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

TL;DR

This paper explores how index theory of Dirac operators on non-spin manifolds with spin^c or pin^± structures influences the existence of positive scalar curvature metrics, especially near certain submanifolds.

Contribution

It establishes a link between Dirac operator indices and positive scalar curvature metrics on non-spin manifolds with specific structures, extending known results to broader classes.

Findings

Index of Dirac operators controls psc-metric existence near special submanifolds.

Manifolds with fibered singularities can admit well-adapted psc-metrics.

Results apply to manifolds with spin^c or pin^± structures, not just spin.

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.