Spaces of non-degenerate maps between complex projective spaces
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RESEARCH
Spaces of non-degenerate maps between complex projective spaces Claudio Gómez-Gonzáles * Correspondence:
[email protected] Department of Mathematics, University of Chicago, Chicago, IL 60637, USA 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
Abstract We study the space Hold (CPm , CPn ) of degree d algebraic maps CPm → CPn , 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 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 → CPn 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 Ratd (CPm , CPm )/ PGLm+1 (C) under the action on the target is Q-acyclic up through dimension d − 2, partially generalizing a calculation of Milgram (Topology 36(5):1155–1192, 1997).
1 Introduction and main results Any holomorphic map f : CPm → CPn , m ≤ n, can be represented as f (z) = [f0 (z) : · · · : fn (z)],
(1.1)
where all fi ∈ C[z0 , . . . , zm ] are homogeneous of a common degree d and together have no common root. The degree of f is also characterized by a topological formula: f ∗ (ωCPn ) = d · ωCPm ,
(1.2)
where ωX ∈ H 2 (X; R) denotes the symplectic form of a Kähler manifold X, in this case corresponding to the Fubini–Study metric for specificity. This representation (1.1) is unique up to scaling, so the space of all such maps Hold (CPm , CPn ) is a projective resultant complement of complex dimension (n + 1) m+d − 1; see [6] for details. d In the m = 1 case, these functions are historically called rational maps and the notation Ratnd (C):= Hold (CP1 , CPn ) is used. In 1979, based on intuition from Morse theory, Segal [25] proved that the inclusion Ratnd (C) → Mapd (CP1 , CPn )
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(1.3)
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is a homotopy equivalence through dimension (2n − 1)d, where Mapd (CPm , CPn ) is the space of continuous maps satisfying (1.2) equipped with the compact-open topology. This inspired many generalizations, for example, extending the domain to genus g ≥ 1 curves and the target to Grassmannians or toric varieties; see, e.g., [2,4,11–13,15]. The works of Kozlowski and Yamaguchi [16] and Sasao [24] on linear maps, together with Segal-style stability due to Mostovoy [20] and Munguia-Villanueva [22], are apparently the only result
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