varieties, compactified Jacobians, symplectic varieties, Fourier–Mukai transform, perverse sheaves.