Generalized positive scalar curvature on spin\(^c\) manifolds

Abstract

Let (M, L) be a (compact) non-spin spin\(^c\) manifold. Fix a Riemannian metric g on M and a connection A on L, and let \(D_L\) be the associated spin\(^c\) Dirac operator. Let \(R^{\text {tw }}_{(g,A)}:=R_g + 2ic(\Omega )\) be the twisted scalar curvature (which takes values in the endomorphisms of the spinor bundle), where \(R_g\) is the scalar curvature of g and \(2ic(\Omega )\) comes from the curvature 2-form \(\Omega \) of the connection A. Then the Lichnerowicz-Schrödinger formula for the square of the Dirac operator takes the form \(D_L^2 =\nabla ^*\nabla + \frac{1}{4}R^{\text {tw }}_{(g,A)}\). In a previous work we proved that a closed non-spin simply-connected spin\(^c\)-manifold (M, L) of dimension \(n\ge 5\) admits a pair (g, A) such that \(R^{\text {tw }}_{(g,A)}>0\) if and only if the index \(\alpha ^c(M,L):={\text {ind}}D_L\) vanishes in \(K_n\). In this paper we introduce a scalar-valued generalized scalar curvature \(R^{\text {gen }}_{(g,A)}:=R_g - 2|\Omega |_{op}\), where \(|\Omega |_{op}\) is the pointwise operator norm of Clifford multiplication \(c(\Omega )\), acting on spinors. We show that the positivity condition on the operator \(R^{\text {tw }}_{(g,A)}\) is equivalent to the positivity of the scalar function \(R^{\text {gen }}_{(g,A)}\). We prove a corresponding trichotomy theorem concerning the curvature \(R^{\text {gen }}_{(g,A)}\), and study its implications. We also show that the space \(\mathcal {R}^{{\textrm{gen}+}}(M,L)\) of pairs (g, A) with \(R^{\text {gen }}_{(g,A)}>0\) has non-trivial topology, and address a conjecture about non-triviality of the “index difference” map.