On the conjectural Hodge index theorem for finite quotient spaces of singular varieties

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 $X\mathrm{/}G$, 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 $X\mathrm{/}G$. Without assuming the equivariant $K$-theoretical version, we show that the characteristic class conjecture holds for $X\mathrm{/}G$, 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.