On embeddings of the Grassmannian $Gr(2,m)$ into the Grassmannian $Gr(2,n)$

TL;DR

This paper investigates holomorphic embeddings of Grassmannians of 2-planes, characterizing when such embeddings are necessarily linear by analyzing Chern classes and Schubert cycles.

Contribution

It provides a classification of holomorphic embeddings of $Gr(2,m)$ into $Gr(2,n)$, identifying conditions under which all embeddings are linear.

Findings

Embeddings are linear under certain conditions on m and n.

Relations between Chern classes and Schubert cycles are key to classification.

Provides criteria for linearity of embeddings.

Abstract

In this {paper}, we consider holomorphic embeddings of $Gr(2,m)$ into $Gr(2,n)$. We study such embeddings by finding all possible total Chern classes of the pull-back of the universal bundles under these embeddings. Using the relations between Chern classes of the universal bundles and Schubert cycles, together with properties of holomorphic vector bundles of rank $2$ on complex Grassmannians, we find conditions on $m$ and $n$ for which any holomorphic embedding of $Gr(2,m)$ into $Gr(2,n)$ is linear.