紧共形Einstein Kähler流形的一条模曲线
A moduli curve for compact conformally-Einstein Kähler manifolds
摘要：
我们分类四元组$(M,g,m,τ)$，其中$(M,g)$是一个紧凑的凯勒流形的复杂尺寸$m>2$与一个非常数的功能$τ$ $M$，使共形相关的度量$g/τ^2$，定义在任何地方$τ\ne 0$，是爱因斯坦。结果是$M$是紧Kähler-Einstein流形$(N,h)$上的全纯$CP^1$丛的全空间。所讨论的四元组构成了四个不相交的族：一个是众所周知的，具有局部可约的Kähler度量$g$;第二个是由Bérard Bergery（1982）发现的，并且处处具有$τ\ne 0$;第三个是通过解析延拓的形式与第二个相关的，并且类似于一些已知的Kähler曲面度量;第四个族，只存在于奇数复维数$m\ge 9$中。我们的分类使用{ \it模量曲线}，它是$R^2$中代数曲线的子集$\mathcal{C}$，取决于$m$。$\mathcal{C}$中的一个点$(u,v)$自然地与任何具有上述所有性质的$(M,g,m,τ)$相关联，除了$M$的紧性，被"垂直“紧性的较弱要求所取代。人们可以从这个$(u,v)$再加上一些其他数据，其中包括一个用于$CP^1$束$M$的Kähler-Einstein基$(N,h)$，依次重建$M,g$和$τ$。以这种方式从具有紧$M$的$(M,g,m,τ)$产生的点$(u,v)$形成$\mathcal{C}$的可数无限子集。
英文摘要：
We classify quadruples $(M,g,m,τ)$ in which $(M,g)$ is a compact Kähler manifold of complex dimension $m>2$ with a nonconstant function $τ$ on $M$ such that the conformally related metric $g/τ^2$, defined wherever $τ\ne 0$, is Einstein. It turns out that $M$ then is the total space of a holomorphic $CP^1$ bundle over a compact Kähler-Einstein manifold $(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with Kähler metrics $g$ that are locally reducible; a second, discovered by Bérard Bergery (1982), and having $τ\ne 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kähler surface metrics; and a fourth family, present only in odd complex dimensions $m\ge 9$. Our classification uses a {\it moduli curve}, which is a subset $\mathcal{C}$, depending on $m$, of an algebraic curve in $R^2$. A point $(u,v)$ in $\mathcal{C}$ is naturally associated with any $(M,g,m,τ)$ having all of the above properties except for compactness of $M$, replaced by a weaker requirement of ``vertical'' compactness. One may in turn reconstruct $M,g$ and $τ$ from this $(u,v)$ coupled with some other data, among them a Kähler-Einstein base $(N,h)$ for the $CP^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m,τ)$ with compact $M$ form a countably infinite subset of $\mathcal{C}$.
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