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    On the Negative KK-theory of Singular Varieties

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    Let XX be an nn-dimensional variety over a field kk of characteristic zero, regular in codimension 1 with singular locus ZZ. In this paper we study the negative KK-theory of XX, showing that when ZZ is sufficiently nice, K1βˆ’n(X)K_{1-n}(X) is an extension of KH1βˆ’n(X)KH_{1-n}(X) by a finite dimensional vector space, which we compute explicitly. We also show that KH1βˆ’n(X)KH_{1-n}(X) almost has a geometric structure. Specifically, we give an explicit 1-motive [Lβ†’G][L \rightarrow G] and a map G(k)β†’KH1βˆ’n(X)G(k) \rightarrow KH_{1-n}(X) whose kernel and cokernel are finitely generated abelian groups.Comment: 24 page
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