Singularities and noncommutative geometry (Q2754588)

The author and Legrand study differential forms on singular varieties and obtain a `mixed complex' whose cyclic homology is related to intersection homology [\textit{J. P. Brasselet} and \textit{A. Legrand}, Ann. Scoula Normale Sup. Pisa, Ser. IV, XXI, 213-234 (1994; Zbl 0839.55005) and London Math. Soc. Lect. Notes 263 Cambridge, 175-187 (1999; Zbl 0949.55003) hereafter referred to as [1], Algebras of differential functions on singular varieties, preprint no. 118, Univ. Paul Sabatier (1998), hereafter referred to as [2]]. This paper deals on these results.NEWLINENEWLINENEWLINEThe author's research aims to get an index theorem on singular varieties. So in Section 1, the Atiyah-Singer index theorem is reviewed. Section 2 is devoted to the definition of intersection homology. To get a de Rham theorem for intersection homology shadow forms, generalization of the Whitney forms, are defined in Section 3 [\textit{J. P. Brasselet}, \textit{M. Goresky} and \textit{R. MacPherson}, Am. J. Math. 113, 1019-1052 (1991; Zbl 0748.55002)]. Inspired by the properties of shadow forms, controlled differential forms on a singular pseudovariety endowed with a Thom-Mather stratification are defined in Section 4. For a cone \(c(L)=L \times[0,1]/L \times\{0\}\), \(L\) a compact Riemannian manifold, the algebra of intersection functions \(IC^\infty (cL)\) is defined by \(rA\widehat\otimes C^\infty L+A\), where NEWLINE\[NEWLINEA=\biggl\{ a\in C^\infty\bigl( (0,1)\bigr):\forall k\sup_{r\in (0,1)} \bigl |(r\partial_r)^k a(r)\bigr |< \infty\biggr\}.NEWLINE\]NEWLINE In the algebra generated by \(IC^\infty (cL)\) and \(d\), the subalgebra of differential forms which do not contain terms with element \(dr\) is denoted by \(\widetilde \Omega^* (cL)\). The de Rham complex of forms with poles of order \(\beta\) is defined by NEWLINE\[NEWLINEI \Omega^*_\beta (cL)=B^*_\beta\widehat \otimes_A\widetilde \Omega^*(cL), \quad B^*_\beta= r^{-\beta} \left(A\oplus A{dr\over r}\right).NEWLINE\]NEWLINE The author refers [1] for the subtle discussions on the topologies of these spaces. Then it is shown \(H^k(I\Omega^*_\beta (cL))=H^k (\Omega^*L)\) if \(k\leq [\beta]-1\), and \(H^k (I\Omega^*_\beta (cL))\) vanishes if \(k>[\beta] -1\), provided \(\beta>0\) is not an integer. These constructions and results are extended to iterated cone. Then by using sheaf theory, de Rham theorem for intersection homology is obtained (Th. 4.7, cf. [1]). In Section 5, Hochschild homology, cyclic homology and periodic cyclic homology \(PHC_{o/e}({\mathcal A})\) are reviewed and Connes' result \(PHC_{o/e}(C^\infty (M))\cong \oplus_{o/e} H^*_{dR}(M)\) is stated. In Section 6, the last Section, by using previous results, a mixed complex on a cone on a smooth manifold is defined, and Connes' result is extended to this cone (Theorem 6.3). The author says this result is extended to the stratified varieties in [2] and will be useful to extend index theorem to singular varieties.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00034].