论文标题
歧管布置的同事环
Cohomology ring of manifold arrangements
论文作者
论文摘要
我们研究了平滑的歧管$ m $，在没有边界的平滑歧管中，研究了comply $ \ mathcal {m} $ \ mathcal {m}（\ Mathcal {a}）$。我们首先在局部几何poset $ \ mathfrak {l} $上给出了单型cosheaf的概念，然后定义了通用的orlik-- solomon代数$ a^*（\ mathfrak {l}，\ m nathcal {c}）$，由单位构建的通勤环上的单位，该单位构建，该单位构建，该单位构建，该单位构建的allik-solair coosh and colla col。 $ \ Mathcal {C} $作为系数。此外，我们构造了一个单型cosheaf $ \ hat {\ Mathcal {c}}（\ Mathcal {a}）$与$ \ Mathcal {a} $相关的$ \ hat {\ Mathcal {c}}（\ Mathcal {a}））$变为具有合适的乘法结构的双重复合体，并且相关的总综合$ TOT（a^*（\ Mathfrak {l}，\ hat {\ hat {\ Mathcal {c}}}}（c}}}}（c}}}（\ mathcal {a} a} a}））$ Albra。我们的主要结果是$ h^*（tot（a^*（\ mathfrak {l}，\ hat {\ mathcal {c}}}}（\ mathcal {a}）））$是异摩态至$ h^*（\ natcal {m}（m}（m}（\ nathcal）（\ Mathcal {a} a} a}）$。我们的论点是使用与$ \ Mathcal {a} $相关的几何过滤引起的光谱序列的拓扑。特别是，如果$ M $和$ \ Mathcal {a} $中的所有元素是复杂的平滑品种，我们还讨论了模型上混合的Hodge复数结构，并表明它诱导了$ \ Mathcal {M}的规范混合杂货结构（\ Mathcal {a}）$。作为应用程序，我们计算了色构型空间的共同体学，在某些特殊情况下，它与许多已知结果一致。此外，还给出了有关庞加莱多项式和色度多项式的一些显式公式。
We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset $\mathfrak{L}$, and then define the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \mathcal{C})$ over a commutative ring with unit, which is built by the classical Orlik--Solomon algebra and a monoidal cosheaf $\mathcal{C}$ as coefficients. Furthermore, we construct a monoidal cosheaf $\hat{\mathcal{C}}(\mathcal{A})$ associated with $\mathcal{A}$, so that the generalized Orlik--Solomon algebra $A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A}))$ becomes a double complex with suitable multiplication structure and the associated total complex $Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A})))$ is a differential algebra. Our main result is that $H^*(Tot(A^*(\mathfrak{L}, \hat{\mathcal{C}}(\mathcal{A}))))$ is isomorphic to $H^*(\mathcal{M}(\mathcal{A}))$ as algebras. Our argument is of topological with the use of a spectral sequence induced by a geometric filtration associated with $\mathcal{A}$. In particular, we also discuss the mixed Hodge complex structure on our model if $M$ and all elements in $\mathcal{A}$ are complex smooth varieties, and show that it induces the canonical mixed Hodge structure of $\mathcal{M}(\mathcal{A})$. As an application, we calculate the cohomology of chromatic configuration spaces, which agrees with many known results in some special cases. In addition, some explicit formulas with respect to Poincaré polynomial and chromatic polynomial are also given.
