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The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

Part of: Deformations of analytic structures Compact analytic spaces Birational geometry

Published online by Cambridge University Press:  15 March 2021

Patrick Graf  and
Martin Schwald
Patrick Graf
Affiliation:
Lehrstuhl für Mathematik I, Universität Bayreuth, 95440Bayreuth, Germany; E-mail: patrick.graf@uni-bayreuth.de URL: www.graficland.uni-bayreuth.de
Martin Schwald
Affiliation:
Fakultät für Mathematik, Universität Duisburg–Essen, 45117Essen, Germany; E-mail: martin.schwald@uni-due.de URL: www.esaga.uni-due.de/martin.schwald/

Abstract

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Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

MSC classification

Primary: 32J27: Compact Kähler manifolds
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 32G05: Deformations of complex structures
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e24
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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