A note on Bondal's conjecture

TL;DR

This paper demonstrates that certain connection vector fields are not locally Hamiltonian unless the Poisson structure vanishes, providing evidence for Bondal's conjecture on degeneracy loci dimensions.

Contribution

It establishes a key property of connection vector fields in Poisson geometry and supports Bondal's conjecture with new theoretical insights.

Findings

Connection vector fields are not locally Hamiltonian unless the Poisson structure is zero.

Provides evidence supporting Bondal's conjecture on degeneracy loci.

Advances understanding of Poisson structures on Fano manifolds.

Abstract

We prove that the connection vector fields associated to ample Poisson line bundles are not locally hamiltonian unless the Poisson structure is zero. We use this result to provide further evidence on Bondal's conjecture regarding the dimensions of the degeneracy loci of a holomorphic Fano Poisson manifold.