Complex symplectic structures and the ∂∂¯-lemma

In this paper, we study complex symplectic manifolds, i.e., compact complex manifolds X which admit a holomorphic (2, 0)-form σ which is d -closed and non-degenerate, and in particular the Beauville–Bogomolov–Fujiki quadric Q σ associated with them. We will show that if X satisfies the ∂ ∂ ¯ -lemma,...

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Bibliographic Details
Published inAnnali di matematica pura ed applicata Vol. 197; no. 1; pp. 139 - 151
Main Authors Cattaneo, Andrea, Tomassini, Adriano
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 2018
Springer Nature B.V
Subjects
Manifolds
Mathematics
Mathematics and Statistics
32C35
Complex symplectic
53C15
Beauville–Bogomolov–Fujiki form
53C56
Lemma
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Summary:In this paper, we study complex symplectic manifolds, i.e., compact complex manifolds X which admit a holomorphic (2, 0)-form σ which is d -closed and non-degenerate, and in particular the Beauville–Bogomolov–Fujiki quadric Q σ associated with them. We will show that if X satisfies the ∂ ∂ ¯ -lemma, then Q σ is smooth if and only if h 2 , 0 ( X ) = 1 and is irreducible if and only if h 1 , 1 ( X ) > 0 .
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-017-0672-1