$\def\partialol{\bar{\partial}}$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 [$\href{https://www.worldscientific.com/doi/abs/10.1142/S0129167X18500957}{29}$], 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.

中文翻译：

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具有辛$(1,1)$-形式的复流形的上同调

$\def\partialol{\bar{\partial}}$设$(X, J)$是一个复流形，具有非退化光滑$d$闭$(1,1)$-形式$\omega$ 。那么我们有一个自然的双复数 $\partialol+ \partialol^\Lambda$，其中 $\partialol^\Lambda$ 表示 $\partialol$ 运算符的辛伴随。我们研究 $X$ 的 Dolbeault 上同调群关于辛形式 $\omega$ 的硬 Lefschetz 条件。在[$\href{https://www.worldscientific.com/doi/abs/10.1142/S0129167X18500957}{29}$]中，我们证明了这样的条件等价于$\partialol\partialol的某种辛类似物$-Lemma，即$\partialol\partialol^\Lambda$-Lemma，可以用与上述双复形相关的Bott–Chern和Aeppli上同调来表征。我们获得了 Bott–Chern 和 Aeppli 上同调的 Nomizu 型定理，并证明 $\partialol\partialol^\Lambda$-引理在 $\omega$ 小变形下是稳定的，但在复杂结构小变形下不稳定。然而，如果我们进一步假设$X$满足$\partialol\partialol$-引理，那么$\partialol\partialol^\Lambda$-引理是稳定的。