Prof. dr. Frank Kutzschebauch: Holomorphic factorization of maps into the special linear group
Source: Mathematics colloquium
Univerza v Bernu, Švica
It is standard material in a Linear Algebra course that the group SLm(C) is generated by elementary matrices E + α eij i ≠ j, i.e., matrices with 1's on the diagonal and all entries outside the diagonal are zero, except one entry. The same question for matrices in SLm(R) where R is a commutative ring instead of the field C is much more delicate, interesting is the case that R is the ring of complex valued functions (continuous, smooth, algebraic or holomorphic) from a space X.
For m≥ 3 (and any n) it is a deep result of Suslin that any matrix in SLm(C[Cn]) decomposes as a finite product of unipotent (and equivalently elementary) matrices. In the case of continuous complex valued functions on a topological space X the problem was studied and solved by Thurston and Vaserstein. For rings of holomorphic functions on Stein spaces, in particular on Cn, this problem was explicitly posed as the Vaserstein problem by Gromov in the 1980's. In this talk we explain a complete solution to Gromov's Vaserstein Problem; from a joint work with B. Ivarsson. The proof uses a very advanced version of the Oka-principle proposed by Gromov and proved in recent years by Forstnerič: An elliptic stratified submersion over a Stein space admits a holomorphic section iff it admits a continuous section.