Generalized positive scalar curvature on spin$^c$ manifolds
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| Formato: | Preprint |
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2024
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| author | Botvinnik, Boris Rosenberg, Jonathan |
| author_facet | Botvinnik, Boris Rosenberg, Jonathan |
| contents | 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 $R^{tw}_{(g,A)}:=R_g + 2ic(Ω)$ 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 $2ic(Ω)$ comes from the curvature $2$-form $Ω$ of the connection $A$. Then the Lichnerowicz-Schrödinger formula for the square of the Dirac operator takes the form $D_L^2 =\nabla^*\nabla+\frac{1}{4}R^{tw}_{(g,A)}$. In a previous work we proved that a closed non-spin simply-connected spin$^c$-manifold $(M,L)$ of dimension $n\geq 5$ admits a pair $(g,A)$ such that $R^{tw}_{(g,A)}>0$ if and only if the index $α^c(M,L):=\text{ind}\, D_L$ vanishes in $K_n$. In this paper we introduce a scalar-valued generalized scalar curvature $R^{gen}_{(g,A)}:=R_g - 2|Ω|_{op}$, 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 $R^{tw}_{(g,A)}$ is equivalent to the positivity of the scalar function $R^{gen}_{g,A}$. We prove a corresponding trichotomy theorem concerning the curvature $R^{gen}_{(g,A)}$, and study its implications. We also show that the space $\mathcal{R}^{gen+}(M,L)$ of pairs $(g,A)$ with $R^{gen}_{(g,A)}>0$ has non-trivial topology, and address a conjecture about non-triviality of the ``index difference'' map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized positive scalar curvature on spin$^c$ manifolds Botvinnik, Boris Rosenberg, Jonathan Differential Geometry Algebraic Topology 53C21 (Primary) 53C27, 58J22, 55N22, 19L41 (Secondary) 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 $R^{tw}_{(g,A)}:=R_g + 2ic(Ω)$ 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 $2ic(Ω)$ comes from the curvature $2$-form $Ω$ of the connection $A$. Then the Lichnerowicz-Schrödinger formula for the square of the Dirac operator takes the form $D_L^2 =\nabla^*\nabla+\frac{1}{4}R^{tw}_{(g,A)}$. In a previous work we proved that a closed non-spin simply-connected spin$^c$-manifold $(M,L)$ of dimension $n\geq 5$ admits a pair $(g,A)$ such that $R^{tw}_{(g,A)}>0$ if and only if the index $α^c(M,L):=\text{ind}\, D_L$ vanishes in $K_n$. In this paper we introduce a scalar-valued generalized scalar curvature $R^{gen}_{(g,A)}:=R_g - 2|Ω|_{op}$, 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 $R^{tw}_{(g,A)}$ is equivalent to the positivity of the scalar function $R^{gen}_{g,A}$. We prove a corresponding trichotomy theorem concerning the curvature $R^{gen}_{(g,A)}$, and study its implications. We also show that the space $\mathcal{R}^{gen+}(M,L)$ of pairs $(g,A)$ with $R^{gen}_{(g,A)}>0$ has non-trivial topology, and address a conjecture about non-triviality of the ``index difference'' map. |
| title | Generalized positive scalar curvature on spin$^c$ manifolds |
| topic | Differential Geometry Algebraic Topology 53C21 (Primary) 53C27, 58J22, 55N22, 19L41 (Secondary) |
| url | https://arxiv.org/abs/2404.00703 |