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The index theorem for almost elliptic systems. (English) Zbl 0575.58029
Ennio de Giorgi Colloq., H. Poincaré Inst., Paris 1983, Res. Notes Math. 125, 17-29 (1988).
[For the entire collection see Zbl 0563.00011.]
Let X be a compact real analytic manifold with analytic boundary \(\partial X\), \(\tilde X\) be a complexification of X. The author constructs the compact strictly pseudoconvex manifold \(X_{\epsilon}\), where \(\epsilon >0\) and \(\{X_{\epsilon}\}\) form a fundamental system of neighbourhoods of X in \(\tilde X.\) Let D be a complex of analytic differential operators on X. Then D extends to \(X_{\epsilon}\) for small \(\epsilon >0\). A complex D is called almost elliptic if its extension to \(X_{\epsilon}\) is elliptic for all small \(\epsilon >0\). An aim of this paper is the index theorem for almost elliptic systems. The proof is otherwise modelled on Grothendieck’s proof of the Riemann-Roch theorem.
Reviewer: V.Deundjak

58J20 Index theory and related fixed-point theorems on manifolds
58J10 Differential complexes