某些谱序列在$E_2$处的退化
Degeneration at $E_2$ of Certain Spectral Sequences
摘要：
本文对第二步的特征空间$E_2^{p,\,q}$提出了一个Hodge理论，该空间要么是紧致复流形$X$的Frölicher谱序列，要么是紧致复流形上的一对互补正则全纯叶理$(N,\,F)$的谱序列.其主要思想是在$X$上引入一个与给定的厄米特度量相关联的拉普拉斯型算子，该算子的核在每一个二度$(p,\,q)$中同构于$E_2^{p,\,q}$。令人惊讶的是，这个算子不是微分算子，因为它涉及调和投影，尽管它依赖于某些微分算子。然后，我们使用此霍奇同构的$E_2^{p,\,q}$ $E_2$考虑在每两种情况下，在$X$上的某些度量的存在的谱序列的退化给出充分条件。例如，在Frölicher的情况下，我们证明在$E_2$退化，如果存在一个SKT度量$ω$（即，使得$\partial\bar\partialω=0$），其挠率与由$ω$定义的椭圆算子$Δ' + Δ"$的谱间隙相比是小的。在叶状的情况下，我们得到退化$E_2$的假设下，涉及拉普拉斯$Δ'_N$和$Δ'_F$与分裂$\partial = \partial_N + \partial_F$诱导的叶状结构。
英文摘要：
We propose a Hodge theory for the spaces $E_2^{p,\,q}$ featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold $X$ or in the spectral sequence associated with a pair $(N,\,F)$ of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on $X$ whose kernel in every bidegree $(p,\,q)$ is isomorphic to $E_2^{p,\,q}$ in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for $E_2^{p,\,q}$ to give sufficient conditions for the degeneration at $E_2$ of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on $X$. For example, in the Frölicher case we prove degeneration at $E_2$ if there exists an SKT metric $ω$ (i.e. such that $\partial\bar\partialω=0$) whose torsion is small compared to the spectral gap of the elliptic operator $Δ' + Δ"$ defined by $ω$. In the foliated case, we obtain degeneration at $E_2$ under a hypothesis involving the Laplacians $Δ'_N$ and $Δ'_F$ associated with the splitting $\partial = \partial_N + \partial_F$ induced by the foliated structure.
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