Construction of Kähler surfaces with constant scalar curvature
Construction of Kähler surfaces with constant scalar curvature
复制标题
具有恒定标量曲率的凯勒曲面的构造
DOI：
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发表时间：
2009
期刊：
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通讯作者：
M. Singer
中科院分区：
文献类型：
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作者：
Yann Rollin;M. Singer
本文的目的是给出复曲面上常数量曲率（CSC）的Kähler度量的一种新构造.为了介绍我们的结果，让我们引入术语“正CSC”表示“常数正标量曲率”，“零CSC”表示“（常数）零标量曲率”，“负CSC”表示“常数负标量曲率”。我们的构造产生了许多例子，但在本介绍中，我们将集中讨论CP 1 × CP 1的Xk：= k重爆破。我们注意到，如果k ≥ 1，则Xk也可以看作是CP 2的k + 1重爆破。当然，上面对Xk的描述并没有固定它的复杂结构：这将取决于爆破中心的位置。我们的第一个结果给出了积极的CSC Kähler度量在一个家庭的Kähler类Xk，k = 6，7，8，并为某些选择复杂的结构。我们注意到，如果k ≤ 7，则Xk是Fano，Tian [13]和其他人的工作给出了Xk上的正Kähler-Einstein度量。我们的结果是新的，因为它产生了X8上的CSC度量以及X6和X7上的CSC度量，这些度量在Kähler类中与c1（X）“任意远”：定理A。对k = 6，7，8，存在CP 1 × CP 1的k点爆破X，不存在非平凡全纯向量场，且具有以下性质.设F = {x} × CP 1是CP 1 × CP 1的一般有理曲线。对任意常数c > 0和ε > 0，存在X上的严格正常数量曲率的Kähler度量ω，使得F [ω]2 − c ≤ ε。（1.1）
The aim of this note is to present a new construction of Kähler metrics of constant scalar curvature (CSC) on complex surfaces. In order to introduce our results, let us introduce the terms “positive CSC” to mean “constant positive scalar curvature”, “zero CSC” for “(constant) zero scalar curvature” and “negative CSC” for “constant negative scalar curvature”. Our construction gives rise to many families of examples, but in this introduction we shall focus on Xk := k-fold blow-up of CP1 × CP1. We note that if k ≥ 1 then Xk can also be viewed as a k + 1-fold blow-up of CP2. Of course, the above description of Xk does not fix its complex structure: this will depend on the location of the centres of the blow-ups. Our first result gives positive CSC Kähler metrics in a family of Kähler classes on Xk , for k = 6, 7, 8, and for certain choices of complex structure. We note that if k ≤ 7 then Xk is Fano and the work of Tian [13] and others gives positive Kähler–Einstein metrics on Xk . Our result is new in that it produces CSC metrics on X8 as well as CSC metrics on X6 and X7 in Kähler classes that are “arbitrarily far” from c1(X): Theorem A. For k = 6, 7, 8, there exists a k-point blow-up X of CP1 × CP1 with no non-trivial holomorphic vector field and the following properties. Let F = {x} ×CP1 be a generic rational curve of CP1 × CP1. For every constant c > 0 and ε > 0, there exists a Kähler metric ω of strictly positive constant scalar curvature on X such that ∣∣∣∣ [ω] · F √[ω]2 − c ∣∣∣∣ ≤ ε. (1.1)