Equivariant L-Classes of Atiyah-Singer-Zagier Type for Singular Spaces

TL;DR

This paper develops equivariant L-classes for singular spaces with group actions, extending classical invariants to Witt pseudomanifolds and providing tools for computing orbit space invariants.

Contribution

It introduces Atiyah-Singer-Zagier type equivariant L-classes for Witt pseudomanifolds with group actions, enabling computation of orbit space invariants.

Findings

Proves orbit spaces of Witt pseudomanifolds are Witt pseudomanifolds.

Constructs equivariant L-classes for these spaces.

Establishes an averaging formula for these classes.

Abstract

If a finite group $G$ acts on a rational homology manifold, then the orbit space is well-known to be a rational homology manifold again. We consider here actions on spaces that may be much more singular. If the $G$-space is a Witt pseudomanifold, which includes all arbitrarily singular complex pure-dimensional algebraic varieties, then we prove that the orbit space is again a Witt pseudomanifold. In the compact oriented situation, this implies that the orbit space possesses characteristic L-classes, as defined by Goresky and MacPherson. We then construct Atiyah-Singer-Zagier type equivariant L-classes for such $G$-pseudomanifolds which serve, as we show by establishing an averaging formula, as a tool to compute the Goresky-MacPherson L-class of the orbit space. The construction of the equivariant class builds on intersection homological transfer properties and on recent joint K-theoretic work with Eric Leichtnam and Paolo Piazza, which established a G-signature theorem on Witt pseudomanifolds.