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arXiv · 2103.00617

Positive scalar curvature on $\mathbf{Pin}^\pm$- and $\mathbf{Spin}^c$-manifolds

Boris Botvinnik·Jonathan Rosenberg
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Abstract

It is well-known that spin structures and Dirac operators play a crucial role in the study of positive scalar curvature metrics (psc-metrics) on compact manifolds. Here we consider a class of non-spin manifolds with "almost spin" structure, namely those with spin$^c$ or pin$^\pm$-structures. It turns out that in those cases (under natural assumptions on such a manifold $M$), the index of a relevant Dirac operator completely controls existence of a psc-metric which is $S^1$- or $C_2$-invariant near a "special submanifold" $B$ of $M$. This submanifold $B\subset M$ is dual to the complex (respectively, real) line bundle $L$ which determines the spin$^c$ or pin$^\pm$ structure on $M$. We also show that these manifold pairs $(M,B)$ can be interpreted as "manifolds with fibered singularities" equipped with "well-adapted psc-metrics". This survey is based on our recent work as well as on our joint work with Paolo Piazza.

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Boris Botvinnik, Jonathan Rosenberg. 2021-02-28. Positive scalar curvature on $\mathbf{Pin}^\pm$- and $\mathbf{Spin}^c$-manifolds. https://doi.org/10.1142/12644

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