SNC纤维的等奇异分解与簇的组合型
Equisingular resolution with SNC fibers and combinatorial type of varieties
摘要：
我们引入了组合型簇$X$的概念，它推广了SNC因子的对偶复形的概念。它是一个唯一的，直到同伦，有限单纯复形$Σ(X)$，这是函的关于态射的品种。其复射影簇的上同调$H^i(Σ(X),Q)$与Deligne滤子$W_0(H^i(X,Q))$的权零部分一致.这个概念可以理解为有限型代数概型的奇异性的拓扑度量。我们还证明，任何品种的特征零承认Hironaka desingularization与所有的纤维SNC。此外，纤维的对偶复合体在地层上是同构的。同样，对于任何态射$f:X\to Y$，都存在类似的去奇异化$\tilde{X}\to X$，其诱导态射$\tilde{X}\to Y$具有SNC纤维。结果之一是，对于任何射影态射$f:X\to Y$，纤维的组合类型是可构造函数。特别地，$\dim(W_0H^i(f^{-1}(y))$是可构建的。
英文摘要：
We introduce the notion of combinatorial type of varieties $X$ which generalizes the concept of the dual complex of SNC divisors. It is a unique, up to homotopy, finite simplicial complex $Σ(X)$ which is functorial with respect to morphisms of varieties. Its cohomology $H^i(Σ(X),Q)$ for complex projective varieties coincide with weight zero part of the Deligne filtration $W_0(H^i(X,Q))$. The notion can be understood as a topological measure of the singularities of algebaric schemes of finite type. We also prove that any variety in characteristic zero admits the Hironaka desingularization with all fibers having SNC. Moreover the dual complexes of the fibers are isomorphic on strata. Also for any morphism $f:X\to Y$ there exists a similar desingularization $\tilde{X}\to X$ for which the induce morphism $\tilde{X}\to Y$ has SNC fibers. One of the consequence is that for any projective morphism $f:X\to Y$ the combinatorial type of the fiber is a constructible function. In particular $\dim(W_0H^i(f^{-1}(y))$ is constructible.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
