Configuration spaces and the space of rational curves on a toric variety
Author:
M. A. Guest
Journal:
Bull. Amer. Math. Soc. 31 (1994), 191196
MSC:
Primary 55P99; Secondary 14M25, 55Q99
MathSciNet review:
1260521
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Abstract: The space of holomorphic maps from to a complex algebraic variety X, i.e. the space of parametrized rational curves on X, arises in several areas of geometry. It is a well known problem to determine an integer such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension , where D denotes the homotopy class of the maps. The solution to this problem is known for an important but special class of varieties, the generalized flag manifolds: such an integer may be computed, and as . We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, may be computed, and as . For other singular toric varieties, however, it turns out that cannot always be made arbitrarily large by a suitable choice of D.
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 R. L. Cohen, J. D. S. Jones, and G. B. Segal, Morse theory and classifying spaces, preprint.
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 R. L. Cohen and D. H. Shimamoto, Rational functions, labelled configurations, and Hilbert schemes, J. London Math. Soc. (2) 43 (1991), 509528. MR 1113390 (93c:55009)
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 S. I. Epshtein, Fundamental groups of spaces of coprime polynomials, Funct. Anal. Appl. 7 (1973), 8283. MR 0345126 (49:9865)
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 W. Fulton, Introduction to toric varieties, Ann. of Math. Stud., vol. 131, Princeton Univ. Press, Princeton, NJ, 1993. MR 1234037 (94g:14028)
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 M. A. Guest, A. Kozlowski, and K. Yamaguchi, The topology of spaces of coprime polynomials, Math. Z. (to appear). MR 1306670 (95i:55014)
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 J. Graveson, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989), 247286. MR 989398 (90g:32023)
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 M. A. Guest, Topology of the space of absolute minima of the energy functional, Amer. J. Math. 106 (1984), 2142. MR 729753 (85h:58047)
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 [Ki2]
 , Geometric invariant theory and the AtiyahJones conjecture, Proc. Sophus Lie Memorial Conf., Oslo, 1992 (to appear).
 [Mi]
 J. Milnor, Morse theory, Ann. of Math. Stud., vol. 51, Princeton Univ. Press, Princeton, NJ, 1963.
 [MM1]
 B. M. Mann and R. J. Milgram, Some spaces of holomorphic maps to complex Grassmann manifolds, J. Differential Geom. 33 (1991), 301324. MR 1094457 (93e:55022)
 [MM2]
 , On the geometry of monopoles and holomorphic maps to flag manifolds, J. Differential Geom. 38 (1993), 39103. MR 1231702 (95c:58031)
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 [Ta]
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Additional Information
DOI:
http://dx.doi.org/10.1090/S027309791994005154
PII:
S 02730979(1994)005154
Article copyright:
© Copyright 1994
American Mathematical Society
