Line bundles associated with normal surface singularities
Line bundles associated with normal surface singularities
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与法向表面奇点相关的线束
DOI：
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发表时间：
2003
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影响因子：
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通讯作者：
A. Némethi
中科院分区：
文献类型：
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作者：
A. Némethi
最近，L.Nicolaescu和作者提出了一个猜想，该猜想将复解析法面奇点(其环$M$是有理同调球面)的几何亏格与$M$的Seiberg-Witten不变量联系在一起，并与$M$的“正则”$自旋^c$结构有关。由于Seiberg-Witten链接理论为任何$Spin^c$结构提供了一个有理数，因此寻找一组完整的猜想有效的恒等式是一个自然的挑战，这涉及到所有的Seiberg-Witten不变量(给出了对它们的分析--即奇点理论--解释)。
拟订这一套身份是本条的目标之一。事实上，我们建立了猜想有效的不等式，在特殊的刚性情况下，这些不等式变成了等式。
通过这种方法，Seiberg-Witten不变量确定了依赖于分辨率的线丛的第一层上同调的最优拓扑上界。此外，对于$\q$-Gorenstein奇点和一些“自然”线丛，等价成立。
本文的第一部分构造了这些“自然”的全纯线丛。线束结构与阿贝尔覆盖相兼容。这使得我们可以重新表述第二个版本中的猜想，该猜想将外生变化的几何亏格(与奇异胚芽的泛阿贝尔覆盖有关)与链接$M$的Seiberg-Witten不变量联系起来。在最后一节中，我们验证了有理奇点猜想。
Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$ associated with the ``canonical'' $spin^c$ structure of $M$. Since the Seiberg-Witten theory of the link $M$ provides a rational number for any $spin^c$ structure it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all the Seiberg-Witten invariants (giving an analytic -- i.e. singularity theoretical -- interpretation of them).
The formulation of this set of identities is one of the goals of the present article. In fact, we formulate conjecturally valid inequalities which became equalities in special rigid situations.
In this way, the Seiberg-Witten invariants determine optimal topological upper bounds for the dimensions of the first sheaf-cohomology of line bundles living on the resolution. Moreover, for $\Q$-Gorenstein singularities and some ``natural'' line bundles equality holds.
The first part of the article constructs these ``natural'' holomorphic line bundles. The line-bundle construction is compatible with abelian covers. This allows us to reformulate the conjecture in its second version which relates the echivariant geometric genus (associated with the universal abelian cover of the singular germ) with the Seiberg-Witten invariants of the link $M$. In the last section we verify the conjecture for rational singularities.