Log-Lipschitz regularity and Hölder regularity imply smoothness for complex analytic sets

Abstract

In this paper, we prove metric analogues, in any dimension and in any codimension, of the famous theorem of Mumford on smoothness of normal surfaces and the beautiful theorem of Ramanujam that gives a topological characterization of ${\mathbb{C}}^2$ as an algebraic surface. For instance, we prove that a complex analytic set that is log-Lipschitz regular at 0 (i.e., a complex analytic set that has a neighbourhood of the origin which is bi-log-Lipschitz homeomorphic to a Euclidean ball) must be smooth at 0. We prove even more: if a complex analytic set $X$ is such that, for each $0<\alpha <1$, $(X,0)$ and $({\mathbb{R}}^k,0)$ are bi-$\alpha$-Hölder homeomorphic, then $X$ must be smooth at 0. These results generalize the Lipschitz regularity theorem, which says that a Lipschitz regular complex analytic set must be smooth. Global versions of these results are also presented, and in particular we obtain a characterization of an affine linear subspace as a pure-dimensional entire complex analytic set.