Lipschitz regular complex algebraic sets are smooth
  The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction on the dimension is needed. No restriction of singularity to be isolated is needed. 
Lev Birbrair, Alexandre Fernandes, Edson Sampaio, L\^e D. Trang
math.AG