The author starts with the following question: suppose a smooth compact manifold \(M\) admits a Kähler-Einstein metric; is any other Einstein metric on \(M\) also Kähler-Einstein? If \(\dim_{\mathbb{R}} M>4\), the answer is generally no. Examples: \(M=\mathbb{C} P_3\), \(M=\mathbb{C} P_3\times\mathbb{C} P_1\times\cdots\times\mathbb{C} P_1\). For \(\dim_{\mathbb{R}} M=4\), N. J. Hitchin [J. Differ. Geom. 9, 435-441 (1974; Zbl 0281.53039)] answered the question affirmatively for complex surfaces which admit Ricci-flat Kähler metrics. Now the question is raised: is any Einstein metric on a compact complex surface Kähler? For surfaces of Kodaira dimension \(\geq 0\), Seiberg-Witten theory makes an affirmative answer plausible, but in general the answer is no. This leads the author to study the classification of compact Einstein Hermitian surfaces \((M^4,J,h)\).
Main result: let \((M^4,J)\) be a compact complex surface with an Einstein metric \(h\), Hermitian and not Kähler with respect to \(J\); then \((M,J)\) is obtained from \(\mathbb{C} P_2\) by blowing up one, two or three points in general position. With regard to the initial question, the author concludes: a compact complex surface \((M,J)\) cannot admit both a Kähler-Einstein and a non-Kähler Einstein Hermitian metric, except perhaps \(M=\mathbb{C} P_2\#3 \overline{\mathbb{C} P_2}\).
For the entire collection see [Zbl 0855.00020].