Stenzel's Ricci-flat Kähler metrics are not projectively induced (Q824117)

A Kähler metric on a complex manifold is projectively induced if it is the pullback of the Fubini-Study metric on some projective space under a holomorphic map. The problem of determining when a Kähler metric, in particular, a canonical Kähler metric (e.g., a Kähler-Einstein metric), is projectively induced has been the object of intensive research for a long time. It goes back at least to the work of \textit{E. Calabi} who introduced a local criterion in terms of the so-called diastasis function [Ann. Math. (2) 58, 1--23 (1953; Zbl 0051.13103)]. In the article under review, the author applies this criterion to show that \textit{M. B. Stenzel}'s complete Ricci flat Kähler metrics on complexified rank-one symmetric spaces [Manuscr. Math. 80, No. 2, 151--163 (1993; Zbl 0811.53049)] are not projectively induced. More precisely, it is shown that for Stenzel's metric \(g\) on the complexification of \(\mathbb{CP}^n\), \(\mathbb{HP}^n\) or \(\mathbb{S}^n\), the rescaled metric \(cg\) is not projectively induced for \(0<c\) small enough. In the case of \(\mathbb{CP}^n\) and \(\mathbb{HP}^n\), the range of \(c\) allowed contains 1, but the methods used do not allow this in the case of \(\mathbb{S}^n\). The case of the octonionic plane is not mentioned. The proofs rely on explicit descriptions of Stenzel's metrics, obtained by \textit{T.-C. Lee} [Pac. J. Math. 185, No. 2, 315--326 (1998; Zbl 0957.53038)]. Note that explicit Ricci flat (even hyper-Kähler) metrics on the complexification of Hermitian symmetric spaces of arbitrary ranks (including the complexification of \(\mathbb{CP}^n\) here) were known previously thanks to the work of \textit{O. Biquard} and \textit{P. Gauduchon} [C. R. Acad. Sci., Paris, Sér. I 323, No. 12, 1259--1264 (1996; Zbl 0866.58007)].