The combinatorics and topology of proper toric maps
The combinatorics and topology of proper toric maps
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原复曲面映射的组合学和拓扑
DOI：
10.1515/crelle-2015-0104
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Mustaţă
中科院分区：
文献类型：
--
作者：
M. A. Cataldo;L. Migliorini;M. Mustaţă
研究了复曲面映射的拓扑。我们证明了如果$f\colon X\to Y$是一个真环面态射，且X$是单纯的，那么$f$的每一个纤维的上同调是纯的，并且是Hodge-Tate型的.当地图是纤维化，我们给一个明确的公式的Betti数的纤维的相对版本的$f$-向量，扩展通常的公式的Betti数的单纯完全复曲面品种。然后，我们描述了一个复曲面纤维化的分解定理，特别是一个非负的组合不变量附加到每个锥的风扇的$Y$，这是积极的正是当相应的闭子集的$Y$出现作为一个支持的分解定理。这个不变量的描述涉及到$X$和$Y$上的交上同调复形的柄，但是在$X$和$Y$都是单纯形的情况下，有一个关于相对$f$-向量的简单公式。
We study the topology of toric maps. We show that if $f\colon X\to Y$ is a proper toric morphism, with $X$ simplicial, then the cohomology of every fiber of $f$ is pure and of Hodge-Tate type. When the map is a fibration, we give an explicit formula for the Betti numbers of the fibers in terms of a relative version of the $f$-vector, extending the usual formula for the Betti numbers of a simplicial complete toric variety. We then describe the Decomposition Theorem for a toric fibration, giving in particular a nonnegative combinatorial invariant attached to each cone in the fan of $Y$, which is positive precisely when the corresponding closed subset of $Y$ appears as a support in the Decomposition Theorem. The description of this invariant involves the stalks of the intersection cohomology complexes on $X$ and $Y$, but in the case when both $X$ and $Y$ are simplicial, there is a simple formula in terms of the relative $f$-vector.