Suppose V is some analytic variety of $P^n$ given by some equation. Is there a criterion for determining when it is a manifold of a particular dimension ?
2026-03-26 22:31:25.1774564285
Criterion for an analytic variety to be a manifold
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Yes. First of all, every analytic subvariety of complex-projective space is actually algebraic (Chow's theorem). To determine whether the complex points of a variety form a manifold, it is enough to show that the variety is smooth. Determining the dimension of a connected component may then be done by checking the dimension of a local ring of a closed point of your variety (there are certainly other ways to check - there are several equivalent definitions of dimension).