论文标题
具有符号$（1,1）$的复杂流形的共同体 - 表格
Cohomologies of complex manifolds with symplectic $(1,1)$-forms
论文作者
论文摘要
令$（x，j）$为一个复杂的歧管，具有非限制的平滑$ d $ claped $（1,1）$ - 表格$ω$。然后，我们有一个天然的双重复杂$ \叠加{\ partial}+\ edline {\ partial}^λ$，其中$ \ edrowline {\ partial}^λ$表示$ \ edimletline {\ partial} $ - 运算符的$ \ edimeptic eptiont。我们研究了$ x $的Dolbeault共同体组上的硬性左手条件，相对于$ x $ $ω$。 In \cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $\partial\overline{\partial}$-Lemma, namely the $\overline{\partial}\, \overline{\partial}^Λ$-Lemma, which can be characterized in terms of Bott--Chern and Aeppli与上述双重复合物相关的辅助学。我们获得了针对Bott-Chern和Aeppli共同体的Nomizu类型定理，并且我们表明$ \ operline {\ partial} \，\ overline {\ partial}^λ$ -lemma在$ω$的小变形下是稳定的，但在$ω$的小变形下，但在较小的复合结构的小变形下稳定。但是，如果我们进一步假设$ x $满足$ \ partial \ edline {\ partial} $ - lemma，则$ \ overline {\ partial} \，\ + overline {\ partial}^λ$ -lemma是稳定的。
Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $ω$. Then we have a natural double complex $\overline{\partial}+\overline{\partial}^Λ$, where $\overline{\partial}^Λ$ 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 $ω$. In \cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $\partial\overline{\partial}$-Lemma, namely the $\overline{\partial}\, \overline{\partial}^Λ$-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 $\overline{\partial}\, \overline{\partial}^Λ$-Lemma is stable under small deformations of $ω$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $\partial\overline{\partial}$-Lemma then the $\overline{\partial}\, \overline{\partial}^Λ$-Lemma is stable.
