Singular support of categories

          @misc{ pirsa_PIRSA:16040075,
            doi = {10.48660/16040075},
            url = {https://pirsa.org/16040075},
            author = {Arinkin, Dima},
            keywords = {Mathematical physics},
            language = {en},
            title = {Singular support of categories},
            publisher = {Perimeter Institute},
            year = {2016},
            month = {apr},
            note = {PIRSA:16040075 see, \url{https://pirsa.org}}
          }
          

Abstract

In many situations, geometric objects on a space have some kind of singular support, which refines the usual support. For instance, for smooth X, the singular support of a D-module (or a perverse sheaf) on X is as a conical subset of the cotangent bundle; similarly, for quasi-smooth X, the singular support of a coherent sheaf on X is a conical subset of the cohomologically shifted cotangent bundle. I would like to describe a higher categorical version of this notion. Let X be a smooth variety, and let Z be a closed conical isotropic subset of the cotangent bundle of X. I will define a 2-category associated with Z; its objects may be viewed as `categories over X with singular support in Z'. In particular, if Z is the zero section, we simply consider categories over Z in the usual sense. This talk is based on a joint project with D.Gaitsgory. The project is motivated by the local geometric Langlands correspondence; I plan to sketch the relation with the Langlands correspondence in the talk.