Projective Structures in Positive Characteristic

Speakers: Yasuhiro Wakabayashi (Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University)\n\nIn the spirit of Klein’s Erlangen program, a projective structure equipsa manifold with a geometry locally modeled on projective space, withtransition maps given by projective linear transformations. In theholomorphic setting, the existence and classification of such structureshave a long history, with foundational work by Gunning, Kobayashi, andOchiai, and subsequent contributions by Dumitrescu and others.In this talk, I will discuss analogues of projective structures onalgebraic varieties in positive characteristic, formulated usingFrobenius morphisms. These structures share many features with theirholomorphic counterparts, while also exhibiting phenomena peculiar topositive characteristic.A central theme will be the finiteness phenomena that arise for suchstructures on algebraic curves, as illustrated by explicit countingresults due to Mochizuki, Lange–Pauly, and Osserman. A more refinedstudy of these phenomena leads naturally to the moduli theory of suchstructures. The enumerative aspects of this theory have connections withthe combinatorics of graphs and polytopes, conformal field theoryassociated with affine Lie algebras, and the Gromov–Witten theory ofGrassmannians.\n\nhttps://indico.math.cnrs.fr/event/17046/