Floer cohomology and pencils of quadrics

TL;DR

This paper explores the symplectic relationship between hyperelliptic curves and pencils of quadrics, establishing a derived equivalence in Fukaya categories that informs low-dimensional topology and Floer homology computations.

Contribution

It introduces a derived equivalence linking Fukaya categories of curves and quadrics, advancing understanding of Floer homology in 3-manifolds fibred by genus two curves.

Findings

Derived equivalence between Fukaya categories of curves and quadrics

Determination of instanton Floer homology for certain 3-manifolds

Insights into symplectic geometry and low-dimensional topology

Abstract

There is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces. We investigate symplectic aspects of this relationship, with a view to applications in low-dimensional topology. We construct a derived equivalence between the Fukaya category of a curve and the nilpotent summand of the Fukaya category of the associated complete intersection of two quadrics. This essentially determines the instanton Floer homology of a 3-manifold fibred by genus two curves.