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On Einstein, Hermitian 4-manifolds. (English) Zbl 1254.53076

Recall that only two examples of non-Kähler Einstein metrics on compact complex surfaces are known: the Page metric given in [Phys. Lett. 79B, 235–238 (1979)] and the Hermitian-Einstein metric, recently found by by X.-X. Chen, B. Weber and the author [C. LeBrun, “Einstein manifolds and extremal Kähler metrics”, arXiv:1009.1270]. The main result of this paper shows that these are in fact the only \(J\)-Hermitian non-Kähler Einstein metrics on compact complex surfaces. In fact, one has
Theorem. Let \((M^4, J)\) be a compact complex surface and \(h\) a \(J\)-Hermitian Einstein metric on \(M\). Then, either
a) \((M, J, h)\) is Kähler-Einstein or
b) \(M \simeq \mathbb C P^2 \sharp \overline{\mathbb C P^2}\) and \(h\) is, up to homotheties, the Page metric or
c) \(M \simeq \mathbb C P^2 \sharp 2 \overline{\mathbb C P^2}\) and \(h\) is, up to homotheties, the Chen-Lebrun-Weber metric.
This theorem stems from a previous result of the author, which gives a very short list of possibilities for the compact complex surface \((M, J)\) that might admit a \(J\)-Hermitian, non Kähler Einstein metric and states that any such metric is necessarily conformally equivalent to an extremal Kähler metric, invariant under the action of a 2-torus. The proof is then reduced to show that for each of the few candidates \((M, J)\) there is exactly one 2-torus invariant critical point of the squared \(L^2\) norm of the scalar curvature. This is done using information on the Hessian of such functional extracted from machine-assisted explicit computations.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32Q20 Kähler-Einstein manifolds
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