Discrete faithful SL(2,C) representations of surface groups are monodromies of holomorphic connections on trivial bundles, implying compact quotients of SL(2,C) have genus ≥2 holomorphic curves.
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Holomorphic curves in compact quotients of SL(2,C)
Discrete faithful SL(2,C) representations of surface groups are monodromies of holomorphic connections on trivial bundles, implying compact quotients of SL(2,C) have genus ≥2 holomorphic curves.