Holomorphic Factorization of Mappings into the Symplectic Group

TL;DR

This paper proves that complex holomorphic mappings into the symplectic group can be factorized into elementary matrices if and only if they are null-homotopic, with a bound on the number of factors depending on the dimension.

Contribution

It establishes a holomorphic factorization criterion for symplectic matrices over Stein spaces, linking algebraic decomposition to topological null-homotopy.

Findings

Factorization into elementary matrices is equivalent to null-homotopy.

The number of factors is bounded by a constant depending on dimension.

Provides a topological and algebraic characterization of holomorphic symplectic matrices.

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.