Complexes of Differential Operators

This book gives a systematic account of the facts concerning complexes of differential operators on differentiable manifolds. The central place is occupied by the study of general complexes of differential operators between sections of vector bundles. Although the global situation often contains nothing new as compared with the local one (that is, complexes of partial differential operators on an open subset of ]Rn), the invariant language allows one to simplify the notation and to distinguish better the algebraic nature of some questions. In the last 2 decades within the general theory of complexes of differential operators, the following directions were delineated: 1) the formal theory; 2) the existence theory; 3) the problem of global solvability; 4) overdetermined boundary problems; 5) the generalized Lefschetz theory of fixed points, and 6) the qualitative theory of solutions of overdetermined systems. All of these problems are reflected in this book to some degree. It is superfluous to say that different directions sometimes whimsically intersect. Considerable attention is given to connections and parallels with the theory of functions of several complex variables. One of the reproaches avowed beforehand by the author consists of the shortage of examples. The framework of the book has not permitted their number to be increased significantly. Certain parts of the book consist of results obtained by the author in 1977-1986. They have been presented in seminars in Krasnoyarsk, Moscow, Ekaterinburg, and N ovosi birsk.

Contenu

Preface to the English Translation. Preface to the Russian edition. Introduction. List of main notations. 1. Resolution of differential operators. 2. Parametrices and fundamental solutions of differential complexes.3. Sokhotskii-Plemelj formulas for elliptic complexes. 4. Boundary problems for differential complexes. 5. Duality theory for cohomologies of differential complexes. 6. The Atiyah-Bott-Lefschetz theorem on fixed points for elliptic complexes. Bibliography. Name index. Subject index. Index of notation.