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Holomorphic maps of a Riemann surface into a flag manifold. (English) Zbl 0853.58022
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 G. Segal [Acta Math. 143, 39-72 (1979; Zbl 0427.55006)], J. Gravesen [Acta Math. 162, No. 3/4, 247-286 (1989; Zbl 0696.58014)], and C. P. Boyer, the author, B. M. Mann and 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.
Reviewer: T.Sasaki (Kobe)

##### MSC:
 58D15 Manifolds of mappings 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 55P35 Loop spaces
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