Reshetikhin-Turaev representations as Kähler local systems
Freitag, 6.5.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
From a joint work, partially in progress, with Louis Funar. In Orbifold Kähler Groups related to Mapping Class groups, arXiv:2112.06726, we constructed certain orbifold compactifications of the moduli stack of stable pointed curves labelled by an integer p such that the corresponding Reshetikhin Turaev representation of the mapping class group descends to a representation of the orbifold fundamental group. I will explain the construction of that orbifold and why it is uniformizable. I will then report on a work in progress on the uniformization of these orbifolds. I will sketch a proof of the steiness of its universal covering p odd large enough. An interesting new quantum topological consequence is that the image of the fundamental group of the smooth base of a non isotrivial complex algebraic family of smooth complete curves of genus greater than 2 by the Reshetikhin-Turaev representation is infinite (generalizing the Funar-Masbaum and the Koberda-Santharoubane-Funar-Lochak infiniteness theorems).
Lifting globally F-split surfaces over the Witt vectors
Freitag, 20.5.22, 10:30-11:30, Hörsaal II, Albertstr. 23b
Given a projective variety X over an algebraically closed field k of characteristic p, it is natural to understand the possible geometric and arithmetic obstructions to the existence of a lifting to characteristic zero. Motivated by the case of abelian manifolds and K3 surfaces, a folklore conjecture claims that ordinary Calabi-Yau manifolds should admit a lifting over the ring of Witt vectors W(k). I will report a joint work with I. Brivio, T. Kawakami and J. Witaszek where we show that globally F-split surfaces (which can be thought of as log Calabi-Yau surfaces that behave arithmetically well) are liftable over W(k) and we deduce several geometric consequences (as the Bogomolov bound on the number of singular points of klt del Pezzo F-split surfaces).