The purpose of this note is to obtain a restriction on the fundamantel groups of non-elliptic complex surfaces in Kodaira's classification. The main result is that if M is such a surface and N is a Riemannian manifold of constant negative curvature, tehn the image of the fundamental group of M in the fundamental group of N under any homomorphism is either cyclic or trivial.
As a corollary we find that a the fundamental group of a
compact manifold of constant negative curvature
and dimension at least three is not isomorphic to that of any
compact complex surface. We also find that a compact manifold
of constant negative curvature and real dimension four has no complex structure.
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Last modified by jac at 15:51 on 12/27/1997.