Virtual Riemann-Roch Theorems for Almost Perfect Obstruction Theories

TL;DR

This paper establishes virtual Riemann-Roch theorems within the framework of almost perfect obstruction theories, extending previous results and removing technical assumptions for moduli space invariants in algebraic geometry.

Contribution

It proves generalized virtual Riemann-Roch theorems for almost perfect obstruction theories, including equivariant and cosection localized versions, broadening the scope of prior results.

Findings

Proved virtual Riemann-Roch theorems for almost perfect obstruction theories.

Extended theorems to equivariant and cosection localized cases.

Removed technical assumptions from earlier theorems.

Abstract

This is the third in a series of works devoted to constructing virtual structure sheaves and $K$-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in $K$-theory for many moduli stacks of interest, including generalized $K$-theoretic Donaldson-Thomas invariants. In this paper, we prove virtual Riemann-Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann-Roch theorems of Fantechi-G\"{o}ttsche and Ravi-Sreedhar. The main technical ingredients are a treatment of the equivariant $K$-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.