Equivariant holomorphic embeddings from the complex projective line into complex Grassmannian of 2-planes

Abstract

Using gauge theory, we classify $\text{SU}(2)$-equivariant holomorphic embeddings from $\mathbf CP^1$ with the Fubini-Study metric into Grassmann manifold $\mathit{Gr}_{N-2}(\mathbf C^N)$. It is shown that the moduli spaces of those embeddings are identified with the gauge equivalence classes of non-flat invariant connections satisfying semi-positivity on the vector bundles given by \textit{extensions} of line bundles. A topology on the moduli is obtained by means of $L^2$-inner product on Dolbeault cohomology group to which the extension class belongs. The compactification of the moduli is provided with geometric meaning from viewpoint of maps.