Configuration spaces and the space of rational curves on a toric variety
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Abstract:
The space of holomorphic maps from ${S^2}$ 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 $n(D)$ such that the inclusion of this space in the corresponding space of continuous maps induces isomorphisms of homotopy groups up to dimension $n(D)$, 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 $n(D) \to \infty$ as $D \to \infty$. We consider the problem for another class of varieties, namely, toric varieties. For smooth toric varieties and certain singular ones, $n(D)$ may be computed, and $n(D) \to \infty$ as $D \to \infty$. For other singular toric varieties, however, it turns out that $n(D)$ cannot always be made arbitrarily large by a suitable choice of D.References
- M. F. Atiyah and J. D. S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), no. 2, 97–118. MR 503187, DOI 10.1007/BF01609489
- Christopher I. Byrnes and Tyrone E. Duncan, On certain topological invariants arising in system theory, New directions in applied mathematics (Cleveland, Ohio, 1980) Springer, New York-Berlin, 1982, pp. 29–71. MR 661283
- C. P. Boyer, J. C. Hurtubise, B. M. Mann, and R. J. Milgram, The topology of instanton moduli spaces. I. The Atiyah-Jones conjecture, Ann. of Math. (2) 137 (1993), no. 3, 561–609. MR 1217348, DOI 10.2307/2946532 —, The topology of the space of rational maps into generalized flag manifolds, preprint.
- F. R. Cohen, R. L. Cohen, B. M. Mann, and R. J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), no. 3-4, 163–221. MR 1097023, DOI 10.1007/BF02398886 R. L. Cohen, J. D. S. Jones, and G. B. Segal, Morse theory and classifying spaces, preprint.
- Ralph L. Cohen and Don H. Shimamoto, Rational functions, labelled configurations, and Hilbert schemes, J. London Math. Soc. (2) 43 (1991), no. 3, 509–528. MR 1113390, DOI 10.1112/jlms/s2-43.3.509
- Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
- J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68. MR 495450, DOI 10.1112/blms/10.1.1
- S. I. Èpšteĭn, Fundamental groups of the spaces of sets of relatively prime polynomials, Funkcional. Anal. i Priložen. 7 (1973), no. 1, 90–91 (Russian). MR 0345126
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- M. A. Guest, A. Kozlowski, and K. Yamaguchi, The topology of spaces of coprime polynomials, Math. Z. 217 (1994), no. 3, 435–446. MR 1306670, DOI 10.1007/BF02571953
- Jens Gravesen, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989), no. 3-4, 247–286. MR 989398, DOI 10.1007/BF02392839
- M. A. Guest, Topology of the space of absolute minima of the energy functional, Amer. J. Math. 106 (1984), no. 1, 21–42. MR 729753, DOI 10.2307/2374428
- Martin A. Guest, Instantons, rational maps, and harmonic maps, Mat. Contemp. 2 (1992), 113–155. Workshop on the Geometry and Topology of Gauge Fields (Campinas, 1991). MR 1303160
- M. A. Guest, On the space of holomorphic maps from the Riemann sphere to the quadric cone, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 177, 57–75. MR 1269290, DOI 10.1093/qmath/45.1.57
- Frances Kirwan, On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat. 24 (1986), no. 2, 221–275. MR 884188, DOI 10.1007/BF02384399 —, Geometric invariant theory and the Atiyah-Jones conjecture, Proc. Sophus Lie Memorial Conf., Oslo, 1992 (to appear). J. Milnor, Morse theory, Ann. of Math. Stud., vol. 51, Princeton Univ. Press, Princeton, NJ, 1963.
- Benjamin M. Mann and R. James Milgram, Some spaces of holomorphic maps to complex Grassmann manifolds, J. Differential Geom. 33 (1991), no. 2, 301–324. MR 1094457
- Benjamin M. Mann and R. James Milgram, On the moduli space of $\textrm {SU}(n)$ monopoles and holomorphic maps to flag manifolds, J. Differential Geom. 38 (1993), no. 1, 39–103. MR 1231702
- Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
- Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088
- Clifford Henry Taubes, The stable topology of self-dual moduli spaces, J. Differential Geom. 29 (1989), no. 1, 163–230. MR 978084 K. Uhlenbeck, Applications of non-linear analysis in topology, Proc. Internat. Cong. Math., Kyoto, 1990, Springer, Tokyo, 1991, pp. 261-279.
- V. A. Vassiliev, Complements of discriminants of smooth maps: topology and applications, Translations of Mathematical Monographs, vol. 98, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by B. Goldfarb. MR 1168473, DOI 10.1090/conm/478
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 31 (1994), 191-196
- MSC: Primary 55P99; Secondary 14M25, 55Q99
- DOI: https://doi.org/10.1090/S0273-0979-1994-00515-4
- MathSciNet review: 1260521