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Let $X$ be a closed Kaehler manifold of real dimension 4 endowed with a Kaehler--Einstein 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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    $\begingroup$ 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
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    $\begingroup$ Any surface with ample canonical bundle admits a Kähler--Einstein metric (Aubin-Yau), 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
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    $\begingroup$ 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 Bogomolov-Miyaoka-Yau 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

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