AI 中文总结
该研究将Maxim-Schuermann公式从顶点有理的多面体推广至任意凸多面体，依托BBFK的组合相交上同调框架，还探讨了多面体的霍奇指标定理。
AI 中文摘要
环面簇可由有理多面体构造，其若干不变量可通过对应多面体的组合性质表示。Barthel-Brasselet-Fieseler-Kaup（BBFK）引入凸多面体的组合相交上同调，当多面体为有理时，该上同调与对应环面簇的相交上同调一致。Maxim-Schuermann计算了射影环面簇的相交上同调符号，对应顶点为有理的多面体情形。利用BBFK的组合框架，我们证明Maxim-Schuermann公式可推广至任意凸多面体，最后讨论多面体的霍奇指标定理的一种形式。
英文摘要
Toric varieties can be constructed from rational polytopes, and several invariants of toric varieties can be expressed in terms of the combinatorics of the corresponding polytope. Barthel-Brasselet-Fieseler-Kaup (BBFK) introduced combinatorial intersection cohomology for convex polytopes, which agrees with the intersection cohomology of the associated toric variety when the polytope is rational. Maxim-Schuermann computed the intersection cohomology signature of a projective toric variety, corresponding to the case of a polytope with rational vertices. Using the combinatorial framework of BBFK, we show that the Maxim-Schuermann formula extends to arbitrary convex polytopes. Finally, we discuss a version of the Hodge index theorem for polytopes.
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