Equisingular resolution with SNC fibers and combinatorial type of varieties
Equisingular resolution with SNC fibers and combinatorial type of varieties
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SNC 纤维和品种组合类型的等奇异分辨率
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发表时间：
2016
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通讯作者：
J. Włodarczyk
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作者：
J. Włodarczyk
我们引入了组合型的概念$X$推广的概念的对偶复杂的SNC因子。它是一个唯一的，直到同伦，有限单纯复形$Sigma（X）$这是函子关于态射的品种。它对复射影簇的上同调H^i（Sigma（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。对于任何态射$f：X O Y$存在类似的去奇异化$ ilde{X} o X$，其中诱导态射$ ilde{X} 〇 Y具有SNC纤维。其中一个推论是，对于任何投射态射$f：X O Y$纤维的组合类型是可构造函数。特别地，$dim（W_0H^i（f^{-1}（y））$是可构造的。
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 $Sigma(X)$ which is functorial with respect to morphisms of varieties. Its cohomology $H^i(Sigma(X),Q)$ 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 $f:X o Y$ there exists a similar desingularization $ ilde{X} o X$ for which the induce morphism $ ilde{X} o Y$ has SNC fibers. One of the consequence is that for any projective morphism $f:X o Y$ the combinatorial type of the fiber is a constructible function. In particular $dim(W_0H^i(f^{-1}(y))$ is constructible.