Poisson modules and degeneracy loci

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 Poincaré 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\uıgin and Odesski\uı, where the degeneracy loci are given by the secant varieties of elliptic normal curves.

33 pages

References in corpus (1)

• Geometric syzygies of elliptic normal curves and their secant varieties

Cited by in corpus (13)

• Stable generalized complex structures
• Constructions and classifications of projective Poisson varieties
• Elliptic singularities on log symplectic manifolds and Feigin--Odesskii Poisson brackets
• Symplectic groupoids of log symplectic manifolds
• A characterization of diagonal Poisson structures
• Poisson structures of divisor-type
• Mixed Hodge structures in log symplectic geometry
• A local Torelli theorem for log symplectic manifolds
• Deformations of holomorphic pseudo-symplectic Poisson manifolds
• Abelian Complex Structures and Generalizations
• Rational Morita equivalence for holomorphic Poisson modules
• The Orbit Method for Poisson Orders
• Poisson structure on manifolds with corners