Twisted Milnor hypersurfaces (Q2159154)

This article makes a valuable contribution in the study of existence of metrics with positive scalar curvature on Milnor-type hypersurfaces within the framework of index theory. The authors introduce the notion of a twisted Milnor hypersurface, a Milnor hypersurface associated to a projective bundle on \(\mathbb{C}P^n\) with fiber \(\mathbb{C}P^m\) (see Definition 1.1 and Remark 1), and explicit its \(\hat{A}\)-genus (Theorem 3.6) and \(\alpha\)-invariant (Theorem 3.8), which is the mod 2 index of Atiyah-Singer Dirac operator; see also Corollary 1.5. They use Zhang's analytic Rokhlin congruence formula and dyadic expansion coefficients. As a consequence, some spin twisted Milnor hypersurfaces that do not admit metrics with positive scalar curvature are given, see Example 3.3.