Let $X$ be a closed Kaehler manifold of real dimension 4 endowed with a KaehlerEinstein metric of negative curvature. Is it true that $X$ is isomorphic, as a Kaehler manifold, to a quotient of a complex ball?
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1$\begingroup$ Which curvature do you want to be negative? $\endgroup$ – abx Dec 16 '18 at 14:52

$\begingroup$ @abx say scalar curvature $\endgroup$ – geometer Dec 16 '18 at 14:55

4$\begingroup$ Any surface with ample canonical bundle admits a KählerEinstein metric (AubinYau), with constant negative scalar curvature. Those which are quotients of the complex ball form a minuscule part of this huge crowd. $\endgroup$ – abx Dec 16 '18 at 14:59

$\begingroup$ @abx what if we require holomorphic sectional curvature to be negative? What if we require all sectional curvature to be negative? $\endgroup$ – geometer Dec 16 '18 at 15:30

2$\begingroup$ Any compact complex surface of general type satisfies $c_1^2\le 3c_2$, with equality just when the surface is a ball quotient, a result of Yau and also of Miyaoka (the BogomolovMiyaokaYau inequality). For related results on singular ball quotients see Greb et al. arxiv.org/abs/1511.08822 $\endgroup$ – Ben McKay Dec 16 '18 at 16:54