Mathematics – Symplectic Geometry
Scientific paper
2010-10-19
Mathematics
Symplectic Geometry
Scientific paper
Let X be the Fermat quintic threefold. The set of real solutions L forms a Lagrangian submanifold of X. Multiplying the homogeneous coordinates of X by various fifth roots of unity gives automorphisms of X; the images of L under these automorphisms defines a family of 625 different Lagrangian submanifolds, called real Lagrangians. In this paper we try to calculate the Floer cohomology between all pairs of these Lagrangians. We are able to complete most of the calculations, but there are a few cases we cannot do. The basic idea is to explicitly describe some low energy moduli spaces and then use this knowledge to calculate the differential on the E_2 page of the standard spectral sequence for Floer cohomology. It turns out that this is often enough to calculate the cohomology completely. Several techniques are developed to help describe these low energy moduli spaces, including a formula for the Maslov index, a formula for the obstruction bundle, and a way to relate holomorphic strips and discs to holomorphic spheres. The real nature of the Lagrangians is crucial for the development of these techniques.
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