Journal of Symplectic Geometry

Volume 21 (2023)

Number 1

Cohomologies of complex manifolds with symplectic $(1,1)$-forms

Pages: 73 – 109



Adriano Tomassini (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli Studi di Parma, Italy)

Xu Wang (Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway)


$\def\partialol{\bar{\partial}}$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 $\partialol+ \partialol^\Lambda$, where $\partialol^\Lambda$ denotes the symplectic adjoint of the $\partialol$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $\omega$. In [$\href{}{29}$], we proved that such a condition is equivalent to a certain symplectic analogue of the $\partialol\partialol$-Lemma, namely the $\partialol\partialol^\Lambda$-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 $\partialol\partialol^\Lambda$-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 $\partialol\partialol$-Lemma then the $\partialol\partialol^\Lambda$-Lemma is stable.

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The first-named author is partially supported by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, by Project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics”, and by GNSAGA of INdAM.

Received 17 February 2021

Accepted 23 June 2022

Published 27 July 2023