Symplectic cohomology and q-intersection numbers

TL;DR

This paper introduces equivariant Lagrangian submanifolds in symplectic cohomology, defines a deformation of intersection numbers via Euler characteristics, and explores their properties and computations in specific geometric contexts.

Contribution

It defines equivariant Lagrangian submanifolds associated with symplectic cohomology classes and develops a framework for their intersection theory and symmetries, including Dehn twists.

Findings

Deformation of intersection numbers via Euler characteristics.

Examples in cotangent bundles and Lefschetz fibrations.

Dilation condition simplifies computations.

Abstract

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.