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

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