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Review of Poisson Geometry
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Poisson modules and degeneracy loci

Profile image of Marco GualtieriMarco Gualtieri

2013, Proceedings of the London Mathematical Society

https://doi.org/10.1112/PLMS/PDS090
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33 pages

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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 ≤ 2k locus of a Fano Poisson manifold always has dimension ≥ 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 Feȋgin and Odesskiȋ, where the degeneracy loci are given by the secant varieties of elliptic normal curves.

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