论文标题
自然线束的共同体在通用正常表面奇点上
Cohomology of natural line bundles on generic normal surface singularities
论文作者
论文摘要
令$ \ mathcal {t} $为任意分辨率图，$（x，0）$ a通用复杂分析态度正常表面奇点，$ \ tilde {x} $ a的通用分辨率与之相对应。修复在特殊曲线上支持的有效整数周期$ z $，并在l'$中使用任意的Chern类$ z'\。 在本文中，我们旨在计算共同体编号$ h^1（\ Mathcal {o} _ {z}（z'））$。请注意，在\ cite {nna2}中讨论了$ z'_v <0，v \ in | z | $的情况，其中主要定理是在这种特殊情况下，这些共同体学数量等于$ \ text {pic}^{z'}（z）$ quial in | z yv <0 | v <0 |没有这个假设，陈述远非如此。 在本文中，使用\ cite {r}的相对通用线束和相对通用的分析结构的tecniques，我们给出了组合算法来计算所有情况下通用粒子的自然线捆$ h^1（\ calo_ {z}（z'））$的共同体学数。
Let $\mathcal{T}$ be an arbitrary resolution graph and $(X, 0)$ a generic complex analytic normal surface singularity, and $\tilde{X}$ a generic resolution corresponding to it. Fix an effective integer cycle $Z$ supported on the exceptional curve and also an arbitrary Chern class $Z' \in L'$. In this article we aim to compute the cohomology numbers $h^1(\mathcal{O}_{Z}(Z'))$. Notice, that the case $Z'_v < 0, v \in |Z|$ was discussed in \cite{NNA2}, where the main theorem was, that in this special case these cohomology numbers equal to the cohomology numbers of the generic line bundle in $\text{Pic}^{Z'}(Z)$ However the condition $Z'_v < 0, v \in |Z|$ was crucial in the proof and without this assumption the statement is far from being true. In this article using the tecniques of relatively generic line bundles and relatively generic analytic structures from \cite{R} we give combinatorial algorithms to compute the cohomology numbers of natural line bundles $h^1(\calO_{Z}(Z'))$ for generic singularities in all cases.
