Weight filtrations on GKZ-systems
Weight filtrations on GKZ-systems
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DOI：
10.1353/ajm.2022.0033
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发表时间：
2018-09
影响因子：
1.7
通讯作者：
Thomas Reichelt;U. Walther
中科院分区：
文献类型：
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作者：
Thomas Reichelt;U. Walther
给定一个整数矩阵$A\in\Bbb{Z}^{d\times n}$，研究了Gau\ss-Manin系统上由$A$诱导的单项映射$h\colon（\Bbb{C}^*）^d\to\Bbb{C}^n$在Saito意义下的自然混合Hodge模结构.我们完全确定在正常情况下，相关的分级对象的权重过滤，通过计算的交叉复合体与各自的多重性，形成其成分。我们的结果表明，这些数据是纯粹的组合，而不是算术，在这个意义上，他们只依赖于多面体结构的锥$A$，而不是对半群本身。特别地，我们将de Cataldo，Migliorini和Musta\c{t}\v{a}的结果推广到环面嵌入的情形，并给出了分解定理失效的封闭形式。如果$A$是齐次的，并且如果$\beta\in\Bbb{C}^d$是积分但不是强共振参数，我们使用一个单值的Fourier-拉普拉斯变换将Gau\ss-Manin系统的混合Hodge模结构转移到GKZ-系统上，并附加到$A$和$\beta$上.在$A$是从正规自反Gorenstein多面体$P$导出的情况下，Batyrev和Stienstra将GKZ-系统的一般纤维上的某些滤子与$P$诱导的投射环面簇内的一般超平面截面上同调上的混合霍奇结构联系起来。我们的公式，措辞在诱导相对环面品种的交叉上同调群，提供必要的校正条款，全球化的计算。特别是，我们的文件，在GKZ系统的权重过滤将不同于Batyrev的过滤面每当$P$是不是一个单纯形：相交复合物有助于权重过滤措施失败的$P$是一个simplex. Regardless的同质性，我们得到一个纯粹的组合公式的长度的Gau\ss-Manin系统，从而为相应的GKZ系统。在维数为3时，对于单纯半群，我们给出了权滤子的显式生成元。
abstract:Given an integer matrix $A\in\Bbb{Z}^{d\times n}$, we study the natural mixed Hodge module structure in the sense of Saito on the Gau\ss--Manin system attached to the monomial map $h\colon(\Bbb{C}^*)^d\to\Bbb{C}^n$ induced by $A$. We completely determine in the normal case the associated graded object to the weight filtration, by computing the intersection complexes with respective multiplicities that form its constituents. Our results show that these data are purely combinatorial, and not arithmetic, in the sense that they only depend on the polyhedral structure of the cone of $A$, but not on the semigroup itself. In particular, we extend results of de Cataldo, Migliorini and Musta\c{t}\v{a} to the setting of torus embeddings and give a closed form for the failure of the Decomposition Theorem in our context.If $A$ is homogeneous and if $\beta\in\Bbb{C}^d$ is an integral but not strongly resonant parameter, we use a monodromic Fourier--Laplace transform to carry the mixed Hodge module structure from the Gau\ss--Manin system to the GKZ-system attached to $A$ and $\beta$. In case $A$ is derived from a normal reflexive Gorenstein polytope $P$, Batyrev and Stienstra related certain filtrations on the generic fiber of the GKZ-system to the mixed Hodge structure on the cohomology of a generic hyperplane section inside the projective toric variety induced by $P$. Our formul\ae, phrased in terms of intersection cohomology groups on induced relative toric varieties, provide the necessary correction terms to globalize their computation. In particular, we document that on the GKZ-system the weight filtration will differ from Batyrev's filtration-by-faces whenever $P$ is not a simplex: the intersection complexes contributing to the weight filtration measure the failure of $P$ to be a simplex.Irrespective of homogeneity, we obtain a purely combinatorial formula for the length of the Gau\ss--Manin system, and thus for the corresponding GKZ-system. In dimension up to three, and for simplicial semigroups, we give explicit generators of the weight filtration.