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Holomorphic factorization of mappings into \(\mathrm{SL}_n(\mathbb{C})\). (English) Zbl 1243.32007
In the present paper, the authors solve Gromov’s Vaserstein problem. Namely, they show that a null-homotopic holomorphic mapping from a finite-dimensional reduced Stein space into \(\text{SL}_m(\mathbb{C})\) can be factored into a finite product of unipotent matrices with holomorphic entries.
After the Gromov-Eliashberg embedding theorem for Stein manifolds [Y. Eliashberg and M. Gromov, “Embeddings of Stein manifolds of dimension \(n\) into the affine space of dimension \(3n/2 + 1\)”, Ann. Math. (2) 136, No. 1, 123–135 (1992; Zbl 0758.32012); J. Schürmann, “Embeddings of Stein spaces into affine spaces of minimal dimension”, Math. Ann. 307, No. 3, 381–399 (1997; Zbl 0881.32007)], this is another deep application of Gromov’s holomorphic h-principle. M. Gromov [“Oka’s principle for holomorphic sections of elliptic bundles”, J. Am. Math. Soc. 2, No. 4, 851–897 (1989; Zbl 0686.32012)] extended the classical Oka-Grauert principle from bundles with homogeneous fibers to fibrations with elliptic fibers. The authors use the Oka-Grauert-Gromov-principle in its strongest form as elaborated by F. Forstnerič [“The Oka principle for sections of stratified fiber bundles”, Pure Appl. Math. Q. 6, No. 3, 843–874 (2010; Zbl 1216.32005)] and F. Forstnerič and J. Prezelj [“Extending holomorphic sections from complex subvarieties”, Math. Z. 236, No. 1, 43–68 (2001; Zbl 0968.32005)].
Let us describe the result in more detail. It is well known that the group \(\mathrm{SL}_m(\mathbb{C})\) is generated by elementary matrices \(E+\alpha e_{ij}\), \(i\neq j\), i.e., matrices with 1’s on the diagonal and all entries outside the diagonal are zero, except one entry. Equivalently, every matrix \(A\in \mathrm{SL}_m(\mathbb{C})\) can be written as a finite product of upper and lower diagonal matrices (in interchanging order). The same question for matrices in \(\mathrm{SL}_m(R)\), where \(R\) is a commutative ring (e.g. the ring of complex valued functions on a space \(X\) which are continuous, smooth, algebraic or holomorphic) instead of the field \(\mathbb{C}\), is much more delicate. For results in the algebraic setting, we refer to [P. M. Cohn, “On the structure of the \(\mathrm{GL}_2\) of a ring”, Publ. Math., Inst. Hautes Étud. Sci. 30, 365–413 (1966; Zbl 0144.26301); A. A. Suslin, “On the structure of the special linear group over polynomial rings”, Izv. Akad. Nauk SSSR, Ser. Mat. 41, 235–252 (1977; Zbl 0354.13009); F. Grunewald, J. Mennicke and L. Vaserstein, “On the groups \(\mathrm{SL}_2 (\mathbb{Z}[x])\) and \(\mathrm{SL}_2 (k[x,y])\)”, Isr. J. Math. 86, No. 1–3, 157–193 (1994; Zbl 0805.20042); D. Wright, “The amalgamated free product structure of \(\mathrm{GL}_2(k[X_1,\dots ,X_n ])\) and the weak Jacobian theorem for two variables”, J. Pure Appl. Algebra 12, 235–251 (1978; Zbl 0387.20039)].
We restrict our attention to the ring of holomorphic functions on a reduced Stein space \(X\). So, the problem amounts to factorizing a given holomorphic map \(X \rightarrow \mathrm{SL}_m(\mathbb{C})\) as a product of upper and lower diagonal unipotent matrices. Since any product of such matrices is homotopic to a constant map, one has to assume that the given map \(f\) is homotopic to a constant map, say null-homotopic. The main result of the present paper is a complete positive solution of Gromov’s Vaserstein problem as posed in [Gromov, loc.cit.]:
Let \(X\) be a finite dimensional reduced Stein space and \(f: X\rightarrow \mathrm{SL}_m(\mathbb{C})\) be a holomorphic mapping that is null-homotopic. Then there exists a natural number \(K\) and holomorphic mappings \(G_1, \dots, G_K: X\rightarrow \mathbb{C}^{m(m-1)/2}\) such that \(f\) can be written as a product of upper and lower diagonal unipotent matrices \[ f(x) = \left(\begin{matrix} 1 & 0\\ G_1(x) & 1\end{matrix}\right) \left(\begin{matrix} 1 & G_2(x)\\ 0 & 1\end{matrix}\right)\cdots \left(\begin{matrix} 1 & G_K(x)\\ 0 & 1\end{matrix}\right) \] for every \(x\in X\).
As indicated above, the proof relies on an application of the Oka-Grauert-Gromov-principle to certain stratified fibrations. For this, a topological solution to the problem is required, i.e., a factorization of \(f\) into a finite product of unipotent matrices with continuous entries. That was achieved by L. N. Vaserstein [“Reduction of a matrix depending on parameters to a diagonal form by addition operations”, Proc. Am. Math. Soc. 103, No. 3, 741–746 (1988; Zbl 0657.55005)] after some preliminary results by W. P. Thurston and L. N. Vaserstein [“On \(K_1\)-theory of the Euclidean space”, Topology Appl. 23, 145–148 (1986; Zbl 0611.18007)].
See [the authors, “On the number of factors in the unipotent factorization of holomorphic mappings into \(\text{SL}_2(\mathbb C)\)”, Proc. Am. Math. Soc. 140, No. 3, 823–838 (2012; Zbl 1250.32009)] for an effecitive version of the main theorem with a bound on the number of factors \(K\) for the group \(\mathrm{SL}_2(\mathbb{C})\).
The results of the paper under review had been announced in [“A solution of Gromov’s Vaserstein problem”, C. R., Math., Acad. Sci. Paris 346, No. 23–24, 1239–1243 (2008; Zbl 1160.32017)].

MSC:
32E10 Stein spaces, Stein manifolds
32Q55 Topological aspects of complex manifolds
15A23 Factorization of matrices
15A54 Matrices over function rings in one or more variables
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