Weakly-Einstein hermitian surfaces

Apostolov, Vestislav; Muškarov, Oleg

Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1673-1692.

Le théorème de Goldberg-Sachs riemannien a pour conséquence le fait que toute surface complexe, hermitienne, d’Einstein satisfait la condition $*$-Einstein disant que la forme de Kähler est forme propre de l’opérateur de courbure. Dans cet article nous obtenons la classification complète des surfaces hermitiennes localement homogènes qui satisfont la condition $*$-Einstein précédente. Nous construisons aussi des exemples de métriques hermitiennes non homogènes qui sont $*$-Einstein (mais non Einstein) sur $ℂ{\mathbb{P}}^2♯{\overline{ℂ\mathbb{P}}}^2$, $ℂ{\mathbb{P}}^1\times ℂ{\mathbb{P}}^1$ et sur le produit d’une courbe de genre supérieur à 0 et d’une courbe de genre supérieur à 1.

A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kähler form of an Einstein Hermitian surface is an eigenform of the curvature operator. Referring to this property as $*$-Einstein condition we obtain a complete classification of the compact locally homogeneous $*$-Einstein Hermitian surfaces. We also provide large families of non-homogeneous $*$-Einstein (but non-Einstein) Hermitian metrics on $ℂ{\mathbb{P}}^2♯{\overline{ℂ\mathbb{P}}}^2$, $ℂ{\mathbb{P}}^1\times ℂ{\mathbb{P}}^1$, and on the product surface $X\times Y$ of two curves $X$ and $Y$ whose genuses are greater than 1 and 0, respectively.

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