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The Lawson homology for Fulton-MacPherson configuration spaces. (English) Zbl 1166.14014

If \(X\) is a complex projective variety, let \(F(X,n)\) be the dense open set in \(X^n\) consisting of pairwise distinct ordered \(n\)-tuples in \(X\). For every integer \(n\), the Fulton-MacPherson configuration space \(X^{[n]}\) is the Zariski closure of \(F(X,n)\) in \(X^n \times \prod_{|I|\geq 2}Bl_{\Delta _I}(X^I)\), where \(I\) runs through the subsets of \(\{1,\ldots, n\}\) of length at least two and \(\Delta _I\) is the small diagonal in \(X^I\). The variety \(X^{[n]}\) is a smooth compactification of \(F(X,n)\) and \(X^{[n]}\setminus F(X,n)\) is a simple normal crossing divisor in \(X^{[n]}\). Furthemore, \(X^{[n]}\) can be obtained in an algorithmic way by blowing up in a suitable order the strict transforms of the diagonals in \(X^n\).
The main result of the article is the computation of the Lawson homology groups of the Fulton-MacPherson configuration spaces, they are proved to be direct sum of Lawson homology groups of cartesian products of X. The proof relies on the structure theorem of Lawson homology for a blowup. Using the same method, the authors also compute the Deligne-Beilinson cohomology groups of \(X^{[n]}\).

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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