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.

0.0.1 Timeliness.- 0.0.2 Directions.- 0.0.3 Purpose.- 0.0.4 Methods.- 0.0.5 Approach.- 0.0.6 Results.- 0.0.7 Authorship.- List of Main Notations.- 1 Resolution of Differential Operators.- 1.1 Differential Complexes and Their Cohomology.- 1.2 The Hilbert Resolution of a Differential Operator with Constant Coefficients.- 1.3 The Spencer Resolution of a Formally Integrable Differential Operator.- 1.4 Tensor products of differential complexes and Künneth’s formula.- 1.5 Cochain mappings of differential complexes.- 2 Parametrices and Fundamental Solutions of Differential Complexes.- 2.1 Parametrices of Differential Complexes.- 2.2 Hodge Theory for Elliptic Complexes on Compact Manifolds.- 2.3 Fundamental Solutions of Differential Complexes.- 2.4 Green Operators for Differential Operators and the Homotopy Formula on Manifolds with Boundary.- 2.5 The Most Immediate Corollaries and Examples.- 3 Sokhotskii-Plemelj Formulas for Elliptic Complexes.- 3.1 Formally Non-characteristic Hypersurfaces for Differential Complexes. The Tangential Complex.- 3.2 Sokhotskii-Plemelj Formulas for Elliptic Complexes of First Order Differential Operators.- 3.3 Generalization of the Sokhotskii-Plemelj Formulas to the Case of Arbitrary Elliptic Complexes.- 3.4 Integral Formulas for Elliptic Complexes. Morera’s Theorem.- 3.5 Multiplication of Currents via Their Harmonic Representations.- 4 Boundary Problems for Differential Complexes.- 4.1 The Neumann-Spencer Problem.- 4.2 The L2-Cohomologies of Differential Complexes and the Bergman Projector.- 4.3 The Mayer-Vietoris sequence.- 4.4 The Cauchy problem for cohomology classes of differential complexes.- 4.5 The Kernel Approach to Solving the Equation Pu = f.- 5 Duality Theory for Cohomologies of Differential Complexes.- 5.1 The Poincaré Dualityand the Alexander-Pontryagin Duality.- 5.2 The Weil Homomorphism.- 5.3 Integral Formulas Connected by the Weil Homomorphism.- 5.4 Grothendieck’s Theorem on Cohomology Classes Regular at Infinity.- 5.5 Grothendieck Duality for Elliptic Complexes.- 6 The Atiyah-Bott-Lefschetz Theorem on Fixed Points for Elliptic Complexes.- 6.1 The Argument Principle for Elliptic Complexes.- 6.2 An Integral Formula for the Lefschetz Number.- 6.3 The Atiyah-Bott Formula for Simple Fixed Points.- 6.4 Isolated Components of the Set of Fixed Points.- 6.5 Some Examples for the Classical Complexes.- Name Index.- Index of Notation.