Configuration spaces and the space of rational curves on a toric variety

Author:
M. A. Guest

Journal:
Bull. Amer. Math. Soc. **31** (1994), 191-196

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*.

**[AJ]**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****[BD]**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****[BHMM1]**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**, 10.2307/2946532**[BHMM2]**-,*The topology of the space of rational maps into generalized flag manifolds*, preprint.**[CCMM]**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**, 10.1007/BF02398886**[CJS]**R. L. Cohen, J. D. S. Jones, and G. B. Segal,*Morse theory and classifying spaces*, preprint.**[CS]**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**, 10.1112/jlms/s2-43.3.509**[DT]**Albrecht Dold and René Thom,*Quasifaserungen und unendliche symmetrische Produkte*, Ann. of Math. (2)**67**(1958), 239–281 (German). MR**0097062****[EL]**J. Eells and L. Lemaire,*A report on harmonic maps*, Bull. London Math. Soc.**10**(1978), no. 1, 1–68. MR**495450**, 10.1112/blms/10.1.1**[Ep]**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****[Fu]**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****[GKY]**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**, 10.1007/BF02571953**[Gr]**Jens Gravesen,*On the topology of spaces of holomorphic maps*, Acta Math.**162**(1989), no. 3-4, 247–286. MR**989398**, 10.1007/BF02392839**[Gu1]**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**, 10.2307/2374428**[Gu2]**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****[Gu3]**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**, 10.1093/qmath/45.1.57**[Ki1]**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**, 10.1007/BF02384399**[Ki2]**-,*Geometric invariant theory and the Atiyah-Jones 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]**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****[MM2]**Benjamin M. Mann and R. James Milgram,*On the moduli space of 𝑆𝑈(𝑛) monopoles and holomorphic maps to flag manifolds*, J. Differential Geom.**38**(1993), no. 1, 39–103. MR**1231702****[Od]**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****[Se]**Graeme Segal,*The topology of spaces of rational functions*, Acta Math.**143**(1979), no. 1-2, 39–72. MR**533892**, 10.1007/BF02392088**[Ta]**Clifford Henry Taubes,*The stable topology of self-dual moduli spaces*, J. Differential Geom.**29**(1989), no. 1, 163–230. MR**978084****[Uh]**K. Uhlenbeck,*Applications of non-linear analysis in topology*, Proc. Internat. Cong. Math., Kyoto, 1990, Springer, Tokyo, 1991, pp. 261-279.**[Va]**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**

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DOI:
https://doi.org/10.1090/S0273-0979-1994-00515-4

Article copyright:
© Copyright 1994
American Mathematical Society