A characterization of cohomological Einstein Kaehler manifolds and applications (Q579645)

Let M be a compact Kaehler manifold of complex dimension \(n>1\). Let \(\omega\) be the cohomology class represented by the fundamental 2-form \(\phi\). M is said to be cohomologically Einsteinian if \(c_ 1=b\omega\) for some constant b, where \(c_ 1\) is the first Chern class of M [see \textit{K. Ogiue}, J. Differ. Geom. 10, 201-205 (1975; Zbl 0301.53035)]. The class of cohomological Einstein Kaehler manifolds is wider than the class of Einstein Kaehler manifolds. The author characterizes the complex 2- dimensional cohomological Einstein Kaehler manifolds. Let \(a_ 1\) be the real number such that \(\omega^{n-1}c_ 1=a_ 1\omega^ n\). The main result of the paper is the following Theorem: Let M be a compact Kaehler manifold of complex dimension \(n=2\). Denote by \({\mathcal A}(M)\) the arithmetic genus of M and by \(\tau\) the scalar curvature. One has \[ (A)^ 2\chi (M)+3 sign(M)=8{\mathcal A}(M)+sign(M)=12{\mathcal A}(M)-\chi (M)\leq 2 vol(M)a^ 2_ 1 \] with equality if and only if the Kaehler metric is cohomologically Einsteinian with \(c_ 1=a_ 1\). Moreover \[ (B)\quad 3 sign(M)+\chi (M)\leq vol(M)a^ 2_ 1+(1/192\pi^ 2)\int_{M}\tau^ 2dv \] \[ sign(M)+6{\mathcal A}(M)\leq (3/2)vol(M)a^ 2_ 1+(1/1152\pi^ 2)\int_{M}\tau^ 2dv, \] where the equalities hold, respectively, only if the Kaehler metric is of constant holomorphic sectional curvature \(c=(4/3)\cdot \pi a_ 1.\) From this theorem the author obtains some interesting restrictions for the constructions of cohomological Einstein metrics on complex 2- manifolds as well as a characterization of Kaehler metrics of constant holomorphic sectional curvature in terms of \(\int_{M}\tau dv\) and \(\int_{M}\tau^ 2dv\).