Multidimensional Residue Theory and Applications

Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry. This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra. Contents Preface Chapter 1. Residue calculus in one variable 1.1. An analytic approach 1.1.1. Laurent expansions and meromorphic functions 1.1.2. Residue of a (1, 0) meromorphic form 1.1.3. Jordan’s lemma and Mellin transform 1.2. The geometric point of a view 1.2.1. Factorization of integration currents through residue currents 1.2.2. Residue currents and Lagrange’s interpolation formula 1.2.3. Residue currents and Bergman–Weil developments 1.2.4. Residue currents and multiplicative calculus 1.2.5. Residues of sections of hermitian bundles 1.3. The algebraic point of view 1.3.1. Cauchy’s formula in ..... or ... and division in K[X] 1.3.2. Currential Cauchy’s formula on compact Riemann surfaces. 1.3.3. Green currents and heights of arithmetic cycles 1.3.4. Residues, residue currents, and Gorenstein K-algebras Chapter 2. Residue currents: a multiplicative approach 2.1. De Rham complex and iterated residues 2.1.1. Division of forms and de Rham’s lemma 2.1.2. Leray coboundary morphism, cohomological residue formulae 2.1.3. Meromorphic multilogarithmic forms 2.2. Coleff–Herrera sheaves of currents 2.2.1. Integration currents, pseudo-meromorphicity, and holonomy 2.2.2. The sheaf CH_(X,V) 2.2.3. Coleff–Herrera currents with prescribed polar set. 2.2.4. Extension-restriction of pseudo-meromorphic currents 2.3. Coleff–Herrera’s original construction revisited 2.3.1. Essential intersection and Coleff–Herrera’s original construction 2.3.2. Coleff–Herrera currents attached to sections of line bundles 2.3.3. Coleff-Herrera theory in the complete intersection setting Chapter 3. Residue currents: a bundle approach 3.1. A toric complex geometry digest 3.1.1. Rational fans 3.1.2. Complete simplicial fans and homogeneous coordinate rings 3.1.3. Kähler cone and moment maps 3.1.4. Projective toric manifolds 3.2. Bochner–Martinelli currents attached to bundle sections 3.2.1. The currents R.... and P.... for s ......... 3.2.2. The complete intersection setting revisited 3.2.3. Transformation laws from the geometric point of view 3.3. Bochner–Martinelli currents and generically exact complexes 3.3.1. Generically exact Koszul and Eagon–Northcott complexes 3.3.2. The currents R.... and P.... for a metrized complex ... 3.3.3. Residual obstruction for exactness and duality theorems 3.4. Bochner–Martinelli residue currents: structural results 3.4.1. Decomposition of R...... along distinguished varieties 3.4.2. R... and the Buchsbaum–Eisenbud–Fitting sequence 3.4.3. Residue currents and structure forms 3.4.4. The Cohen–Macaulay case Chapter 4. Bochner–Martinelli kernels and weights 4.1. Diagonal submanifold and Koszul complex .... 4.1.1. Koszul complex over the duplication of a good manifold 4.1.2. Nabla operators and L^ν sheaves of currents for goodmanifolds 4.2. Affine and projective Bochner–Martinelli kernels 4.2.1. The affine case X = C^N. 4.2.2. The projective case X= (P_C)^N 4.3. Chern connection and Koppelman’s formulae on good manifolds 4.4. Bochner–Martinelli weights, definition, and constructions 4.4.1. Global weights 4.4.2. Local weights 4.4.3. Global weights attached to metrized complexes 4.5. Bochner–Martinelli weighted integral representation formulae 4.5.1. The affine case X= Ω ⊂ C^N. 4.5.2. The projective case X= (P_C)^N 4.5.3. Weighted Koppelman’s formulae in the general setting 4.6. Solving the F_•-Hefer problem in specific situations 4.6.1. Complexes of trivial bundles over Stein manifolds 4.6.2. Projective Hefer forms and Koszul complex over (P_C)^N 4.6.3. Hefer problem, syzygies for homogeneous polynomial ideals Chapter 5. Integral closure, Briançon–Skoda type theorems 5.1. Integral closure of ideals and valuative criterion 5.1.1. Integral closure of ideals, the analytic reduced case 5.1.2. The nonreduced case, valuative criterion, noetherian operators 5.2. Briançon–Skoda theorem in A = O_0 5.3. Briançon–Skoda theorem in A =...... 5.4. Briançon–Skoda theorem for purely dimensional A =...... 5.5. Primary ideals in O_0 Chapter 6. Residue calculus and trace formulae 6.1. Trace in analytic polyhedra 6.1.1. Analytic Weil polyhedron, nondegeneracy, skeleton 6.1.2. The case when m = n 6.1.3. The case when m > n. 6.1.4. Hardy spaces of strongly nondegenerate analytic Weil polyhedra 6.2. Algebraic approach to residue symbols 6.2.1. Quasi-regular sequences in a commutative K-algebra A 6.2.2. Algebraic residue symbols as traces 6.2.3. Wiebe’s theorem and the algebraic transformation law 6.3. Residue symbols in K-polynomial algebras 6.3.1. Residue symbols Res_(K[X]/K) 6.3.2. Residue symbols Res....... 6.3.3. Residues symbols in K[......] 6.4. Trace in K[X] or K_J[X, Y ] 6.4.1. The algebraic trace function in the K[X]-setting 6.4.2. The algebraic trace function in the K_J[X, Y ] setting. 6.5. The characteristic zero case 6.5.1. Algebraic residue symbols realized as currents 6.5.2. Properness and algebraic residue symbols in characteristic 0 6.5.3. Properness and toric residue symbols (characteristic 0) 6.6. Residue symbols and arithmetics 6.6.1. The univariate case 6.6.2. Geometric and arithmetic markers for complexity 6.6.3. Size estimates of rational algebraic residue symbols 6.6.4. Rational algebraic residue symbols: a dynamic approach Chapter 7. Miscellaneous applications: intersection, division 7.1. Hilbert’s nullstellensatz in K[X1, . . .,Xn] and residue calculus 7.1.1. Rabinowitsch’s trick and Grete Hermann’s algorithmic procedure 7.1.2. Lipman–Teissier theorem and N-solvability of Bézout identity 7.1.3. N-solvability of Bézout identity in C[X] and Bochner–Martinelli currents 7.1.4. N-solvability in C[X] of Bézout identity via a diagram chase 7.1.5. Sharp geometric nullstellensatz 7.1.6. An arithmetic Perron theorem in its parametric form 7.1.7. Hilbert’s nullstellensatz and P = NP 7.2. Jacobi–Lagrange–Kronecker (JLK) parametric identities 7.2.1. Parametric Bézout identity of the JLK form 7.2.2. Parametric membership identities of the JLK form. 7.3. Effective geometric Briançon–Skoda–Huneke type theorems 7.3.1. Global algebraic Briançon–Skoda–Huneke exponents μ....... 7.3.2. An effective Briançon–Skoda–Huneke type theorem on X ⊂ C^N. 7.3.3. Global Briançon–Skoda or Lipman–Teissier theorems in K^n. 7.4. Algebraic residues and tropical considerations 7.4.1. Rational polyhedral complexes and tropical cycles 7.4.2. The tropical current trop... 7.4.3. Tropicalization and (p, q)-supercurrents on R^n 7.5. Jacobian and socle, radical and top radical 7.6. Multivariate residue calculus and the exponential function 7.6.1. Weighted algebras of entire functions, tempered currents 7.6.2. Trace formulae in weighted algebras of entire functions 7.6.3. Ehrenpreis–Palamodov’s fundamental principle revisited 7.7. Residue calculus outside the commutative setting 7.7.1. Univariate quaternionic setting and relevant regularity concepts. 7.7.2. Principal value and residue currents in the H-setting Appendix A. Complex manifolds and analytic spaces A.1. Complex manifold, structural sheaf O_X, sheaves of O_X-modules A.1.1. Definitions A.1.2. Examples (........) A.2. Coherence, Stein manifolds, free resolutions A.2.1. Coherent sheaves of O_X-modules A.2.2. Stein manifolds and Cartan Theorems A and B. A.2.3. Free resolutions of coherent sheaves A.3. Closed analytic subsets of a complex manifold A.3.1. Local dimension A.3.2. Splitting V........., global dimension. A.3.3. Free resolutions of sheaves and codimension of supports A.3.4. Local presentation in the purely dimensional case A.3.5. Weakly holomorphic functions and Oka universal denominator A.3.6. Example: closed analytic subsets of (P_C)^N, Chow’s theorem A.4. Complex analytic spaces, normalization, and log resolutions A.4.1. Complex analytic spaces A.4.2. Normalization and blowup of complex analytic spaces A.4.3. Log resolution and the Hironaka theorem Appendix B. Holomorphic bundles over complex analytic spaces B.1. Analytic cocycles, holomorphic bundles, isomorphism classes B.1.1. Cocycles on a complex analytic space, examples B.1.2. Linear algebra and construction of cocycles B.1.3. Holomorphic bundles over a complex analytic space B.1.4. Isomorphism classes of holomorphic bundles. B.1.5. Local holomorphic frames and sheaves of holomorphic sections. B.1.6. The concepts of local versus global complete intersection B.1.7. The bundles T..... over a complex manifold X. B.2. Sheaves of bundle-valued differential forms or currents B.2.1. Differential forms and currents on a complex manifold B.2.2. Differential forms and currents on a complex analytic space B.2.3. Dolbeault complex, Serre duality on a complex manifold B.2.4. Strongly holomorphic (p, 0) forms on V ↪ X. B.2.5. The Barlet sheaves on V ↪ X B.2.6. Dolbeault complex, Serre duality on a complex analytic space B.3. Hermitian bundles B.3.1. Hermitian complex bundles over a complex manifold B.3.2. Hermitian holomorphic bundles over a complex analytic space B.3.3. Chern and Segre forms and classes Appendix C. Positivity on complex analytic spaces C.1. Positive forms, positive currents C.1.1. Positivity of differential forms on a C-vector space. C.1.2. Positive currents on a complex manifold C.1.3. Trivial extension of positive closed (p, p)-currents C.2. Lelong numbers and integration currents C.2.1. Lelong numbers of positive closed currents on Stein manifolds C.2.2. Lelong numbers of positive closed currents on analytic spaces C.2.3. Analytic subvarieties and corresponding integration currents C.2.4. Integration currents attached to sections of holomorphic bundles. C.2.5. Holonomy of integration currents and consequences C.3. Positivity of holomorphic hermitian bundles Appendix D. Various concepts in algebraic or analytic geometry D.1. Cycles on a complex analytic space D.1.1. Cycles, effectivity, splitting along a dimension and cutting out. D.1.2. Lelong–Poincaré equation and classes of generalized cycles D.1.3. Local multiplicities of classes of generalized cycles. D.2. Cartier divisors and Čech cohomology D.2.1. Cartier divisors and Picard group on a complex analytic space. D.2.2. Toric Cartier divisors on a toric variety D.2.3. Picard group in terms of Čech cohomology with values in O... D.2.4. The first Chern characteristic class of a complex line bundle D.3. Weil divisors and Chow groups D.3.1. Weil divisors on complex analytic spaces or algebraic varieties D.3.2. Chow groups on complex analytic spaces or algebraic varieties. D.4. Ampleness versus positivity for holomorphic line bundles D.4.1. Globally generated holomorphic bundles D.4.2. Semi-ampleness of holomorphic bundles D.4.3. Very ampleness/ampleness of line bundles (compact setting). D.4.4. Very ampleness and ampleness of higher rank bundles D.4.5. Examples. D.5. Local versus global intersection theory D.5.1. Local extended multi-index of intersection, Crofton formula D.5.2. Crofton’s averaging and metrized toric divisors D.5.3. The join product in the projective setting D.5.4. Global intersection in (P_C)^N and classes of generalized cycles Bibliography Index