Index Theorem for Elliptic Pairs

Index Theorem for Elliptic Pairs

Index Theorem for Elliptic Pairs
  • Année : 1994
  • Tome : 224
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
  • Class. Math. : 58G05, 32C99, 58G10, 14F12, 46M20, 46H25
  • Nb. de pages : 113
  • ISSN : 0303-1179
  • DOI : 10.24033/ast.270

In this book, we investigate the relative index theorem in the framework of algebraic analysis. An elliptic pair on a complex analytic manifold is the data of a coherent $\mathcal {D}$-module (i.e., a system of partial differential equations) and an $R$-constructible sheaf (for example, a local system on a subanalytic subset) satisfying some transversality condition. A natural problem is to find conditions under which the complex of holomorphic solutions of such a pair has finite-dimensional cohomology and then to compute the corresponding Euler-Poincaré characteristic. Here, we solve a relative version of this problem. We give finiteness and duality theorems unifying and extending ical results for coherent $\mathcal {O}$-modules (e.g., Grauert's theorem) or coherent $\mathcal {D}$-modules, as well as for elliptic systems. Then, to an elliptic pair, we attach a characteristic cohomology on the cotangent bundle and prove it behaves well under direct and inverse images. In particular, we establish a new kind of index formula.

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