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The homotopy type of spaces of rational curves on a toric variety. (English) Zbl 06963141
Summary: Spaces of holomorphic maps from the Riemann sphere to various complex manifolds have played an important role in several areas of mathematics (e.g. linear control theory and mathematical physics ([2], [3])). G. Segal [22] investigated the homotopy type of spaces of holomorphic maps on complex projective spaces and M. Guest [10] generalized Segal’s result for compact smooth toric varieties. Recently Mostovoy-Villanueva [20] improved the homology stability dimension obtained by Guest. In this paper we generalize their result [20] for certain non-compact smooth toric varieties by the careful analysis of toric varieties with the scanning maps.
55P10 Homotopy equivalences in algebraic topology
55R80 Discriminantal varieties and configuration spaces in algebraic topology
55P35 Loop spaces
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI
[1] Adamaszek, M.; Kozlowski, A.; Yamaguchi, K., Spaces of algebraic and continuous maps between real algebraic varieties, Q. J. Math., 62, 771-790, (2011) · Zbl 1245.14060
[2] Atiyah, M. F.; Jones, J. D.S., Topological aspects of Yang-Mills theory, Commun. Math. Phys., 59, 97-118, (1978) · Zbl 0387.55009
[3] Atiyah, M. F., Instantons in dimension two and four, Commun. Math. Phys., 93, 437-451, (1984) · Zbl 0564.58040
[4] Buchstaber, V. M.; Panov, T. E., Torus actions and their applications in topology and combinatorics, Univ. Lecture Note Series, vol. 24, (2002), Amer. Math. Soc. Providence · Zbl 1012.52021
[5] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A., The double suspension and exponents of the homotopy groups of spheres, Ann. Math., 110, 549-565, (1979) · Zbl 0443.55009
[6] Cox, D. A., The homogenous coordinate ring of a toric variety, J. Algebraic Geom., 4, 17-50, (1995) · Zbl 0846.14032
[7] Cox, D. A., The functor of a smooth toric variety, Tohoku Math. J., 47, 251-262, (1995) · Zbl 0828.14035
[8] Cox, D. A.; Little, J. B.; Schenck, H. K., Toric varieties, Graduate Studies in Math., vol. 124, (2011), Amer. Math. Soc. · Zbl 1223.14001
[9] Guest, M. A., Instantons, rational maps and harmonic maps, Math. Contemp., 2, 113-155, (1992) · Zbl 0875.30005
[10] Guest, M. A., The topology of the space of rational curves on a toric variety, Acta Math., 174, 119-145, (1995) · Zbl 0826.14035
[11] Guest, M. A.; Kozlowski, A.; Yamaguchi, K., The topology of spaces of coprime polynomials, Math. Z., 217, 435-446, (1994) · Zbl 0861.55015
[12] Guest, M. A.; Kozlowski, A.; Yamaguchi, K., Spaces of polynomials with roots of bounded multiplicity, Fundam. Math., 116, 93-117, (1999) · Zbl 1016.55004
[13] Kallel, S., Divisor spaces on punctured Riemann surfaces, Trans. Am. Math. Soc., 350, 135-164, (1998) · Zbl 0889.57038
[14] Kozlowski, A.; Ohno, M.; Yamaguchi, K., Spaces of algebraic maps from real projective spaces to toric varieties, J. Math. Soc. Jpn., 68, 745-771, (2016) · Zbl 1353.55009
[15] Kozlowski, A.; Yamaguchi, K., Simplicial resolutions and spaces of algebraic maps between real projective spaces, Topol. Appl., 160, 87-98, (2013) · Zbl 1276.55012
[16] Kozlowski, A.; Yamaguchi, K., The homotopy type of spaces of coprime polynomials revisited, Topol. Appl., 206, 284-304, (2016) · Zbl 1387.55012
[17] Kozlowski, A.; Yamaguchi, K., The homotopy type of spaces of polynomials with bounded multiplicity, Publ. Res. Inst. Math. Sci. Kyoto Univ., 52, 297-308, (2016) · Zbl 1359.55005
[18] Mostovoy, J., Spaces of rational maps and the stone-Weierstrass theorem, Topology, 45, 281-293, (2006) · Zbl 1086.58005
[19] Mostovoy, J., Truncated simplicial resolutions and spaces of rational maps, Q. J. Math., 63, 181-187, (2012) · Zbl 1237.58012
[20] Mostovoy, J.; Munguia-Villanueva, E., Spaces of morphisms from a projective space to a toric variety, Rev. Colomb. Mat., 48, 41-53, (2014) · Zbl 1350.14037
[21] Panov, T. E., Geometric structures on moment-angle manifolds, Russ. Math. Surv., 68, 503-568, (2013) · Zbl 1283.57028
[22] Segal, G. B., The topology of spaces of rational functions, Acta Math., 143, 39-72, (1979) · Zbl 0427.55006
[23] Vassiliev, V. A., Complements of discriminants of smooth maps, (Topology and Applications, Translations of Math. Monographs, vol. 98, (1992), Amer. Math. Soc.), (revised edition 1994)
[24] Vassiliev, V. A., Topologia dopolneniy K diskriminantam, (1997), Fazis Moskva
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