zbMATH — the first resource for mathematics

Spaces of holomorphic maps between complex projective spaces of degree one. (English) Zbl 1165.55302
Summary: For an integer \(d\geq 0\), let \(\text{Hol}_d(\mathbb CP^k,\mathbb CP^n)\) denote the space consisting of all holomorphic maps \(f:\mathbb CP^k\to\mathbb CP^n\) of degree \(d\). We study the homogeneous space structure of \(\text{Hol}_d(\mathbb CP^k,\mathbb CP^n)\) for the case \(d=1\). In particular we explicitly determine its homotopy type.

55P10 Homotopy equivalences in algebraic topology
55P35 Loop spaces
55P15 Classification of homotopy type
Full Text: DOI
[1] Atiyah, M.F.; Hitchin, N.J., The geometry and dynamics of magnetic monopoles, (1988), Princeton Univ. Press Princeton, NJ · Zbl 0671.53001
[2] Cohen, F.R.; Cohen, R.L.; Mann, B.M.; Milgram, R.J., The topology of rational functions and divisors of surfaces, Acta math., 166, 163-221, (1991) · Zbl 0741.55005
[3] Cohen, F.R.; Moore, J.C.; Neisendorfer, J.A., The double suspension and exponents of the homotopy groups of spheres, Ann. of math., 110, 549-565, (1979) · Zbl 0443.55009
[4] Cohen, R.L.; Shimamoto, D., Rational functions, labelled configurations and Hilbert schemes, J. London math. soc., 43, 509-528, (1991) · Zbl 0756.55005
[5] Donaldson, S.K., Nahm’s equations and the classification of monopoles, Comm. math. phys., 96, 387-407, (1984) · Zbl 0603.58042
[6] Guest, M.A.; Kozlowski, A.; Murayama, M.; Yamaguchi, K., The homotopy type of spaces of rational functions, J. math. Kyoto univ., 35, 631-638, (1995) · Zbl 0862.55011
[7] Guest, M.A.; Kozlowski, A.; Yamaguchi, K., The topology of spaces of coprime polynomials, Math. Z., 217, 435-446, (1994) · Zbl 0861.55015
[8] Guest, M.A.; Kozlowski, A.; Yamaguchi, K., Spaces of polynomials with roots of bounded multiplicity, Fund. math., 116, 93-117, (1999) · Zbl 1016.55004
[9] Guest, M.A.; Kozlowski, A.; Yamaguchi, K., Stable splitting of the space of polynomials with roots of bounded multiplicity, J. math. Kyoto univ., 38, 351-366, (1998) · Zbl 0917.55005
[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] Harris, J., Algebraic geometry, (1993), Springer Berlin
[12] Kozlowski, A.; Yamaguchi, K., Topology of complements of discriminants and resultants, J. math. soc. Japan, 52, 949-959, (2000) · Zbl 0974.55002
[13] Mostovoy, J., Spaces of rational loops on a real projective space, Trans. amer. math. soc., 353, 1959-1970, (2001) · Zbl 0979.55008
[14] Ono, Y.; Yamaguchi, K., Group actions on spaces of rational functions, Publ. RIMS Kyoto univ., 39, 173-181, (2003) · Zbl 1026.55011
[15] Sasao, S., The homotopy of \( Map(C P\^{}\{m\},C P\^{}\{n\})\), J. London math. soc., 8, 193-197, (1974)
[16] Segal, G.B., The topology of spaces of rational functions, Acta math., 143, 39-72, (1979) · Zbl 0427.55006
[17] Yamaguchi, K., Spaces of holomorphic maps with bounded multiplicity, Quart. J. math., 52, 249-259, (2001) · Zbl 0993.32012
[18] Yamaguchi, K., Universal coverings of spaces of holomorphic maps, Kyushu J. math., 56, 381-389, (2002) · Zbl 1041.55005
[19] K. Yamaguchi, Fundamental groups of spaces of holomorphic maps, Preprint · Zbl 1088.55010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.