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accepted Is the homology class of a compact complex submanifold non-trivial?
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comment Is the homology class of a compact complex submanifold non-trivial?
If I understand correctly the wiki section about Hopf surfaces, your $X$ has the structure of a bundle over the projective line with elliptic curves as fibers; I find it a bit annoying that both the fiber and the zero section provide compact complex submanifolds which are trivial in homology! Anyway this is exactly what I was looking for, thank you.
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revised Is the homology class of a compact complex submanifold non-trivial?
context/motivation added
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revised Is the homology class of a compact complex submanifold non-trivial?
fixed an index
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asked Is the homology class of a compact complex submanifold non-trivial?
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comment Are there closed Riemann surfaces without non-constant holomorphic functions?
I think your hint proves the converse, i.e. that on a compact Riemann surface every holomorphic function is constant
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reviewed No Action Needed Does one Lie subgroup imply the existence of another in this situation?
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revised Extension of biholomorphisms between planar domain to Möbius transformations
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