On the moduli space of \(SU(n)\) monopoles and holomorphic maps to flag manifolds. (English) Zbl 0790.53053

Mann, Benjamin M.; Milgram, R. James

J. Differ. Geom. 38, No. 1, 39-103 (1993).

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

Keywords:

holomorphic maps; complex flag manifolds; filtration; stability theorem