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Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?

In general, it is well-known that there are compact complex manifolds which do not admit compact complex submanifolds — take a generic torus of dimension at least two, for instance. Note that the torus is not (Kobayashi) hyperbolic.

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    $\begingroup$ I don't know much about complex submanifolds but e.g. suppose $X$ is a complex hyperbolic manifold with infinite cyclic fundamental group. Thus the universal cover of $X$ is the complex hyperbolic space. Does $X$ contain a compact complex submanifold (of positive dimension)? $\endgroup$
    – Igor Belegradek
    Commented Feb 25, 2022 at 20:14
  • $\begingroup$ What about the open unit disk with its hyperbolic metric? $\endgroup$
    – Jason Starr
    Commented Feb 25, 2022 at 23:37
  • $\begingroup$ More counterexamples arise from the complex hyperbolic spaces: maths.dur.ac.uk/users/j.r.parker/img/NCHG.pdf $\endgroup$
    – Jason Starr
    Commented Feb 26, 2022 at 1:20
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    $\begingroup$ Does the title imply that $X$ should be non-compact? $\endgroup$
    – LeechLattice
    Commented Feb 26, 2022 at 12:59
  • $\begingroup$ Any simply connected complete Kahler manifold with negative sectional curvature has negative holomorphic sectional curvature ,and is a Stein manifold . Any complex submanifold of it has negative holomorphic sectional curvature . $\endgroup$
    – Mohan Ramachandran
    Commented Mar 2, 2022 at 20:39

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