圆作用与调和态射的拓扑限制
Topological restrictions for circle actions and harmonic morphisms
摘要：
设$M^m$是一个紧致光滑流形，它有一个光滑的圆作用，其孤立不动点也是孤立奇点.则$M^m$的所有Pontryagin数为零，其Euler数为非负偶数.特别地，$M^m$具有签名零。由于具有一维纤维的非常数调和态射产生圆作用，我们有以下应用：（i）许多紧致流形，例如$CP^{n}$，$K3$曲面，$S^{2n}\times P_g$（$n\geq2$）其中$P_g$是亏格的闭曲面，$g\geq2$永远不可能是一维纤维的非常数调和态射的域，无论我们在它们上放置什么度量;（ii）设$(M^4,g)$是紧致可定向四维流形，$ϕ:(M^4,g)\to(N^3,h)$是非常数调和态射.假设下列断言之一成立：（1）$(M^4,g)$是半共形平坦的，其标量曲率为零，（2）$(M^4,g)$是爱因斯坦的，并且是半共形平坦的，（3）$(M^4,g,J)$是厄米-爱因斯坦的。然后，直到同素异义和黎曼覆盖，$ϕ$是平坦环面之间的正则投影$T^4\to T^3$。
英文摘要：
Let $M^m$ be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of $M^m$ are zero and its Euler number is nonnegative and even. In particular, $M^m$ has signature zero. Since a non-constant harmonic morphism with one-dimensional fibres gives rise to a circle action we have the following applications: (i) many compact manifolds, for example $CP^{n}$, $K3$ surfaces, $S^{2n}\times P_g$ ($n\geq2$) where $P_g$ is the closed surface of genus $g\geq2$ can never be the domain of a non-constant harmonic morphism with one-dimensional fibres whatever metrics we put on them; (ii) let $(M^4,g)$ be a compact orientable four-manifold and $ϕ:(M^4,g)\to(N^3,h)$ a non-constant harmonic morphism. Suppose that one of the following assertions holds: (1) $(M^4,g)$ is half-conformally flat and its scalar curvature is zero, (2) $(M^4,g)$ is Einstein and half-conformally flat, (3) $(M^4,g,J)$ is Hermitian-Einstein. Then, up to homotheties and Riemannian coverings, $ϕ$ is the canonical projection $T^4\to T^3$ between flat tori.
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