The Kernel of the $\hat{A}$-Genus in Rational Spin Bordism is Generated by Ricci-Positive Manifolds

Abstract

We prove that, in every degree, the rational Spin bordism classes represented by manifolds admitting metrics with positive Ricci curvature span exactly the kernel of the $\hat{A}$-genus. More precisely, for \[ R=Ω_*^{Spin}\otimes\mathbb{Q},\qquad J=\ker(\hat A:R\longrightarrow\mathbb{Q}[u]),\] the $\mathbb{Q}$-span of bordism classes of Ricci-positive Spin manifolds equals $J$ in each degree. This answers, in the differentiable rational Spin category, a question about rational bordism obstructions to positive Ricci curvature which was raised in the context of complex elliptic genera. The proof uses smooth complete intersections of an odd number $\ell$ of quadrics \[ Y_{m,\ell}\subset \mathbb{CP}^{2m+\ell}, \qquad \ell=1,\, 3,\, \ldots,\, 2m-1. \] These manifolds have real dimension $4m$, are Spin and Fano, and therefore admit metrics with positive Ricci curvature. A first-order thickening of the $\hat{A}$-genus induces $m-1$ linear functionals on $(J/J^2{)}_{4m}$. Their values on the classes $[Y_{m,\ell }]$ are governed by polynomials $P_{m,q}(\ell )$ of strictly increasing degrees $q+1=1$, $2$, $\dots$, $m-1$. This gives full rank by a polynomial-interpolation argument.

10 pages, LaTeX