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 R^tw₆, ₀: =Rg + 2ic () 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 () comes from the curvature 2-form of the connection A. Then the Lichnerowicz-Schr\"odinger formula for the square of the Dirac operator takes the form DL² =^*+14R^tw₆, ₀. 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 R^tw₆, ₀>0 if and only if the index ᶜ (M, L): =ind\, DL vanishes in Kₙ. In this paper we introduce a scalar-valued generalized scalar curvature R^gen₆, ₀: =Rg - 2||₎₏, where ||₎₏ is the pointwise operator norm of Clifford multiplication c (), acting on spinors. We show that the positivity condition on the operator R^tw₆, ₀ is equivalent to the positivity of the scalar function R^gen₆, ₀. We prove a corresponding trichotomy theorem concerning the curvature R^gen₆, ₀, and study its implications. We also show that the space R^gen+ (M, L) of pairs (g, A) with R^gen₆, ₀>0 has non-trivial topology, and address a conjecture about non-triviality of the ``index difference'' map.