Symplectic topology of Lagrangian submanifolds of ℂPn with intermediate minimal Maslov numbers

Abstract

Abstract We examine symplectic topological features of a certain family of monotone Lagrangian submanifolds in ℂPn . First we give cohomological constraints on a Lagrangian submanifold in ℂPn whose first integral homology is p-torsion. In the case where (n, p) = (5,3), (8, 3), we prove that the cohomologies with coefficients in ℤ2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3)ℤ3) and SU(3)/ℤ3, respectively. Then we calculate the Floer cohomology with coefficients in ℤ2 of a monotone Lagrangian submanifold SU(p)/ℤ p in C P p 2 − 1 . ${\mathbb C}P^{p^2-1}.$