Shifted Symplectic Structures on Derived Foliations
par
Salle Johnson
Institut de Mathématiques de Toulouse
Principal link: https://rendez-vous.renater.fr/private/Victor_Alfieri_thesis_id_20cbd2eb90e34da8efc21f59eec8ba25ee2a4838e60dccef1d631c916d8af955__w46vhpkfdz8q_f40e35-542c2e-aa381a
Password: 796206
Jury members:
M. Pavel SAFRONOV, Rapporteur, The University of Edinburgh
M. Joost NUITEN, Examinateur, Université de Toulouse
Mme Wendy LOWEN, Examinatrice, Universiteit Antwerpen
M. Jean-Baptiste TEYSSIER, Examinateur, Sorbonne Université
M. Frédéric DÉGLISE, Examinateur, CNRS Rhône Auvergne
M. Bertrand TOEN, Directeur de thèse, CNRS Occitanie Ouest
Invited member:
M. Marco ROBALO, Co-encadrant de thèse, Sorbonne Université
Abstract:
In this thesis, we work in derived algebraic geometry in characteristic zero and study derived foliations and shifted symplectic structures on such objects. Our first main result is an existence theorem for derived foliations on mapping stacks. More precisely, if S is a base derived stack, X is a proper-schematic, flat, and local complete intersection derived stack over S, and Y is a relative derived Deligne-Mumford stack over S equipped with a derived foliation, then the derived mapping stack Map_S(X,Y) itself inherits, under natural representability hypotheses, a derived foliation. In order to establish this result, we first show that the infinity-category of relative derived foliations can be realized as a full subcategory of an infinity-category of derived stacks equipped with extra structure, and then prove that the Weil restriction along a proper-schematic, flat, and local complete intersection morphism preserves this structure. This yields, more generally, a push-forward functor for derived foliations, together with an explicit description of their tangent complexes after push-forward. As applications, we obtain derived foliations on certain derived moduli spaces, including derived moduli spaces of stable curves and derived Hilbert schemes.
The second part of the thesis introduces the new notions of shifted symplectic structures on derived foliations and Lagrangian derived foliations. We then prove existence theorems for these structures: an intersection theorem for Lagrangian derived foliations in an n-shifted symplectic derived stack, producing an (n-1)-shifted symplectic derived foliation, and a symplectic push-forward theorem, showing that the push-forward of an n-shifted symplectic derived foliation along a d-oriented morphism naturally carries an (n-d)-shifted symplectic structure.