Complexes of Differential Operators by Nikolai N. Tarkhanov Institute of Physics, Siberian Academy of Sciences, Krasnoyarsk, Russia KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
Contents Preface to the English Translation Preface to the Russian Edition xv xvii Introduction 1 0.0.1 Timeliness 1 0.0.2 Directions 2 0.0.3 Purpose 4 0.0.4 Methods 4 0.0.5 Approach 6 0.0.6 Results 6 0.0.7 Authorship 7 List of Main Notations 9 1 Resolution of Differential Operators 11 1.1 Differential Complexes and Their Cohomology 11, 1.1.1 Manifolds countable at infinity 11 1.1.2 Partitions of unity i 12 1.1.3 Sections of vector bundles 13 1.1.4 Spaces of smooth sections 13 1.1.5 Holder spaces 14 1.1.6 Sobolev spaces 15 1.1.7 Besov spaces 16 1.1.8 Spaces of generalized sections 17 1.1.9 Differential operators.... 18 1.1.10 Information from homological algebra 20 1.1.11 Differential complexes 23 1.1.12 Elliptic complexes 24 1.1.13 Transposed complex 25 1.2 The Hilbert Resolution of a Differential Operator with Constant Coefficients 26 1.2.1 Compatibility problem for overdetermined systems of differential equations 26 1.2.2 Formal theory in the case of constant coefficients (Hilbert complexes) 30
Vlll 1.2.3 Elliptic Hilbert complexes 32 1.2.4 Existence theory in the case of constant coefficients 35 1.2.5 Recovering a solution of an elliptic system by means of a "scalar" component 37 1.2.6 The de Rham complex 41 1.2.7 The Dolbeault complex 44 1.2.8 The Koszul complexes 46 1.2.9 The Asada complexes 47 1.3 The Spencer Resolution of a Formally Integrable Differential Operator 48 1.3.1 Geometrical viewpoint for differential operators 48 1.3.2 Regularity conditions 50 1.3.3 Formally exact differential complexes 51 1.3.4 Formal integrability 53 1.3.5 Involutive differential operators 54 1.3.6 Normalized differential operators 56 1.3.7 Commutativity relations 57 1.3.8 Compatibility complexes for normalized operators 59 1.3.9 Compatibility complexes for arbitrary regular operators... 62 1.3.10 Existence theory in the real analytic case 62 1.3.11 The first Spencer sequence 64 1.3.12 The second Spencer sequence 66 1.3.13 Existence theory in the differentiate case 68 1.4 Tensor products of differential complexes and Kiinneth's formula... 71 1.4.1 Topological tensor product of locally convex spaces 71 1.4.2 Grothendieck's theorem on the topological tensor product of complexes 72 1.4.3 Tensor product of differential complexes. A Kunneth formula, 73 1.4.4 Solvability of systems of differential equations with a parameter 75 1.4.5 Examples 76 1.5 Cochain mappings of differential complexes 77 1.5.1 A brief survey of operator kernels. Schwartz's kernel theorem. 77 1.5.2 Kernel of the identity operator 80 1.5.3 Pseudo-differential operators 80 1.5.4 Kernels of pseudo-differential operators. Seeley's theorem... 82 1.5.5 The complex of homomorphisms 85 1.5.6 A differential version of the complex of homomorphisms.... 85 1.5.7 A Kunneth formula 87 1.5.8 Examples 88 Parametrices and Fundamental Solutions of Differential Complexes 91 2.1 Parametrices of Differential Complexes 91 2.1.1 Parametrices and the homotopy formula 91 2.1.2 Parametrices and cohomologies of differential complexes.... 93 2.1.3 Formal adjoint operator 95
Table of Contents ix 2.1.4 Parametrices of elliptic complexes. A theorem of Atiyah and Bolt 96 2.1.5 Hypoellipticity of elliptic complexes 99 2.1.6 Analyticity of cohomologies of elliptic complexes. 100 2.2 Hodge Theory for Elliptic Complexes on Compact Manifolds 101 2.2.1 Harmonic spaces 101 2.2.2 Hodge theory 101 2.2.3 Orthogonal decompositions 104 2.3 Fundamental Solutions of Differential Complexes 105 2.3.1 Fundamental solutions 105 2.3.2 Existence of local fundamental solutions 107 2.3.3 Fundamental solution at degree q. A generalization of a theorem of Malgrange 108 2.3.4 Kernels of convolution type 112 2.3.5 Fundamental solutions of convolution type of complexes of differential operators with constant coefficients in R 113 2.4 Green Operators for Differential Operators and the Homotopy Formula on Manifolds with Boundary 119 2.4.1 Poly differential operators with values in the space of differential forms 119 2.4.2 Green operator of a differential operator 123 2.4.3 The homotopy formula on manifolds with boundary 125 2.4.4 Conservation laws for solutions of overdetermined systems.. 128 2.5 The Most Immediate Corollaries and Examples 130 2.5.1 The homotopy formula for elliptic complexes on a manifold with boundary 130 2.5.2 Fundamental solution of convolution type for the de Rham ; complex inr n 131 2.5.3 Fundamental solution of convolution type for the Dolbeault complex inc 133 2.5.4 Green formulas 136 2.5.5 Fundamental solution of convolution type for the Koszul complexl37 2.5.6 Homotopy formula on manifolds with boundary for Asada complexes 138 3 Sokhotskii-Plemelj Formulas for Elliptic Complexes 141 3.1 Formally Non-characteristic Hypersurfaces for Differential Complexes. The Tangential Complex 141 3.1.1 Preliminaries 141 3.1.2 Quotient complexes 142 3.1.3 Complexes of jet spaces on S 143 3.1.4 Cauchy data complexes on S 145 3.1.5 Tangential complexes 148 3.1.6 Formally non-characteristic hypersurfaces 151 3.1.7 Decomposition of a complex 152
3.1.8 Existence and uniqueness of formal solutions to a non-characteristic local Cauchy problem 154 3.1.9 A condition for a hypersurface to be formally non-characteristic locally 154 3.1.10 Completion of the proof of Theorem 3.1.30 157 3.2 Sokhotskii-Plemelj Formulas for Elliptic Complexes of First Order Differential Operators 158 3.2.1 Historical reference 158 3.2.2 Tangential and normal components of sections 158 3.2.3 The Sokhotskii-Plemelj formulas 160 3.2.4 Structure of the Cauchy data for the differential operator A;. 161 3.2.5 Formal version of the Cauchy-Kovalevskaya Theorem for the differential operator A 162 3.2.6 Green's identity for the differential operator A 162 3.2.7 Proof of Theorem 3.2.6 162 3.2.8 Further results 164 3.2.9 A description of the tangential complex 166 3.3 Generalization of the Sokhotskii-Plemelj Formulas to the Case of Arbitrary Elliptic Complexes 167 3.3.1 Special parametrix of an elliptic complex 167 3.3.2 Behaviour of potentials near the surface of integration 168 3.3.3 Jump formulas 169 3.3.4. Formal version of the Cauchy-Kovalevskaya Theorem for the differential operator A (general case) 173 3.3.5 Decomposition in jet spaces 174 3.3.6 Dual decompositions 175 3.3.7 Formal version of the Cartan-Kahler Theorem for elliptic complexes 176 3.3.8 A description of the tangential complex for an arbitrary elliptic): complex 178 3.3.9 Jump formulas for the volume potential 179 3.3.10 Jump formulas for potentials with smooth densities 180 3.4 Integral Formulas for Elliptic Complexes. Morera's Theorem 182 3.4.1 Historical remarks 182 3.4.2 Preliminary results 182 3.4.3 A complete description of cocycles of the complex (E\y) 184 3.4.4 The spaces W( F ) 186 3.4.5 A complete description of harmonic sections of E'\y 187 3.4.6 Morera's theorem 190 3.5 Multiplication of Currents via Their Harmonic Representations... 195 3.5.1 A brief survey of representations of distributions 195 3.5.2 Weak boundary values of harmonic forms of finite order of growth 195 3.5.3 Harmonic representations of currents with compact support.. 199 3.5.4 Harmonic representation of arbitrary currents 202
Table of Contents xi 3.5.5 On multiplication of distributions 204 3.5.6 Exterior product of currents 204 3.5.7 Index of intersection of currents.. -.- 208 4 Boundary Problems for Differential Complexes 211 4.1 The Neumann-Spencer Problem 211 4.1.1 Representation of cohomologies of differential complexes on manifolds with boundary 211 4.1.2 I?-version of the Neumann problem 215 4.1.3 Examples 220 4.1.4 Reducing the question of solvability to a subelliptic estimate.. Ill 4.1.5 A geometrical condition for solvability of the Neumann problem.223 4.2 The Z 2 -Cohomologies of Differential Complexes and the Bergman Projector 231 4.2.1 L 2 -cohomologies 231 4.2.2 Adjoint complex 232 4.2.3 Weak version of the Neumann problem 233 4.2.4 Weak orthogonal decomposition 233 4.2.5 The Neumann operator. 234 4.2.6 The Dirichlet norm 235 4.2.7 A representation of I?-cohomologies of differential complexes. 238 4.2.8 The Bergman projector and a generalization of a theorem of Bungart 239 4.3 The Mayer-Vietoris sequence 242 4.3.1 Basic example 242 4.3.2 Long cohomological sequence 242 4.3.3 The Mayer-Vietoris sequence for open sets (the case of arbi- i trary supports). 243 4.3.4 The Mayer- Vietoris sequence for open sets (the case of compact supports) 245 4.3.5 The Mayer-Vietoris sequence for closed sets 246 4.3.6 The Mayer-Vietoris sequence for elliptic complexes 248 4.3.7 Examples 250 4.4 The Cauchy problem for cohomology classes of differential complexes 251 4.4.1 The Cauchy problem for sections 251 4.4.2 Reducing to boundary differential complexes f... 252 4.4.3 The Cauchy problem for cohomology classes 254 4.4.4 Representation of cohomologies of a differential complex by means of cohomologies of the tangential complex 257 4.4.5 Removable singularities of solutions of overdetermined systems and a generalization of Bochner 's theorem 257 4.4.6 The Dirichlet problem and representation of cohomologies of the complex of spaces of sections with zero Cauchy data... 260 4.4.7 A generalization of the theorem of Kohn and Rossi 261 4.5 The Kernel Approach to Solving the Equation Pu f 262
Xll 4.5.1 General remark 262 4.5.2 Decomposition of the fundamental solution 263 4.5.3 Special homotopy formulas on manifolds with boundary... 265 4.5.4 Kernels for solutions of the equation Pu = f 266 Duality Theory for Cohomologies of Differential Complexes 269 5.1 The Poincare Duality and the Alexander-Pontryagin Duality 269 5.1.1 Duality Lemma 269 5.1.2 Poincare duality 271 5.1.3 Alexander-Pontryagin duality 274 5.2 The Weil Homomorphism 276 5.2.1 Preliminaries 276 5.2.2 The Weil homomorphism 277 5.2.3 Properties of the Weil homomorphism 280 5.2.4 A localized version of the Weil homomorphism 283 5.2.5 The adjoint Weil homomorphism 285 5.2.6 Properties of the adjoint Weil homomorphism 289 5.2.7 A localized version of the adjoint Weil homomorphism 292 5.2.8 Pairing at sequences of Weil representatives 293 5.2.9 A remark on hypoelliptic complexes 295 5.3 Integral Formulas Connected by the Weil Homomorphism 295 5.3.1 U-resolution of a singular cycle 295 5.3.2 Homologies generated by cycles and conservation laws 297 5.3.3 Abstract integral representations for solutions of overdetermined systems 298 5.3.4 Generalization of the Cauchy- Weil formula 299 5.3.5 C polyhedra in general position with respect to a covering.. 301 5.3.6 Overdetermined homogeneous systems of differential equations with constant coefficients '". 302 5.3.7 A Cauchy-Fantappie formula for Koszul complexes 307 5.3.8 Cech parametrices 310 5.4 Grothendieck's Theorem on Cohomology Classes Regular at Infinity. 312 5.4.1 History of the question 312 5.4.2 Sections regular at infinity 312 5.4.3 Cohomology classes regular at infinity 315 5.4.4 Example 317 5.5 Grothendieck Duality for Elliptic Complexes 318 5.5.1 Survey of results 318 5.5.2 The space H q ( {E\ Y )) 319 5.5.3 The space HimSiE'lxw)) 320 5.5.4 The approximation condition 321 5.5.5 Grothendieck duality 321 5.5.6 A preparatory lemma 322 5.5.7 Proof of the-grothendieck duality theorem 324 5.5.8 Some consequences 326
TaWe of Contents xiii 6 The Atiyah-Bott-Lefschetz Theorem on Fixed Points for Elliptic Complexes 329 6.1 The Argument Principle for Elliptic Complexes 329 6.1.1 A brief history 329 6.1.2 Chains defined by smooth mappings 329 6.1.3 The argument principle 331 6.1.4 Proof of Theorem 6.1.5 332 6.1.5 The logarithmic residue formula for holomorphic functions..336 6.1.6 A generalization of the Kronecker formula 337 6.1.7 Mappings preserving the dimension 339 6.1.8 An analog of the Cauchy-Fantappie formula for closed differential forms 339 6.2 An Integral Formula for the Lefschetz Number 343 6.2.1 A brief survey of the Lefschetz theory for elliptic complexes.. 343 6.2.2 The Lefschetz number of an endomorphism of an elliptic complex344 6.2.3 Main integral formula for the Lefschetz number 345 6.2.4 Geometric endomorphisms 346 6.2.5 A modification of the main formula for geometric endomorphisms 347 6.2.6 Localization of the Lefschetz number 348 6.2.7 Local index of an isolated component of the set of fixed points. 350 6.3 The Atiyah-Bott Formula for Simple Fixed Points 352 6.3.1 Preliminaries 352 6.3.2 The Atiyah-Bott formula 355 6.4 Isolated Components of the Set of Fixed Points 357 6.4.1 General remarks 357 6.4.2 Uniqueness of the local index 357 6.4.3 A cohomological formula for the local index 360 6.5 Some Examples for the Classical Complexes 361 6.5.1 The classical Lefschetz formula 361 6.5.2 The holomorphic Lefschetz formula 363 6.5.3 Other examples and applications 367 Bibliography 369 Name Index 387 Subject Index 390 Index of Notation 394