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arXiv · 1203.4293

Poisson modules and degeneracy loci

Marco Gualtieri·Brent Pym
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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 \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.

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Marco Gualtieri, Brent Pym. 2012-03-20. Poisson modules and degeneracy loci. https://doi.org/10.1112/plms%2Fpds090

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