Classification of spin$^c$ manifolds with generalized positive scalar curvature

TL;DR

This paper characterizes when closed spin$^c$ manifolds admit metrics with positive generalized scalar curvature, linking geometric conditions to algebraic invariants and extending surgery techniques to this setting.

Contribution

It establishes a topological criterion for positive generalized scalar curvature on spin$^c$ manifolds using the generalized $eta$-invariant and develops an analogue of Stolz's sequence for these metrics.

Findings

Positive generalized scalar curvature exists iff the generalized $eta$-invariant vanishes.

Develops surgery techniques adapted to spin$^c$ manifolds for constructing positive curvature metrics.

Connects the existence problem to the analytic surgery sequence of Roe and Higson.

Abstract

Suppose $M$ is a closed $n$-dimensional spin$^c$ manifold with spin$^c$ structure $\sigma$ and associated spin$^c$ line bundle $L$. If one fixes a Riemannian metric $g$ on $M$ and a connection $\nabla_L$ on $L$, the generalized scalar curvature $R^{\text{gen}}$ of $(M,L)$ is $R_g - 2|\Omega_L|_{\text{op}}$, where $|\Omega_L|_{\text{op}}$ is the pointwise operator norm of the curvature $2$-form $\Omega_L$ of $\nabla_L$, acting on spinors. In a previous paper, we showed that positivity of $R^{\text{gen}}$ is obstructed by the non-vanishing of the index of the spin$^c$ Dirac operator on $(M,g,L,\nabla_L)$, and that in some cases, the vanishing of this index guarantees the existence of a pair $(g,\nabla_L)$ with positive generalized scalar curvature. Building on this and on surgery techniques inspired by those that have been developed in the theory of positive scalar curvature on spin manifolds, we show that if $\dim M = n \ge 5$, if the fundamental group $\pi$ of $M$ is in a large class including surface groups and finite groups with periodic cohomology, and if $M$ is totally non-spin (meaning that the universal cover is not spin), then $(M,L)$ admits positive generalized scalar curvature if and only if the generalized $\alpha$-invariant of $(M,L)$ vanishes in the $K$-homology group $K_n(B\pi)$. We also develop an analogue of Stolz's sequence for computing the group of concordance classes of positive generalized scalar curvature metrics, and connect this to the analytic surgery sequence of Roe and Higson. Finally, we give a number of applications to moduli spaces of positive generalized scalar curvature metrics.