On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems

Abstract

In this article, we approach the Arnold corank problem posed by Arnold himself in 1975, which asks whether the corank of a holomorphic function is a topological invariant. Here, the corank of a holomorphic function is the corank of its Hessian matrix at 0, which is one of the most basic and fundamental invariants of a holomorphic function. Here, we prove the topological invariance of the corank mod 2. Consequently, we obtain another proof of the Arnold corank problem in dimension 3, which was recently proved by Fernandes, Jelonek, and the author of this article. We also prove that the corank is constant in a continuous family of holomorphic functions with isolated singularities and constant Milnor number. We also approach a conjecture recently posed by Budur, Fernández de Bobadilla, Lê, and Nguyen (2022), which states that if two germs of holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. We prove that a (much) weaker version of this conjecture, which states that if two holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms have the same Euler characteristic, is equivalent to the Zariski multiplicity conjecture for right topological equivalence. Some other results are also presented here. For instance, the multiplicity $> 2$ of generalized semi-Bireskorn-Pham hypersurfaces is an invariant for the right topological equivalence.