Several Complex Variables Problem List — Problem 61

Definitions and mathematical statement

Let $n$ be a positive integer and write $z=(w_1,\ldots,w_n)\in{\mathbb{C}}^n$. Let $g,f\in{\mathbb{C}}[w_1,\ldots,w_n]$ be homogeneous polynomials of degrees $2$ and $3$, respectively, so that \[ g(tz)=t^2g(z),\qquad f(tz)=t^3f(z)\qquad(t\in{\mathbb{C}}). \] Define the affine algebraic hypersurface \[ X_{f,g}=\left\{(w,z)\in{\mathbb{C}}^{n+1}:w^3+w g(z)+f(z)=1\right\}. \] This is a smooth complex manifold of dimension $n$: the polynomial $H(w,z)=w^3+w g(z)+f(z)$ is homogeneous of degree $3$, and Euler's identity gives \[ w\frac{\partial H}{\partial w}+\sum_{j=1}^n w_j\frac{\partial H}{\partial w_j}=3H=3 \quad\text{on }X_{f,g}, \] so its differential does not vanish there.

A holomorphic immersion into ${\mathbb{C}}^n$ means a holomorphic map $\Phi:X_{f,g}\to{\mathbb{C}}^n$ such that \[ \operatorname{rank}_{{\mathbb{C}}}({\,\mathrm{d}}\Phi_p)=n\qquad\text{for every }p\in X_{f,g}. \] Injectivity is not required.

Question (R. Narasimhan; supplied Problem 61). Does every hypersurface of the displayed form admit a holomorphic immersion into ${\mathbb{C}}^n$? Under the general reading of the question, the quantifiers are: for every positive integer $n$ and every pair $f,g$ satisfying the stated homogeneity assumptions, does such a map $\Phi$ exist?

Unresolved scope and source qualification. The supplied statement does not explicitly specify the range of $n$ or quantify over $f,g$; the universal formulation above records the apparent general question and requires confirmation against the intended source scope. The homogeneity identities use the convention allowing the zero polynomial; the source does not specify this convention. The source appends the qualification [Not true for algebraic immersion]. This is retained as a supplied assertion that the algebraic analogue fails, without an independently verified claim about which dimensions or pairs $f,g$ it covers.

Short English statement

R. Narasimhan asks whether the complex hypersurface $w^3+w g(z)+f(z)=1$, with $g$ quadratic homogeneous and $f$ cubic homogeneous, admits a holomorphic immersion into ${\mathbb{C}}^n$. Such a map must have full differential rank everywhere but need not be injective. The source says the algebraic analogue fails; the precise scope of that qualification and the intended quantifiers require review.

Sources

• Local UnsolvedMath registry: permanent ID $800262$, exact source problem code catalogue entry, title Several Complex Variables Problem List --- Problem 61. The supplied registry labels the problem open; this is source metadata, not an independently verified assessment of current mathematical status.
• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 61. Supplied source URL: https://arxiv.org/abs/math/9509202. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access or verification.
• The problem text attributes the question to R. Narasimhan, Department of Mathematics, University of Chicago, Chicago, IL 60637. The registry's proposed_by field instead lists the source-list authors, Buzzard, Fornaess, Krantz, and gives the year 1995. These distinct attribution roles are preserved.
• The source includes the qualification [Not true for algebraic immersion], without a supporting reference, proof, or specified range of counterexamples in the supplied record.