A Bogomolov unobstructedness theorem for log-symplectic manifolds in general position

2017/05/23 by Ziv Ran, Ran, Ziv
Mathematics · #14J40 #32G07 #32J27 #53D17 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1705.08366

openalex publication_date 2017/05/23 · openalex created_date 2017/06/05 · openalex updated_date 2026/07/28

Abstract

We consider compact Kählerian manifolds X of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure Π which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor D(Π). We prove that (X, Π) has unobsrtuced deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on H² of the open symplectic manifold X∖ D(Π), and in fact coincides with this H² provided the Hodge number h^2,0_X=0, and finally that the degeneracy locus D(Π) deforms locally trivially under deformations of (X, Π). It has been pointed out that the general position hypothesis in the original paper is not strong enough and this is corrected in an appended erratum/corrigendum to the revised version.

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