A Splitting Theorem for Extremal Kähler Metrics

Based on recent work of S.K. Donaldson (J. Differ. Geom. 59:479–522, 2001 ; Q. J. Math. 56:345–356, 2005 ) and T. Mabuchi (Osaka J. Math. 41:563–582, 2004 ; Invent. Math. 159:225–243, 2005 ; Osaka J. Math. 46:115–139, 2009 ), we prove that any extremal Kähler metric in the sense of E. Calabi (in Seminar on Differential Geometry, pp. 259–290. Princeton Univ. Press, Princeton, 1982 ), defined on the product of polarized compact complex projective manifolds is the product of extremal Kähler metrics on each factor, provided that either the polarized manifold is asymptotically Chow semi-stable or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau (Commun. Anal. Geom. 1:473–486, 1993 ) about the splitting of a Kähler–Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kähler metrics.