On the curvature of Einstein-Hermitian surfaces (Q669470)

In this paper the authors, as a part of the classification program of compact complex Einstein-Hermite surfaces, give a detailed study of the Page gravitational instanton, i.e., an Einstein metric on the connected sum \(\mathbb{C}P^2\#\overline{\mathbb{C}P^2}\). \par Let \(X\) be a complex manifold with underlying (integrable) real almost complex manifold \((M,J)\). Recall that a real Riemannian metric \(g\) on \(X\) is called an Einstein-Hermite metric if \(g\) is Einstein, i.e., \(\mathrm{Ric}_g=\Lambda g\) and \(J\) is orthogonal for \(g\), i.e., \(g(Ju,Jv)=g(u,v)\) for all real vector fields \(u,v\) on \(M\) and the triple \((M,J,g)\) is called an \textit{Einstein-Hermite space}; if moreover \(g\) is Kähler then \((M,J,g)\) is an Einstein-Kähler space. The \textit{holomorphic bisectional curvature} on these spaces is defined by \(H(u,v):=\mathrm{Riem}_g(u,Ju,v,Jv)\). Compact Einstein-Kähler spaces with positive \(H\) have been classified by Berger and Goldberg-Kobayashi a long time ago. One can ask for further classification by relaxing the Kähler condition. Regarding this, the authors' first main result is the following: Assume \((M,J,g)\) is a compact Einstein-Hermite (complex) surface with positive holomorphic bisectional curvature. Then \((M,J,g)\) is biholomorphically isometric to \(\mathbb{C}P^2\) with its Fubini-Study metric (see Theorem 1.3 in the article, as well as the related Theorem 1.4 and Corollary 1.5). One can then proceed even further by relaxing the positivity assumption as well. This has been done in the case of complex surfaces by \textit{C. LeBrun} [Lect. Notes Pure Appl. Math. 184, 167--176 (1997; Zbl 0876.53024)] whose result is that these spaces are either isometric to (i) a Kähler-Einstein surface, or (ii) the explicitly known Page metric on \(\mathbb{C}P^2\#\overline{\mathbb{C}P^2}\), or (iii) the transcendental Chen-LeBrun-Weber metric on \(\mathbb{C}P^2\# 2\overline{\mathbb{C}P^2}\). \par Motivated by this classification program, the authors also give a detailed, explicit study of the Page geometry with special attention to its various submanifolds. Among other things, they prove that neither the Page nor the Chen-LeBrun-Weber metric can have positive biholomorphic sectional curvature (see Corollaries 1.5 and 1.6 in the article).