Bounds on Wahl singularities from symplectic topology

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2017-08-07 DOI:10.14231/ag-2020-003

J. Evans, I. Smith

引用次数: 10

Abstract

Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$ embeds symplectically in a quintic surface, then $p \leq 12$, partially answering the symplectic version of a question of Kronheimer.

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辛拓扑中Wahl奇点的界

设X是具有正几何亏格（$b_+>1$）的一般类型的极小曲面，设$K^2$是其规范类的平方。在Khodorovskiy和Rana工作的基础上，我们证明了如果X在Q-Gorenstein退化中发展出长度为$\ell$的Wahl奇点，那么$\ell\leq4K^2+7$。这改善了李目前最著名的上限（$\ell\leq 400（K^2）^4$）。我们的界来自于一个更强的定理，该定理约束了一般类型曲面中某些有理同调球的辛嵌入。特别地，我们证明了如果有理同调球$B_{p，1}$辛嵌入五次曲面，那么$p\leq12$，部分回答了Kronheimer问题的辛版本。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.