听到加权的射影平面的权重
Hearing the weights of weighted projective planes
摘要：
我们可以"听到“轨道的哪些性质，即，一个轨道的哪些拓扑和几何性质是由它的拉普拉斯谱决定的？我们考虑了一类四维Kähler轨道折叠的问题：带三个孤立奇点的加权射影平面$M:=\C P^2(N_1,N_2,N_3)$.我们表明，在$M$上的0-和1-形式的拉普拉斯算子的谱确定的权重$N_1$，$N_2$，和$N_3$。证明涉及使用几种技术，包括本地化等变上同调的热不变量的分析。我们证明了我们可以用欧拉示性数的知识来代替1-形式上的谱的知识，并得到相同的结果。最后，在确定了$N_1$、$N_2$和$N_3$的值之后，我们就可以知道$M$是否被赋予了极值凯勒度量。
英文摘要：
Which properties of an orbifold can we ``hear,'' i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes $M:=\C P^2(N_1,N_2,N_3)$ with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on $M$ determine the weights $N_1$, $N_2$, and $N_3$. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of $N_1$, $N_2$, and $N_3$, we can hear whether $M$ is endowed with an extremal Kähler metric.
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