发表机构
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明奇异簇有限商空间满足Brasselet-Schürmann-Yokura提出的Hodge指标定理特征类猜想，通过等变解析K-同调与代数G-理论识别局部化类，并给出无需等变K-理论版本时的充分条件。
AI 中文摘要
我们研究了由Brasselet-Schürmann-Yokura提出的关于奇异复代数簇的Hodge指标定理的特征类类比猜想，该猜想将Goresky-MacPherson同调$L$-类用适当的Hodge理论$L$-类表示，适用于商空间$X/G$，其中$G$是有限群，$X$是纯维复射影簇。假设该猜想的等变$K$-理论版本对$X$成立，我们证明特征类猜想对$X/G$成立。在不假设等变$K$-理论版本的情况下，我们证明特征类猜想对$X/G$成立，前提是它对所有不动点集$X^g$成立，并且满足适当的正规非奇异包含假设。我们的方法是在等变解析$K$-同调和等变代数$G$-理论中工作，并识别相应的局部化类。在解析方面，我们将Banagl-Zagier等变$L$-类与由Banagl-Leichtnam-Piazza定义的签名算子的等变$K$-同调类的局部化Chern特征等同。在代数方面，我们计算交空间Hodge模的等变motivic Hodge-Chern类变换的局部化。
英文摘要
We study the conjectural characteristic class analogue of the Hodge index theorem for singular complex algebraic varieties, formulated by Brasselet-Schürmann-Yokura which expresses the Goresky-MacPherson homology $L$-classes in terms of suitable Hodge-theoretic $L$-classes, for quotient spaces $X/G$, where $G$ is a finite group and $X$ is a pure-dimensional complex projective variety. Assuming that an equivariant $K$-theoretical version of the conjecture holds for $X$, we show that the characteristic class conjecture holds for $X/G$. Without assuming the equivariant $K$-theoretical version, we show that the characteristic class conjecture holds for $X/G$, provided that it holds for all fixed point sets $X^g$ and that a suitable normally nonsingular inclusion assumption is satisfied. Our approach is to work in equivariant analytic $K$-homology and equivariant algebraic $G$-theory and to identify the corresponding localized classes. On the analytic side, we identify the Banagl-Zagier equivariant $L$-classes with localized Chern character of the equivariant $K$-homology class of the signature operator defined by Banagl-Leichtnam-Piazza. On the algebraic side, we compute the localization of the equivariant motivic Hodge-Chern class transformation of the intersection Hodge module.
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