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

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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