Weakly-Einstein Hermitian surfaces

A consequence of the Riemannian Goldberg-Sachs theorem is the fact that the Kahler form of an Einstein Hermitian surface is an eigenform of the curvat ure operator. Referring to this property as *-Einstein condition we obtain a complete classification of the compact locally homogeneous *-Einstein Her mitian surfaces. We also provide large families of non-homogeneous *-Einste in (but non-Einstein) Hermitian metrics on CP2#<(CP)over bar>(2), CP1 x CP1 , and on the product surface X x Y of two curves X and Y whose genuses are greater than 1 and 0, respectively.