Let (M, L) be a (compact) non-spin spinc\documentclass[12pt]{minimal}
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\begin{document}$$^c$$\end{document} manifold. Fix a Riemannian metric g on M and a connection A on L, and let DL\documentclass[12pt]{minimal}
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\begin{document}$$D_L$$\end{document} be the associated spinc\documentclass[12pt]{minimal}
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\begin{document}$$^c$$\end{document} Dirac operator. Let R(g,A)tw:=Rg+2ic(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {tw }}_{(g,A)}:=R_g + 2ic(\Omega )$$\end{document} be the twisted scalar curvature (which takes values in the endomorphisms of the spinor bundle), where Rg\documentclass[12pt]{minimal}
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\begin{document}$$R_g$$\end{document} is the scalar curvature of g and 2ic(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$2ic(\Omega )$$\end{document} comes from the curvature 2-form Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} of the connection A. Then the Lichnerowicz-Schrödinger formula for the square of the Dirac operator takes the form DL2=∇∗∇+14R(g,A)tw\documentclass[12pt]{minimal}
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\begin{document}$$D_L^2 =\nabla ^*\nabla + \frac{1}{4}R^{\text {tw }}_{(g,A)}$$\end{document}. In a previous work we proved that a closed non-spin simply-connected spinc\documentclass[12pt]{minimal}
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\begin{document}$$^c$$\end{document}-manifold (M, L) of dimension n≥5\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 5$$\end{document} admits a pair (g, A) such that R(g,A)tw>0\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {tw }}_{(g,A)}>0$$\end{document} if and only if the index αc(M,L):=indDL\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ^c(M,L):={\text {ind}}D_L$$\end{document} vanishes in Kn\documentclass[12pt]{minimal}
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\begin{document}$$K_n$$\end{document}. In this paper we introduce a scalar-valued generalized scalar curvatureR(g,A)gen:=Rg-2|Ω|op\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {gen }}_{(g,A)}:=R_g - 2|\Omega |_{op}$$\end{document}, where |Ω|op\documentclass[12pt]{minimal}
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\begin{document}$$|\Omega |_{op}$$\end{document} is the pointwise operator norm of Clifford multiplication c(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$c(\Omega )$$\end{document}, acting on spinors. We show that the positivity condition on the operator R(g,A)tw\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {tw }}_{(g,A)}$$\end{document} is equivalent to the positivity of the scalar function R(g,A)gen\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {gen }}_{(g,A)}$$\end{document}. We prove a corresponding trichotomy theorem concerning the curvature R(g,A)gen\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {gen }}_{(g,A)}$$\end{document}, and study its implications. We also show that the space Rgen+(M,L)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {R}^{{\textrm{gen}+}}(M,L)$$\end{document} of pairs (g, A) with R(g,A)gen>0\documentclass[12pt]{minimal}
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\begin{document}$$R^{\text {gen }}_{(g,A)}>0$$\end{document} has non-trivial topology, and address a conjecture about non-triviality of the “index difference” map.
