Poisson modules and degeneracy loci

TL;DR

This paper introduces a new notion of residue for Poisson modules, explores its relation to degeneracy loci, and provides evidence supporting Bondal's conjecture, especially for Fano fourfolds.

Contribution

It defines a novel residue concept for Poisson modules and applies it to study degeneracy loci, offering new insights and confirming Bondal's conjecture in specific cases.

Findings

Residue of Poisson modules relates to degeneracy loci.

Bondal's conjecture holds for Fano fourfolds.

Degeneracy loci include secant varieties of elliptic curves.

Abstract

In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincar\'e residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical line bundle of a Poisson manifold and its degeneracy loci---where the rank of the Poisson structure drops. As an application, we provide new evidence in favour of Bondal's conjecture that the rank \leq 2k locus of a Fano Poisson manifold always has dimension \geq 2k+1. In particular, we show that the conjecture holds for Fano fourfolds. We also apply our techniques to a family of Poisson structures defined by Fe\u{\i}gin and Odesski\u{\i}, where the degeneracy loci are given by the secant varieties of elliptic normal curves.