The space of linear maps into a Grassmann manifold (Q2868535)

For a complex projective variety \(M\), denote by \(\text{Hol}(M)\) the space of all holomorphic maps from the Riemann sphere \(\mathbb{P}^1\) into \(M\). It is known that \(\text{Hol}(M)\) can be given the structure of a quasiprojective variety and hence it has the homotopy type of a finite CW complex. In general \(\text{Hol}(M)\) is not connected. Computing the exact homotopy type of various components in \(\text{Hol}(M)\) in terms of well known spaces therefore presents itself as an interesting problem. In the present paper, the authors study the case of \(M =\text{Gr}(n, m)\), the Grassmann manifold of \(n\)-dimensional complex linear subspaces in \(\mathbb{C}^{n+m}\). For a holomorphic map \(f:\mathbb{P}^1 \to\text{Gr}(n, m)\), the degree \(d\geq 0\) is the multiplicative factor for the induced map between the second homology groups (\(\mathbb{Z}\) for both spaces). It turns out that two maps in \(\text{Hol}(\text{Gr}(n, m))\) are in the same connected component if and only if they have the same degree. Hence the components of \(\text{Hol}(\text{Gr}(n, m))\) are determined by the degree \(d\) of maps. Denote by \(\text{Hol}_d(\text{Gr}(n, m))\) the component of maps of degree \(d\). Clearly \(\text{Hol}_0(\text{Gr}(n, m))=\text{Gr}(n, m)\) are the constant maps. The space \(\text{Hol}_1(\text{Gr}(n, m))\), which is the main object of a study in the paper, is the component of linear maps. In one of the main theorems, a very complete description is given of \(\text{Hol}_1(\text{Gr}(n, m))\) as a sphere bundle over a flag manifold. For the quadric Grassmann manifold \(\text{Gr}(2, 2)\) the paper contains a detailed study of the homology of \(\text{Hol}_1(\text{Gr}(2, 2))\). The paper is well structured and adequately referenced.