Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles

Abstract

We study intersections of complex Lagrangian in complex symplectic manifolds, proving two main results. First, we construct global canonical perverse sheaves on complex Lagrangian intersections in complex symplectic manifolds for any pair of {\it oriented} Lagrangian submanifolds in the complex analytic topology. Our method uses classical results in complex symplectic geometry and some results from arXiv:1304.4508. The algebraic version of our result has already been obtained by the author et al. in arXiv:1211.3259 using different methods, where we used, in particular, the recent new theory of algebraic d-critical loci introduced by Joyce in arXiv:1304.4508. This resolves a long-standing question in the categorification of Lagrangian intersection numbers and it may have important consequences in symplectic geometry and topological field theory. Our second main result proves that (oriented) complex Lagrangian intersections in complex symplectic manifolds for any pair of Lagrangian submanifolds in the complex analytic topology carry the structure of (oriented) analytic d-critical loci in the sense of arXiv:1304.4508.