Rational blow-downs and smoothings of surface singularities

Rational homology disk smoothings of surface singularities; the exceptional cases

It is known (Stipsicz-Szabó-Wahl) that there are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a rational homology disk ($\mathbb{Q}$HD) smoothing, i.e., having a Milnor fibre with Milnor number zero. Some examples are found by an explicit "quotient construction", while others require the "Pinkham method". The fundamental group of the Milnor fibre has been known for all except the three exceptional families $\mathcal B_2^3(p), \mathcal C^3_2(p),$ and $\mathcal C^3_3(p)$. In this paper, we settle these cases. We p…

Weighted homogeneous singularities and rational homology disk smoothings

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown that a (negative definite) starshaped plumbing tree gives rise to a weighted homogeneous singularity admitting a rational homology disk smoothing if and only if the Milnor fillable contact structure of the link admits a weak symplectic rational homology disk filling.

Complex structure on the rational blowdown of sections in E(4)

We show that there is a complex structure on the symplectic 4-manifold $W_{4, k}$ obtained from the elliptic surface E(4) by rationally blowing down $k$ sections for $2\le k\le 9$. And we interpret it via ${\mathbb Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.

最小有理图允许QHD光滑的条件

Minimal rational graphs admitting a QHD smoothing

本文通过De Jong-Van Straten的图变形技术，证明了所有解析图有3或4个大节点（即满足d(v)+e(v)≤-2的节点）的奇点均不具有QHD光滑。通过提供一个通用的减少算法和枚举方法，作者展示了新的例子和家族，这些例子允许组合QHD光滑，即沙漏呈现的接触关系可以被满足。此外，作者还用尖点奇点的方法给出了Bhupal-Stipsicz定理的新证明，该定理分类了允许QHD光滑的加权同质奇点。

意外的有理吹管操作

An Unexpected Rational Blowdown

本文提出了一种新的辛有理吹管操作，通过构建Stein有理同调盘填充，展示了非加权齐次有理奇点的联系链接的辛填充方法。

A note on symplectic rational blow--downs

Due to a significant error in the main result (pointed out by J. Wahl), the paper has been withdrawn by the authors. A corrected and expanded version is 'Rational blow-downs and smoothings of surface singularities' by A. Stipsicz, Z. Szabo and J. Wahl.

Thin-thick decomposition for real definable isolated singularities

Two subset germs of Euclidean spaces are called blow-spherically equivalent, if their spherical modifications are homeomorphic and the homeomorphism induces homeomorphic tangent links. Blow-spherical equivalence is stronger than the topological equivalence but weaker than the Lipschitz equivalence. We introduce the thin-thick decomposition of an isolated singularity germ - which happens to be a natural blow-spherical invariant. This decomposition is a generalization of the thin-thick decomposition of normal complex surface singularity germs introduced in [7]

Symplectic surgeries and normal surface singularities

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of s…

椭球体奇异交点的光滑化：pyramitoid

Smoothing of singular intersections of ellipsoids: pyramitoid

本文研究通过非简单直角柯西特多面体轨道的微覆盖获得的三维流形光滑化，提出n-pyramitoid概念以推广n-pyramid。

Combinatorial properties of the K3 surface: Simplicial blowups and slicings

The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL K3 surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover we …

Rational homology disk smoothings and Lefschetz fibrations

In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.

椭球相交的通用奇点：八面体

On generic singularities of intersections of ellipsoids: the octahedron

研究共轴椭球相交的通用奇点光滑化，重点分析三维情形下的八面体结构。