Existence of further applications of the local Lefschetz–Riemann–Roch (LRR) formula

Ascertain whether there exist applications, beyond the one developed in this work for deriving a local index density on complex manifolds with holomorphic C*-actions, of the local Lefschetz–Riemann–Roch formula in the existing literature; specifically, identify and characterize any other contexts in which a local LRR density has been used or can be rigorously applied.

Background

In developing a local index formula for complex manifolds with holomorphic C*-actions, the authors analyze a contribution that, as t → 0, is essentially recognized as the local density of the Lefschetz–Riemann–Roch (LRR) formula. This observation leverages the local version of LRR (in the sense of Berline–Getzler–Vergne) within their heat-kernel-based approach.

While this provides a concrete application of local LRR to obtain an index density and ultimately a Hirzebruch–Riemann–Roch type formula including strata contributions, the authors explicitly note uncertainty regarding whether such a local LRR construction has been used elsewhere. Clarifying this would situate their contribution within the broader literature and potentially uncover additional settings where local LRR is applicable.