Spaces of non-degenerate maps between complex projective spaces

Abstract

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).

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Acknowledgements

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.

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Correspondence to Claudio Gómez-Gonzáles.

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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.

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Gómez-Gonzáles, C. Spaces of non-degenerate maps between complex projective spaces. Res Math Sci 7, 26 (2020). https://doi.org/10.1007/s40687-020-00224-5

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