Symplectic cohomology and q-intersection numbers

Details Symplectic cohomology and q-intersection numbers Symplectic cohomology and q-intersection numbers

2010-05-27

arxiv.org/abs/1005.5156v3

Mathematics

Symplectic Geometry

32 pages, 9 figures, expanded introduction, added details of example 7.5, added discussion of signs

Scientific paper

Given a symplectic cohomology class of degree 1, we define the notion of an equivariant Lagrangian submanifold. The Floer cohomology of equivariant Lagrangian submanifolds has a natural endomorphism, which induces a grading by generalized eigenspaces. Taking Euler characteristics with respect to the induced grading yields a deformation of the intersection number. Dehn twists act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz fibrations give fully computable examples. A key step in computations is to impose the "dilation" condition stipulating that the BV operator applied to the symplectic cohomology class gives the identity. Equivariant Lagrangians mirror equivariant objects of the derived category of coherent sheaves.

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