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A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION

Part of: Surfaces and higher-dimensional varieties Deformations of analytic structures Compact analytic spaces Symplectic geometry, contact geometry

Published online by Cambridge University Press:  09 November 2018

Ziv Ran
Ziv Ran*
Affiliation:
UC Math Department, Big Springs Road Surge Facility, Riverside CA 92521 US, USA (ziv.ran@ucr.edu). URL: http://math.ucr.edu/∼ziv/
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Abstract

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

Keywords

Poisson structurelog-symplectic manifolddeformation theorylog complexmixed Hodge theory

MSC classification

Primary: 32G07: Deformations of special (e.g. CR) structures 53D17: Poisson manifolds; Poisson groupoids and algebroids
Secondary: 14J40: $n$-folds ($n>4$) 32J27: Compact Kähler manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1509 - 1519
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Copyright
© Cambridge University Press 2018

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