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On the moduli space of $$SU(n)$$ monopoles and holomorphic maps to flag manifolds. (English) Zbl 0790.53053
Summary: In this paper, motivated by questions in mathematical physics, we study the geometry of the components in the spaces of based holomorphic maps from the Riemann sphere to complex flag manifolds, which we denote by $$\text{Rat}_ C({\mathbb{F}}(J))$$. We decompose these spaces into smooth, in fact, complex strata each having a complex normal bundle. Using a modification of this filtration we study the forgetful map $$l_ *: H_ *(\text{Rat}_ C({\mathbb{F}}(J))\to H_ *(\Omega^ 2{\mathbb{F}}(J))$$ and prove an Atiyah-Jones type stability theorem. We also use the filtrations to determine the basic groups $$H_ *(\text{Rat}_{\vec 1}({\mathbb{F}}(J));{\mathbb{Z}})$$ and show that $$l_ *$$ has a nontrivial kernel for general flag manifolds.

MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58C10 Holomorphic maps on manifolds
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