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Spaces of algebraic maps from real projective spaces to toric varieties. (English) Zbl 1353.55009
Given manifolds \(X\) and \(Y\), write \(\text{Map}^\ast(X,Y)\) for the space of bas-point preserving continuous maps from \(X\) to \(Y\). If \(X\) and \(Y\) are complex manifolds, J. Mostovoy [Topology 45, No. 2, 281–293 (2006; Zbl 1086.58005); Q. J. Math. 63, No. 1, 181–187 (2012; Zbl 1237.58012)] determined an integer \(n_D\) such that the inclusion map \(j_D : \text{Hol}^\ast_D(X,Y)\to \text{Map}^\ast(X,Y)\) is a homology equivalence through dimension \(n_D\), where \(D=(d_1,\ldots,d_r)\) is a tuple of integers and \(\text{Hol}^\ast_D(X,Y)\) denotes the space of holomorphic maps from \(X\) to \(Y\) of degree \(D\). Recently, J. Mostovoy and E. Munguia-Villanueva [Spaces of morphisms from a projective space to a toric variety, preprint, CP^m\) to a compact smooth toric variety \(X_\Sigma\) associated to a fan \(\Sigma\).
Given algebraic varieties \(X\) and \(Y\), write \(\text{Alg}^\ast_D(X,Y)\) for the space of algebraic (regular) maps from \(X\) to \(Y\) of degree \(D\) and \(A_D(X,Y)\) for the space of tuples of polynomials representing elements of \(\text{Alg}^\ast_D(X,Y)\). For a real projective space \(\mathbb{R}P^m\), the authors study the natural map \(i_D : A_D(m,X_\Sigma)\to \text{Map}^\ast(X,Y)\) and show that the induced map \(i'_D : A_D(m,X_\Sigma;g)\to F(\mathbb{R}R^m,X_\Sigma;g)\simeq\Omega^m X_\Sigma\) is a homology equivalence through a dimension \(n_D(d_1,\ldots,d_r;m)\).

MSC:
55R80 Discriminantal varieties and configuration spaces in algebraic topology
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
55P10 Homotopy equivalences in algebraic topology
55P35 Loop spaces
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