Complete intersections with metrics of positive scalar curvature (Q1028088)

\textit{M. Gromov} and \textit{H. B. Lawson jun.} proved that any simply connected closed \(n\)-manifold of dimension \(n\geq 5\) which is not spin admits a Riemannian metric of positive scalar curvature [Ann. Math., II. Ser. 111, 423--434 (1980; Zbl 0463.53025)]. On the other hand, \textit{A. Lichnerowicz} [C. R. Acad. Sci., Paris, Ser. A-B 257, 7--9 (1963; Zbl 0136.18401)] proved that for spin manifolds of positive scalar curvature, if \(n\cong 0\, \text{mod}\, 4\) then the \({\widehat A}\)-genus vanishes. This result was generalized by \textit{N. J. Hitchin} [Adv. Math. 14, 1--55 (1974; Zbl 0284.58016)] who proved that in fact the Atiyah-Milnor invariant must vanish in all dimensions. \textit{S. Stolz} [Ann. Math. (2) 136, 511--540 (1992; Zbl 0784.53029)] proved the converse; namely that the vanishing of the Atiyah-Milnor invariant in dimensions \(\geq 5\) is sufficient for the existence of such a metric, as conjectured by Gromov and Lawson. This note classifies which complete intersections admit Riemannian metrics of positive scalar curvature. For complex dimensions \(n\geq 3\), this involves calculating the Atiyah-Milnor invariant. Complex dimension 2 is treated separately using Seiberg-Witten theory.