论文标题
紧凑型复杂歧管的高页
Higher-Page Hodge Theory of Compact Complex Manifolds
论文作者
论文摘要
在紧凑型$ \ partial \ bar \ partial $ -manifold $ x $上，一个人具有hodge分解：de rham cohomology群体分为pure-type类的子空间，为$ h_ {dr}^k（x）= \ oplus_ = \ oplus_ {p+q = k = k} $ h^{p，\，q}（x）$在dolbeault cohomology群体上是同构的，$ h _ {\ bar \ partial}^{p，\，\，q}（x）$。对于任意的非负整数$ r $，我们介绍了page-$ r $ - $ \ $ \ poartial \ bar \ partial $ -manifolds，通过需要霍奇分解的类似物来保留紧凑的复杂歧管$ x $，当由$ e_ {r+1}^{p，\，q}（x）$在$（r+1）$ - frölicher频谱序列的$ x $的$（r+1）$中。 Page- $ r $ - $ \ partial \ bar \ partial $ -manifolds的类别与通常的$ \ partial \ bar \ partial $ -manifolds相吻合时，当$ r = 0 $时，但可能会增加$ r $。我们提供两种应用程序。一方面，我们在各种共同体学向量空间的维度方面给出了Page-$ r $ - $ \ partial \ bar \ partial $ - property的纯粹数值表征。另一方面，我们获得了几类示例，包括所有可行的Nilmanifolds以及某些Solvmanifolds和Abelian Nilmanifolds的家族。此外，在标准结构（例如爆破和变形）下，这一新类别的行为有一般的结果。
On a compact $\partial\bar\partial$-manifold $X$, one has the Hodge decomposition: the de Rham cohomology groups split into subspaces of pure-type classes as $H_{dR}^k (X)=\oplus_{p+q=k}H^{p,\,q}(X)$, where the $H^{p,\,q}(X)$ are canonically isomorphic to the Dolbeault cohomology groups $H_{\bar\partial}^{p,\,q}(X)$. For an arbitrary nonnegative integer $r$, we introduce the class of page-$r$-$\partial\bar\partial$-manifolds by requiring the analogue of the Hodge decomposition to hold on a compact complex manifold $X$ when the usual Dolbeault cohomology groups $H^{p,\,q}_{\bar\partial}(X)$ are replaced by the spaces $E_{r+1}^{p,\,q}(X)$ featuring on the $(r+1)$-st page of the Frölicher spectral sequence of $X$. The class of page-$r$-$\partial\bar\partial$-manifolds coincides with the usual class of $\partial\bar\partial$-manifolds when $r=0$ but may increase as $r$ increases. We give two kinds of applications. On the one hand, we give a purely numerical characterisation of the page-$r$-$\partial\bar\partial$-property in terms of dimensions of various cohomology vector spaces. On the other hand, we obtain several classes of examples, including all complex parallelisable nilmanifolds and certain families of solvmanifolds and abelian nilmanifolds. Further, there are general results about the behaviour of this new class under standard constructions like blow-ups and deformations.
