Exact Lagrangians from contracting $\mathbb{C}^*$-actions

TL;DR

This paper constructs families of non-isotopic exact Lagrangian submanifolds in certain symplectic manifolds with contracting $ ext{C}^*$-actions, revealing their Floer cohomologies and implications for symplectic cohomology.

Contribution

It introduces new non-isotopic Lagrangians in holomorphic symplectic manifolds with contracting actions, linking Floer cohomology to topological invariants and providing bounds on symplectic cohomology.

Findings

Floer cohomologies are topological, matching ordinary cohomologies.

Constructed non-isotopic Lagrangians in quasi-projective holomorphic symplectic manifolds.

Derived degree-wise lower bounds on symplectic cohomology.

Abstract

We obtain families of non-isotopic closed exact Lagrangian submanifolds in quasi-projective holomorphic symplectic manifolds that admit contracting $\mathbb{C}^*$-actions. We show that the Floer cohomologies of these Lagrangians are topological in nature, recovering the ordinary cohomologies of their intersection. Moreover, by using these Lagrangians and a version of Carrell-Goresky's integral decomposition theorem, we obtain degree-wise lower bounds on the symplectic cohomology of these spaces.