Structures symplectiques et de Poisson sur les champs en cat\'egories

Abstract

The purpose of this short note is to present two existence results concerning symplectic and lagrangian structures in the derived setting, in situations where the constructions of [Ca] and [PTVV] do not apply. For this we show that symplectic structures can be constructed out of Calabi-Yau structures on sheaves of dg-categories, or out of \emph{orientations} on sheaves of rigid dg-categories. These results follow from two main theorems: the HKR theorem and the cyclotomic aspect of traces in rigid infty-categories.