Holomorphic factorization of mappings into the symplectic group

Abstract

It is shown that any symplectic $2n\times 2n$ matrix, whose entries are complex holomorphic functions on a reduced Stein space, can be decomposed into a finite product of elementary symplectic matrices if and only if it is null-homotopic. Moreover, if this is the case, the number of factors can be bounded by a constant depending only on $n$ and the dimension of the space.