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- W4386624922 abstract "Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $omega$. Then we have a natural double complex $overline{partial}+overline{partial}^Lambda$, where $overline{partial}^Lambda$ denotes the symplectic adjoint of the $overline{partial}$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $omega$. In cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $partialoverline{partial}$-Lemma, namely the $overline{partial}, overline{partial}^Lambda$-Lemma, which can be characterized in terms of Bott--Chern and Aeppli cohomologies associated to the above double complex. We obtain Nomizu type theorems for the Bott--Chern and Aeppli cohomologies and we show that the $overline{partial}, overline{partial}^Lambda$-Lemma is stable under small deformations of $omega$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $partialoverline{partial}$-Lemma then the $overline{partial}, overline{partial}^Lambda$-Lemma is stable." @default.
- W4386624922 created "2023-09-13" @default.
- W4386624922 creator A5027295211 @default.
- W4386624922 creator A5056101043 @default.
- W4386624922 date "2020-04-19" @default.
- W4386624922 modified "2023-09-29" @default.
- W4386624922 title "Cohomologies of complex manifolds with symplectic $(1,1)$-forms" @default.
- W4386624922 doi "https://doi.org/10.48550/arxiv.2004.08841" @default.
- W4386624922 hasPublicationYear "2020" @default.
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