The index theorem for manifolds with corners
Let \(X\) be a complex manifold, \(M\) be a coherent \(D_X\)-module endowed with a good filtration, \(U\) an open subset of \(X\) with smooth boundary. We first prove that the \(K\)-theoretical class associated to \((M, \overline{U})\), constructed by \textit{L. Boutet de Monvel} and \textit{B. Malgrange} [Ann. Sci. École Norm. Supér., IV. Sér. 23, 151-192 (1990; Zbl 0705.58047)] can be obtained as the ``convolution product'' (in the sense of \textit{P. Schapira} and \textit{J. P. Schneiders} [C. R. Acad. Sci., Paris, Sér. I 311, 83-86 (1990; Zbl 0728.32011), ibid. 312, 81-84 (1991; Zbl 0724.32006)] in their theory of elliptic pairs) of the class associated to \(\text{gr}(M)\), the graded module of \(M\), and a class associated to \(\overline{U}\). This allows us to generalize to the case where \(U\) is a real analytic manifold with corners, and to extend the relative index theorem of L. Boutet de Monvel and B. Malgrange to this more general situation.
scientific article; zbMATH DE number 891042