Generalized positive scalar curvature on spinᶜ manifolds

Let $(M,L)$ be a non-spin spinᶜ manifold. Fix a Riemannian metric g on M and a connection A on L, and let DL be the associated spinᶜ Dirac operator. Let Rᵗʷg,A:=Rg + 2ic(Ω) be the twisted scalar curvature (which takes values in the endomorphims of the spinor bundle), where Rg is the scalar curvature of g and 2ic(Ω) comes from the curvature $2$-form Ω of the connection A. Then the Lichnerowicz-Schr\"odinger formula for the square of the Dirac operator takes the form DL² =∇^*∇+1/4Rᵗʷg,A. In a previous work we proved that a closed non-spin simply-connected spinᶜ-manifold $(M,L)$ of dimension n≥ 5 admits a pair $(g,A)$ such that Rᵗʷg,A>0 if and only if the index αᶜ(M,L):=ind\, DL vanishes in Kₙ. In this paper we introduce a scalar-valued generalized scalar curvature Rᵍᵉⁿg,A:=Rg - 2|Ω|ₒₚ, where |Ω|ₒₚ is the pointwise operator norm of Clifford multiplication c(Ω), acting on spinors. We show that the positivity condition on the operator Rᵗʷg,A is equivalent to the positivity of the scalar function Rᵍᵉⁿg,A. We prove a corresponding trichotomy theorem concerning the curvature Rᵍᵉⁿg,A, and study its implications. We also show that the space Rᵍᵉⁿ⁺(M,L) of pairs $(g,A)$ with Rᵍᵉⁿg,A>0 has non-trivial topology, and address a conjecture about non-triviality of the ``index difference'' map.