保持格点多面体的Hodge向量
Preserving Hodge Vectors of Lattice Polytopes
AI总结：
通过Cayley多面体的Hodge向量公式，发现无限多具有相同Hodge向量的高维格点多面体，并应用于构造薄多面体。
摘要：
给定包含在$k$维子空间$U \subseteq \mathbb{R}^d$中的格点多面体$P_1, \ldots, P_k$和一个$d$维格点多面体$Q \subset \mathbb{R}^d$，我们计算Cayley多面体$P_1 * \cdots * P_k * Q$的Hodge向量，并证明它等于$P_1, \ldots, P_k$的混合体积乘以$Q$沿$U$的投影的Hodge向量。这里，格点多面体的Hodge向量是其局部$h^*$-向量去掉首尾零。这一结果允许找到无限多个具有相同Hodge向量但不是自由连接的高维格点多面体。证明依赖于环面上一般完全交的Hodge-Deligne多项式的闭式公式，该公式用双变量/混合$h^*$-多项式表示。我们构造的一个特例是所谓的Lawrence扭结：通过中心对称向量对扩展Gale变换。作为应用，我们可以产生许多新的薄多面体，回答了Borger、Kretschmer和第二作者的问题，并且我们提供了$B_k$-多面体薄性的另一种解释，回答了Selyanin的问题。
英文摘要：
Given lattice polytopes $P_1, \ldots, P_k$ contained in a $k$-dimensional subspace $U \subseteq \mathbb{R}^d$ and a $d$-dimensional lattice polytope $Q \subset \mathbb{R}^d$, we compute the Hodge vector of the Cayley polytope $P_1 * \cdots * P_k * Q$, and show that it equals the mixed volume of $P_1, \ldots, P_k$ times the Hodge vector of the projection of $Q$ along $U$. Here, the Hodge vector of a lattice polytope is its local $h^*$-vector with leading and trailing zeroes removed. This result allows finding infinitely many high-dimensional lattice polytopes with the same Hodge vector that are not free joins. The proof relies on a closed formula for the Hodge-Deligne polynomial of generic complete intersections in the torus in terms of the bivariate/mixed $h^*$-polynomial. A special case of our construction is what we call Lawrence twists: extending the Gale transform by centrally-symmetric pairs of vectors. As applications, we can produce many new thin polytopes answering a question by Borger, Kretschmer and the second author, and we provide an alternative explanation of the thinness of $B_k$-polytopes answering a question of Selyanin.
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