An algebraic compactification for spaces of holomorphic curves in complex Grassmann manifolds

We construct a compactification of the space of holomorphic curves of fixed degree in a finite-dimensional complex Grassmann manifold using basic algebra. The algebraic compactification is defined as the quotient of n-tuples of linearly independent elements in a C[z]-module. The complex analytic structure on the space of holomorphic curves of fixed degree extends to the algebraic compactification. We show that there is a homotopy equivalence through a range increasing with the degree between the compactified spaces and an infinite-dimensional complex Grassmann manifold. These compact spaces form a direct system, indexed by the degree, whose direct limit is homotopy equivalent to an infinite-dimensional complex Grassmann manifold.