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

Abstract

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 $R_{(g,A)}^{tw}:=R_g+2ic(\Omega )$ 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 $D_L^2={\nabla }^{\ast }\nabla +\frac{1}{4}{R}_{(g,A)}^{tw}$. 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 $R_{(g,A)}^{tw}>0$ 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 $R_{(g,A)}^{gen}:=R_g-2|\Omega {|}_{op}$, 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 $R_{(g,A)}^{tw}$ is equivalent to the positivity of the scalar function $R_{g,A}^{gen}$. We prove a corresponding trichotomy theorem concerning the curvature $R_{(g,A)}^{gen}$, and study its implications. We also show that the space ${\mathcal{R}}^{gen+}(M,L)$ of pairs $(g,A)$ with $R_{(g,A)}^{gen}>0$ has non-trivial topology, and address a conjecture about non-triviality of the ``index difference'' map.

21 pages, no figures. No major changes from the last version, but a few proofs have been further clarified

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• Infinite loop spaces and positive scalar curvature in the presence of a fundamental group
• Spin^c Structures and Scalar Curvature Estimates
• Almost complex structures on connected sums of complex projective spaces