Spaces of non-degenerate maps between complex projective spaces


We study the space \({{\,\mathrm{Hol}\,}}_d(\mathbb {CP}^m,\mathbb {CP}^n)\) of degree d algebraic maps \(\mathbb {CP}^m \rightarrow \mathbb {CP}^n\), from the point of view of homological stability as discovered by Segal (Acta Math 143(1–2):39–72, 1979) and later explored by Mostovoy (Topol Appl 45(2):281–293, 2006), Cohen et al. (Acta Math 166:163–221, 1991), Farb and Wolfson (N Y J Math 22:801–821, 2015), and others. In particular, we calculate the \(\mathbb {Q}\)-cohomology ring explicitly in the case \(m=1\), as computed by Kallel and Salvatore (Geom Topol 10:1579–1606, 2006), and stably for when \(m>1\). In doing so, we expand a method, previously studied by Crawford (J Differ Geom 38:161–189, 1993), for analyzing spaces of maps \(X \rightarrow \mathbb {CP}^n\) by introducing subvarieties of non-degenerate functions that approximate the desired cohomologies both integrally and rationally in different ways. We also prove, when \(m=n\), that the orbit space \({{\,\mathrm{Rat}\,}}_d(\mathbb {CP}^m,\mathbb {CP}^m)/{{\,\mathrm{PGL}\,}}_{m+1}(\mathbb {C})\) under the action on the target is \(\mathbb {Q}\)-acyclic up through dimension \(d-2\), partially generalizing a calculation of Milgram (Topology 36(5):1155–1192, 1997).

This is a preview of subscription content, log in to check access.


  1. 1.

    Astey, L., Gitler, S., Micha, E., Pastor, G.: Cohomology of complex projective Stiefel manifolds. Can. J. Math. 51(5), 897–914 (1999)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Boyer, C., Hurtubise, J., Milgram, R.: Stability theorems for spaces of rational curves. Int. J. Math. 12, 223–262 (2001)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bergeron, M., Filom, K., Nariman, S.: Topological aspects of the dynamical moduli space of rational maps. Preprint arXiv:1908.10792 (2019)

  4. 4.

    Cohen, F., Cohen, R., Mann, B., Milgram, R.: The topology of rational functions and divisors of surfaces. Acta Math. 166, 163–221 (1991)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Crawford, T.: Full holomorphic maps from the Riemann sphere to complex projective spaces. J. Differ. Geom. 38, 161–189 (1993)

    MathSciNet  Article  Google Scholar 

  6. 6.

    D’Andrea, C., Dickenstein, A.: Explicit formulas for the multivariate resultant. J. Pure Appl. Algebra. 164, 59–86 (2001)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Deligne, P.: Étale cohomology. Lecture Notes in Mathematics, vol. 569. Springer, Berlin (1977)

    Google Scholar 

  8. 8.

    Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Proc. Sympos. Pure Math. 46, 3–13 (1987)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Farb, B., Wolfson, J.: Topology and arithmetic of resultants. I. N. Y. J. Math. 22, 801–821 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Farb, B., Wolfson, J., Wood, M.: Coincidences between homological densities, predicted by arithmetic. Adv. Math. 352, 670–716 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Guest, M.: The topology of the space of rational curves on a toric variety. Acta Math. 174(1), 119–145 (1995)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Guest, M., Kozlowski, A., Yamaguchi, K.: Spaces of polynomials with roots of bounded multiplicity. Fundam. Math. 161, 93–117 (1999)

    MathSciNet  MATH  Google Scholar 

  13. 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)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kallel, S., Salvatore, P.: Rational maps and string topology. Geom. Topol. 10, 1579–1606 (2006)

    MathSciNet  Article  Google Scholar 

  15. 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)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Kozlowski, A., Yamaguchi, K.: Spaces of holomorphic maps between complex projective spaces of degree one. Topol. Appl. 132, 139–145 (2003)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Milgram, R.: The structure of spaces of Toeplitz matrices. Topology 36(5), 1155–1192 (1997)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Milne, J.S.: Lectures on Etale Cohomology (v2.21), 202 (2013). Available at

  19. 19.

    Møller, J.: On spaces of maps between complex projective spaces. Am. Math. Soc. 91(3), 471–476 (1984)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mostovoy, J.: Spaces of rational maps and the Stone–Weierstrass theorem. Topol. Appl. 45(2), 281–293 (2006)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Mostovoy, J.: Truncated simplicial resolutions and spaces of rational maps. Q. J. Math. 63, 181–187 (2012)

    MathSciNet  Article  Google Scholar 

  22. 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)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Ruiz, C.: The cohomology of the complex projective Stiefel manifold. Am. Math. Soc. 146(12), 541–547 (1969)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sasao, S.: The homotopy of \({\rm Map}({\mathbb{CP}}^m,{\mathbb{CP}}^n)\). J. Lond. Math. Soc. 2(2–8), 193–197 (1974)

    Article  Google Scholar 

  25. 25.

    Segal, G.: The topology of spaces of rational functions. Acta Math. 143(1–2), 39–72 (1979)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Vassiliev, V.: How to calculate homology groups of spaces of nonsingular algebraic projective hypersurfaces. Tr. Mat. Inst. Steklova. 225, 121–140 (1999)

    Google Scholar 

  27. 27.

    Yamaguchi, K.: Fundamental groups of spaces of holomorphic maps and group actions. J. Math. Kyoto Univ. 44(3), 479–492 (2004)

    MathSciNet  Article  Google Scholar 

Download references


The author would like to thank Benson Farb for his invaluable advice and extensive comments. He is also grateful to Oishee Banerjee, Maxime Bergeron, Lei Chen, Ronno Das, Nir Gadish, and Will Sawin for many helpful conversations. The author is also indebted to Sadok Kallel, who pointed out previous homotopy-theoretic work by Crawford on non-degenerate maps under a different name, while the paper was in preprint.

Author information



Corresponding author

Correspondence to Claudio Gómez-Gonzáles.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This material is based on work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1746045 and conducted in space procured via the Jump Trading Mathlab Research Grant.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gómez-Gonzáles, C. Spaces of non-degenerate maps between complex projective spaces. Res Math Sci 7, 26 (2020).

Download citation