Boundary and Milnor obstructions to deformation into a ball

Definitions and mathematical statement

Let $n\ge3$, and let $V\subset{\mathbb{C}}^n$ be a complex analytic hypersurface with compact smooth boundary, containing an isolated singularity at zero. The source question, proposed by Dmitri Zaitsev, asks whether $V$ can be deformed into the unit ball $\mathbb B^{n-1}\subset{\mathbb{C}}^{n-1}$ by a smooth family $(V_t,\partial V_t)$, $t$ in an interval. Its definition is the following local one.

1. Near every interior point $(p_0,t_0)$ of a fiber there is a smooth complex function $F(p,t)$, holomorphic in the ambient variable $p$, such that $V_t=\{p:F(p,t)=0\}$ locally.
2. Near every boundary point there are such an $F$ and a smooth real function $r(p,t)$ with \[ V_t=\{p:F(p,t)=0,\ r(p,t)<0\},\qquad \partial V_t=\{p:F(p,t)=0,\ r(p,t)=0\}, \] and the vectors $\partial F/\partial p$ and $\partial r/\partial p$ are linearly independent. This is the stated boundary transversality condition, interpreted in the usual complex-covector sense.

The requested endpoints are the original variety and a ball. The source writes these in different ambient dimensions. In a fixed-ambient interpretation one can place the terminal ball in the coordinate hyperplane ${\mathbb{C}}^{n-1}\times\{0\}\subset{\mathbb{C}}^n$; this is an explicit interpretation. Singular intermediate fibers are allowed by the printed conditions. Compactness of each individual boundary is not properness of the total boundary over the parameter interval.

The supplied aid reports a negative answer under the additional fixed-ambient and proper-boundary-family hypotheses. Namely, if \[ \mathcal B=\{(p,t):p\in\partial V_t\}\longrightarrow I \] is proper, boundary transversality makes this projection a proper smooth submersion. Its fibers therefore have constant diffeomorphism type. For example the truncated quadratic cone \[ V=\{z\in{\mathbb{C}}^n:z_1^2+\cdots+z_n^2=0,\ \|z\|<\varepsilon\} \] has boundary diffeomorphic to the Stiefel manifold \[ V_2({\mathbb{R}}^n)=\{(u,v)\in{\mathbb{R}}^n\times{\mathbb{R}}^n:\|u\|=\|v\|=1,\ u\cdot v=0\}, \] which the aid records is not diffeomorphic to $S^{2n-3}$, for every $n\ge3$. It thus cannot reach the ball under these additional hypotheses, even if interior singularities are permitted during the family.

Separately, a local Milnor fiber of a genuine isolated hypersurface singularity has nonzero middle homology: in the usual isolated-critical-point setting it has the homotopy type of a bouquet of $\mu>0$ spheres of real dimension $n-1$. Such a Milnor smoothing fiber is not a ball. This local obstruction does not prohibit every more general global family with additional singular topology changes. The printed question omits properness and leaves the ambient interpretation unresolved; neither conditional obstruction is a solution of its weakest literal reading.

Short English statement

Can an isolated-singular hypersurface with compact smooth boundary deform to a ball under the source's local family conditions? Properness of the total boundary would preserve its topology and gives reported quadratic-cone counterexamples. The source omits that hypothesis and permits singular intermediate fibers, so the conditional negative answer does not settle every interpretation.

Sources

• Dmitri Zaitsev, Deformation of varieties, Problem 10 in the AIM workshop Complexity of mappings in CR geometry, https://aimath.org/WWN/complexitycrmap/complexitycrmap.pdf. The record verification says the statement is restored without adding hypotheses.
• Local UnsolvedMath registry, permanent ID 20002892, exact code catalogue entry; canonical location aim-several-complex-variables-notes.json, notes[49], dated 14 August 2026; statement labeled exact.
• Research attempt 1: conditional result, problem status unknown at the run, rigor label conditional, novelty confidence low; registry status partially solved. The boundary and Milnor obstructions are supplied machine-generated research aids, not peer reviewed, requiring independent verification before citation. No separate full bibliography for the proper-submersion or Milnor-fiber results is supplied.