George Ioniţă
University of Bern, Switzerland & ETH Zürich, Switzerland

A classical result from any Linear Algebra course states that the group $\text{SL}_m(\mathbb{C})$ is generated by elementary matrices $\operatorname{Id} + \alpha E_{ij}$, $i \neq j$, i.e., matrices with ones on the diagonal and at most one nonzero element outside the diagonal. The same question for matrices in $\text{SL}_m(R)$, where $R$ is a commutative unital ring is a very complicated and much studied question.

If for example $R$ is the ring of complex valued functions (continuous or holomorphic) from a space $X$, then given a map $f : X \rightarrow \text{SL}_m(\mathbb{C})$ one has to find a factorization into a product of upper and lower diagonal unipotent matrices $$f(x) = \left({\begin{array}{cc} 1 & 0 \\ G_1(x) & 0 \end{array}}\right) \left({\begin{array}{cc} 1 & G_2(x) \\ 0 & 1 \end{array}}\right) \cdots \left({\begin{array}{cc} 1 & G_N(x) \\ 0 & 1 \end{array}}\right), $$ where the $G_i$'s are maps $G_i : X \rightarrow \mathbb{C}^{m(m-1)/2}$. A necessary condition for this factorization to exist is that the map $f$ is homotopic to a constant map (or null-homotopic).

For continuous complex valued functions on a finite dimensional topological space $X$, the problem was studied for a long time and it was finally solved by Vaserstein. In 2012 Ivarsson and Kutzschebauch settled its holomorphic analogue. This is also called Gromov's Vaserstein problem as it was suggested by Gromov in 1989 as a possible application of his h-principle. The first generalization to the case of (special) vector bundle automorphisms was done in the topological case by Hultgren and Wold.

We will present the holomorphic counterpart of Hultgren and Wold's result, but only for holomorphic vector bundles of rank $2$. Our main result is the following: let $X$ be a Stein space and $E \rightarrow X$ be a holomorphic vector bundle of rank $2$ over $X$. Then a special holomorphic automorphism can be written as a (finite) product of unipotent holomorphic automorphisms if and only if it is null-homotopic. In the case of a trivial vector bundle $E \simeq X \times \mathbb{C}^n$, the special holomorphic automorphisms correspond to holomorphic maps $f : X \rightarrow \text{SL}_n(\mathbb{C})$.