Symplectic geometry of orbifolds and Diophantine equations

Abstract

We study symplectic geometry of orbifolds, especially which are necessary to extend the Lagrangian intersection Floer theory to the one of orbifold setting. First, we give another definition of the Maslov indices of bundle pairs via curvature integral of L-orthogonal unitary connection. This definition naturally extends to the one of orbi-bundle pairs with interior singularities. Secondly, we investigate the notion of orbifold embedding. When the target orbifold is a global quotient of a smooth manifold by the action of a Lie group G, we show that orbifold embeddings are equivariant with G-equivariant immersions. In the last part of the dissertation, we compute quantum cohomology of elliptic P1 orbifolds via classifying holomorphic orbi-spheres in those orbifolds. Interestingly, we find that these orbi-spheres have an one-to-one correspondence with the solutions of certain Diophantine equations depending on the lattice structures on the universal covers of elliptic P1 orbifolds constructed from the preimages of three singular points.