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The connectedness of the moduli space of maps to homogeneous spaces. (English) Zbl 1076.14517

Fukaya, K. (ed.) et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14–18, 2000. Singapore: World Scientific (ISBN 981-02-4714-1/hbk). 187-201 (2001).
Summary: We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space \(G/P\) by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to \(G/P\) is a rational variety. The rationality argument depends essentially upon rationality results for quotients of \(\text{SL}_2\) representations proven by P. Katsylo [Mosc. Univ. Math. Bull. 39, No. 5, 80–83 (1984; Zbl 0582.20032)] and F. Bogomolov [in: Algebraic geometry, Proc. Conf. Vancouver 1984, CMS Conf. Proc. 6, 17–37 (1986; Zbl 0593.14018)].
For the entire collection see [Zbl 0980.00036].

MSC:

14H10 Families, moduli of curves (algebraic)
14M17 Homogeneous spaces and generalizations
14E08 Rationality questions in algebraic geometry
14D22 Fine and coarse moduli spaces
14L30 Group actions on varieties or schemes (quotients)
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