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Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.

5 votes

Hyperbolic 3-manifolds inside algebraic varieties

This is a bit of a troll answer but by Whitney embedding theorem a smooth 3-manifold can be smoothly embedded in $\mathbb{R}^6\cong\mathbb{C}^3$. Maybe you want the embedding to be holomorphic but the …
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