Generalized positive scalar curvature on spin$$^c$$ manifolds

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.