Holomorphic maps of a Riemann surface into a flag manifold
Let \(\Sigma\) be a compact surface and \(X\) a complex manifold. The question of comparing two mapping spaces \(\text{Hol}(\Sigma, X)\) and \(\text{Map}(\Sigma, X)\) is an old problem studied from many points of view. One of the interests is knowing topological properties of both spaces and getting a stability result. One can study also natural subspaces of \(\text{Map}(\Sigma, X)\) given as critical sets of certain functionals. The main theorem of this paper is stated as follows. Let \(G/P\) be a flag manifold which has a cell-decomposition by the action of unipotent groups. Let \(Z_\alpha\) be closed cells of codimension 1. For a map \(f\) from a Riemann surface \(\Sigma\) to \(G/P\), define its degree by integer counting the points of \(f^* Z_\alpha\). By fixing these integers, we have a mapping space \(A = \text{Map}_{\{k_\alpha\}} (\Sigma, X)\) of fixed degrees \(k_{\{\alpha\}}\) and its subspace \(B = \text{Hol}_{\{k_\alpha\}} (\Sigma, X)\). The author proves, assuming \(k_{\{\alpha\}} > 0\), that there exist constants \(c_0\) and \(c_1\) depending only on \(G/P\) such that the inclusion \(B \subset A\) is isomorphic in the homology group of order smaller than \(c_0\) \(\min\{k_\alpha\} - c_1\). This result generalizes on the homology level several works by \textit{G. Segal} [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)], \textit{J. Gravesen} [Acta Math. 162, No. 3/4, 247-286 (1989; Zbl 0696.58014)], and \textit{C. P. Boyer}, the author, \textit{B. M. Mann} and \textit{R. J. Milgram} [Acta Math. 173, No. 1, 61-101 (1994; Zbl 0844.57037)], to mention a few. The problem of homotopical stability seems to be open.
scientific article; zbMATH DE number 895019