Several Complex Variables Problem List — Problem 60

Definitions and mathematical statement

For a positive integer $n$, let \[ SL(n,{\mathbb{C}})=\{A\in\operatorname{Mat}_{n\times n}({\mathbb{C}}):\det A=1\}, \] viewed as a complex manifold of dimension $n^2-1$. A holomorphic immersion into ${\mathbb{C}}^{n^2-1}$ is a holomorphic map \[ f:SL(n,{\mathbb{C}})\longrightarrow{\mathbb{C}}^{n^2-1} \] whose complex differential \[ \mathrm{d}f_A:T_A SL(n,{\mathbb{C}})\longrightarrow{\mathbb{C}}^{n^2-1} \] is injective for every $A\in SL(n,{\mathbb{C}})$. Neither global injectivity nor surjectivity is required.

The source asks whether such a holomorphic immersion exists. For a specified $n$, the existence condition is \[ \exists f\in\mathcal O\left(SL(n,{\mathbb{C}}),{\mathbb{C}}^{n^2-1}\right) \quad\forall A\in SL(n,{\mathbb{C}}),\qquad \operatorname{rank}_{{\mathbb{C}}}(\mathrm{d}f_A)=n^2-1. \] The source does not explicitly specify the range of $n$ or state an outer quantifier over $n$; these aspects of the intended family of questions require clarification.

The source adds that the requested immersion cannot be obtained algebraically and that the holomorphic existence statement is true for $n=2$. These are supplied source assertions, not independently verified conclusions. In the usual algebraic interpretation, the first assertion concerns regular algebraic maps with injective differential everywhere, but the source gives neither a definition nor a range of $n$ for that assertion. In particular, its treatment of the degenerate case $n=1$ requires clarification. The registry labels the problem open; this supplied label does not override the source's stated positive case $n=2$ or establish a current literature assessment.

Short English statement

The problem asks whether $SL(n,{\mathbb{C}})$ admits a holomorphic map into complex affine space of the same dimension whose differential is injective everywhere. The source reports a positive answer for $n=2$ and an obstruction to doing this algebraically. The intended range of $n$, the scope of the algebraic assertion, and the current status require review.

Sources

• Local UnsolvedMath registry, permanent ID $800261$, source code catalogue entry, title Several Complex Variables Problem List --- Problem 60. The supplied record labels the problem open and records the year 1995.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 60. 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 source consultation.
• 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 Buzzard, Fornaess, and Krantz; both attribution fields are retained, with the latter associated with the source list.
• The parenthetical assertions concerning algebraic impossibility and the case $n=2$ are supplied by the problem text. No separate supporting references for those assertions are supplied.