POK0

Varieties with trivial canonical bundle, also known as K-trivial varieties, possess an extremely rich geometry and continue to fascinate algebraic and complex geometers in many ways. In recent years, there have been major breakthroughs both in the theory itself (e.g. a generalization of the Beauville-Bogomolov decomposition theorem to the singular setting) and in related areas (e.g. birational geometry, analytic methods, Kähler-Einstein techniques, deformation theory) that have already been applied successfully to the study of K-trivial varieties. The goal of the project is to exploit these recent advances and apply various state-of-the-art techniques in order to attack several fundamental questions concerning K-trivial varieties, thus verifying in this framework general conjectures that guide the study of algebraic varieties. We focus more specifically on various positivity problems such as: (a) the Morrison--Kawamata cone conjecture and applications to minimal models; (b) the study of linear systems on K-trivial varieties and their properties: base loci and global generation, local positivity, abundance-conjecture-type problems, Fujita-conjecture type problems; (c) projective geometry of hyperkähler manifolds: higher syzygies and projective models of such varieties. To achieve our goal, we bring together a group of researchers with different and complementary expertise, relying both on well-established successful collaborations and creating the conditions for new ones, to enhance cooperation between France and Germany.