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Spaces of non-degenerate maps between complex projective spaces. (English) Zbl 1460.55013

For positive integers \(1\leq m\leq n\) and \(d\geq 1\), let \(\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) denote the space of all holomorphic maps \(f:\mathbb{C}\mathrm{P}^m\to \mathbb{C}\mathrm{P}^n\) of degree \(d\). A map \(f\in \mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) is called degenerate if its image \(f(\mathbb{C}\mathrm{P}^m)\) is contained in a hyperplane of \(\mathbb{C}\mathrm{P}^n\), and let \(_{n}\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n) \) be the subspace of \(\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) consisting of all maps \(f\in \mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) such that \(f\) is non-degenerate. In this paper, the author studies the topologies of the two spaces \(_{n}\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) and \(\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) from the point of view of homological stability. In particular, when \(m=1\), he proves that the inclusion map \(_{n}\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^1,\mathbb{C}\mathrm{P}^n)\to \mathrm{Hol}_d(\mathbb{C}\mathrm{P}^1,\mathbb{C}\mathrm{P}^n)\) induces an isomorphism on homology \(H_k(\ ;\mathbb{Z})\) for all \(k\leq 2(d-n)\). Moreover, when \(m>1\), he proves that the inclusion map \(_{n}\mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\to \mathrm{Hol}_d(\mathbb{C}\mathrm{P}^m,\mathbb{C}\mathrm{P}^n)\) induces an isomorphism on homology \(H_k(\ ;\mathbb{Q})\) for all \(k\leq d(2n-2m+1)-1\). He also obtains a similar result for subspaces of most degenerate, non-degenerate, and degenerate holomorphic maps, and he generalizes some calculations previously obtained by R. J. Milgram [Topology 36, No. 5, 1155–1192 (1997; Zbl 0894.57016)].

MSC:

55P62 Rational homotopy theory
55R80 Discriminantal varieties and configuration spaces in algebraic topology
14E05 Rational and birational maps
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables

Citations:

Zbl 0894.57016
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References:

[1] Astey, L.; Gitler, S.; Micha, E.; Pastor, G., Cohomology of complex projective Stiefel manifolds, Can. J. Math., 51, 5, 897-914 (1999) · Zbl 0938.55007
[2] Boyer, C.; Hurtubise, J.; Milgram, R., Stability theorems for spaces of rational curves, Int. J. Math., 12, 223-262 (2001) · Zbl 1110.57302
[3] Bergeron, M., Filom, K., Nariman, S.: Topological aspects of the dynamical moduli space of rational maps. Preprint arXiv:1908.10792 (2019)
[4] Cohen, F.; Cohen, R.; Mann, B.; Milgram, R., The topology of rational functions and divisors of surfaces, Acta Math., 166, 163-221 (1991) · Zbl 0741.55005
[5] Crawford, T., Full holomorphic maps from the Riemann sphere to complex projective spaces, J. Differ. Geom., 38, 161-189 (1993) · Zbl 0784.58015
[6] D’Andrea, C.; Dickenstein, A., Explicit formulas for the multivariate resultant, J. Pure Appl. Algebra., 164, 59-86 (2001) · Zbl 1066.14061
[7] Deligne, P., Étale cohomology. Lecture Notes in Mathematics (1977), Berlin: Springer, Berlin
[8] Eisenbud, D.; Harris, J., On varieties of minimal degree (a centennial account), Proc. Sympos. Pure Math., 46, 3-13 (1987)
[9] Farb, B.; Wolfson, J., Topology and arithmetic of resultants. I, N. Y. J. Math., 22, 801-821 (2015) · Zbl 1379.55016
[10] Farb, B.; Wolfson, J.; Wood, M., Coincidences between homological densities, predicted by arithmetic, Adv. Math., 352, 670-716 (2019) · Zbl 1459.57030
[11] Guest, M., The topology of the space of rational curves on a toric variety, Acta Math., 174, 1, 119-145 (1995) · Zbl 0826.14035
[12] Guest, M.; Kozlowski, A.; Yamaguchi, K., Spaces of polynomials with roots of bounded multiplicity, Fundam. Math., 161, 93-117 (1999) · Zbl 1016.55004
[13] Kallel, S.; Milgram, R., The geometry of the space of holomorphic maps from a Riemann surface to a complex projective space, J. Differ. Geom., 47, 321-375 (1997) · Zbl 0912.58006
[14] Kallel, S.; Salvatore, P., Rational maps and string topology, Geom. Topol., 10, 1579-1606 (2006) · Zbl 1204.58006
[15] Kirwan, F., On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Mat., 24, 1-2, 221-275 (1985) · Zbl 0625.14026
[16] Kozlowski, A.; Yamaguchi, K., Spaces of holomorphic maps between complex projective spaces of degree one, Topol. Appl., 132, 139-145 (2003) · Zbl 1165.55302
[17] Milgram, R., The structure of spaces of Toeplitz matrices, Topology, 36, 5, 1155-1192 (1997) · Zbl 0894.57016
[18] Milne, J.S.: Lectures on Etale Cohomology (v2.21), 202 (2013). Available at www.jmilne.org/math/
[19] Møller, J., On spaces of maps between complex projective spaces, Am. Math. Soc., 91, 3, 471-476 (1984) · Zbl 0514.55011
[20] Mostovoy, J., Spaces of rational maps and the Stone-Weierstrass theorem, Topol. Appl., 45, 2, 281-293 (2006) · Zbl 1086.58005
[21] Mostovoy, J., Truncated simplicial resolutions and spaces of rational maps, Q. J. Math., 63, 181-187 (2012) · Zbl 1237.58012
[22] Mostovoy, J.; Munguia-Villanueva, E., Spaces of morphisms from a projective space to a toric variety, Rev. Colombiana Mat., 48, 1, 41-53 (2014) · Zbl 1350.14037
[23] Ruiz, C., The cohomology of the complex projective Stiefel manifold, Am. Math. Soc., 146, 12, 541-547 (1969) · Zbl 0193.23902
[24] Sasao, S., The homotopy of \({\rm Map}({\mathbb{CP}}^m,{\mathbb{CP}}^n)\), J. Lond. Math. Soc., 2, 2-8, 193-197 (1974) · Zbl 0284.55020
[25] Segal, G., The topology of spaces of rational functions, Acta Math., 143, 1-2, 39-72 (1979) · Zbl 0427.55006
[26] Vassiliev, V., How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces, Tr. Mat. Inst. Steklova., 225, 121-140 (1999) · Zbl 0981.55008
[27] Yamaguchi, K., Fundamental groups of spaces of holomorphic maps and group actions, J. Math. Kyoto Univ., 44, 3, 479-492 (2004) · Zbl 1088.55010
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