Symplectic reduction and lagrangian submanifolds in Gr$(1, n)$

Abstract

We study lagrangian submanifolds of algebraic variety Gr(1, n) equipped with the Kahler form given by the Plucker embedding. We use the correspondence between lagrangian submanifolds of Gr(1, n) and lagrangian submanifolds of variety M_{n-k}, given by symplectic reduction Gr(1, n)//T^k for some specially chosen moment maps, which generate T^k action on Gr(1, n). We establish that in this way one finds many topological types, realized by lagrangian submanifolds, and then one counts that Gr(1, n) admits more than n different topological types of smooth lagrangian submanifolds.