绝热极限与复结构的变形
Adiabatic Limit and Deformations of Complex Structures
AI总结：
研究证明了全纯族的退化极限流形为Moishezon，并引入了与复流形相关的全纯向量束及放松的强Gauduchon度量概念。
摘要：
基于我们最近将绝热极限构造适应到复结构情况的适应，我们证明了任何全纯族的退化极限流形都是Moishezon的。两个新的成分，希望有独立的兴趣，被引入。第一个成分与每一个紧致复流形X，在每一个度数k，关联一个覆盖复数的全纯向量束，其秩等于X的k-阶Betti数。这个向量束，之前在文献中已给出代数构造，显示了Frölicher谱序列的退化页是X的d_h上同调的极限，当h∈C*趋近于0时，其中d_h=h∂+∂¯。然后，这个向量束的相对版本与每一个全纯族的紧致复流形关联。第二个成分是对强Gauduchon（sG）度量概念的放松，我们于2009年引入。对于给定的正整数r，一个在n维紧致复流形X上的Gauduchon度量γ被称为E_r-sG，如果∂γ^{n-1}在X的Frölicher谱序列的第r页上代表零上同调类。强Gauduchon度量与E_1-sG度量相同。
英文摘要：
Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new ingredients, hopefully of independent interest, are introduced. The first one associates with every compact complex manifold $X$, in every degree $k$, a holomorphic vector bundle over $\C$ of rank equal to the $k$-th Betti number of $X$. This vector bundle, previously given an algebraic construction in the literature, shows that the degenerating page of the Frölicher spectral sequence of $X$ is the holomorphic limit, as $h\in\C^\star$ tends to $0$, of the $d_h$-cohomology of $X$, where $d_h=h\partial + \bar\partial$. A relative version of this vector bundle is then associated with every holomorphic family of compact complex manifolds. The second ingredient is a relaxation of the notion of strongly Gauduchon (sG) metric that we introduced in 2009. For a given positive integer $r$, a Gauduchon metric $γ$ on an $n$-dimensional compact complex manifold $X$ is said to be $E_r$-sG if $\partialγ^{n-1}$ represents the zero cohomology class on the $r$-th page of the Frölicher spectral sequence of $X$. Strongly Gauduchon metrics coincide with $E_1$-sG metrics.
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