非Kähler几何中的无限冒泡
Infinite bubbling in non-Kählerian geometry
摘要：
在非Kähler紧流形的全纯族$(X_b)_{b\in B}$中，表示固定2-同调类的全纯曲线一般不构成真族。在非凯勒几何中，这种基本困难的深层根源是{ \it面积爆炸}现象：在固定的2-同调类中，曲线$C_b\subset X_b$的面积可以发散为$b\to b_0$。这种现象经常出现在VII类曲面的变形理论中。例如，众所周知，任何极小GSS曲面$X_0$都是单参数族的简单爆破的主Hopf曲面$(X_z)_{z\in D\setminus\{0\}}$的退化，因此人们得到非真的例外因子族$E_z\subset X_z$，其面积发散为$z\to 0$。我们的主要目标是详细研究这种非properness现象的情况下，第七类表面。我们将证明，在一定的技术假设下，$E_z$在泛覆盖$\widetilde X_z$中的提升$\widetilde E_z$确实收敛到$\widetilde X_0$中的有效因子$\widetilde E_0$，但这个极限因子不是紧的。证明了该极限因子总是有界于$\widetilde X_0$的伪凸端，并且当$X_0$是一个具有整体球壳的极小曲面时，该极限因子是由一个无穷级数的{ \it紧致}有理曲线给出的，其系数可以显式计算.这种现象--紧致曲线族退化为紧致曲线的无限并集--应该称为{PDMATHAMN无限冒泡}。我们相信，这样的分解结果适用于任何家庭的第七类表面的通用纤维是一个吹了主要的霍普夫曲面。这一说法将对第七类表面的分类产生重要影响。
英文摘要：
In a holomorphic family $(X_b)_{b\in B}$ of non-Kählerian compact manifolds, the holomorphic curves representing a fixed 2-homology class do not form a proper family in general. The deep source of this fundamental difficulty in non-Kähler geometry is the {\it explosion of the area} phenomenon: the area of a curve $C_b\subset X_b$ in a fixed 2-homology class can diverge as $b\to b_0$. This phenomenon occurs frequently in the deformation theory of class VII surfaces. For instance it is well known that any minimal GSS surface $X_0$ is a degeneration of a 1-parameter family of simply blown up primary Hopf surfaces $(X_z)_{z\in D\setminus\{0\}}$, so one obtains non-proper families of exceptional divisors $E_z\subset X_z$ whose area diverge as $z\to 0$. Our main goal is to study in detail this non-properness phenomenon in the case of class VII surfaces. We will prove that, under certain technical assumptions, a lift $\widetilde E_z$ of $E_z$ in the universal cover $\widetilde X_z$ does converge to an effective divisor $\widetilde E_0$ in $\widetilde X_0$, but this limit divisor is not compact. We prove that this limit divisor is always bounded towards the pseudo-convex end of $\widetilde X_0$ and that, when $X_0$ is a a minimal surface with global spherical shell, it is given by an infinite series of {\it compact} rational curves, whose coefficients can be computed explicitly. This phenomenon - degeneration of a family of compact curves to an infinite union of compact curves - should be called {\it infinite bubbling}. We believe that such a decomposition result holds for any family of class VII surfaces whose generic fiber is a blown up primary Hopf surface. This statement would have important consequences for the classification of class VII surfaces.
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