论文标题
简单正常越过复杂表面与微不足道的规范捆绑包的差分几何全球平滑表面
Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle
论文作者
论文摘要
令$ x $成为一个简单的普通横码（SNC）紧凑型复杂表面，带有微不足道的规范捆绑包，其中包括三重交叉点。我们证明，如果$ x $是$ d $ semessistable，那么在不同的几何意义上就会存在一个平滑家族。这可以解释为弗里德曼（Friedman），卡瓦玛塔（Kawamata-namikawa），费尔滕·菲利普·鲁达（Felten-Filip-Ruddat，chan-leung-ma等）在代数几何形状中引起的平稳性结果的差异几何类似物。该证明是基于$ x $的单个基因座的明确构造本地平滑的，而第一作者的存在结果是全球平滑$ x $的全球平滑形式的结果。特别是，这些体积形式作为非线性椭圆偏微分方程的解决方案。 作为一个应用程序，我们提供了几个$ d $ semistisss的SNC复合面的示例，这些示例具有微不足道的规范捆绑包，包括双曲线，这些曲线可与复杂的Tori，主要的Kodaira表面和$ K3 $表面相平稳。我们还提供了几个复杂表面，包括三重点，这些示例可使$ k3 $表面平滑。
Let $X$ be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if $X$ is $d$-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of $X$, and the first author's existence result of holomorphic volume forms on global smoothings of $X$. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of $d$-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces and $K3$ surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to $K3$ surfaces.
