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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.

##### MSC:
 55P10 Homotopy equivalences in algebraic topology 55P35 Loop spaces 55P15 Classification of homotopy type
##### Keywords:
mapping space; homotopy type; holomorphic map
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##### References:
 [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
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