Coherent Sheaves, Superconnections, and Riemann-roch-grothendieck (Progress in Mathematics, 347)

This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian. Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource for many researchers in geometry, analysis, and mathematical physics. Acknowledgements 6 Contents 7 Chapter 1 Introduction 11 1.1 The construction of chBC 12 1.2 The Riemann-Roch-Grothendieck theorem 12 1.2.1 The case of embeddings 13 1.2.2 The case of projections 13 1.2.3 The spectral truncations 13 1.3 Earlier work 14 1.4 The organization of the book 15 Chapter 2 Bott-Chern cohomology and characteristic classes 17 2.1 The Bott-Chern cohomology 17 2.2 The Bott-Chern Laplacian of Kodaira-Spencer 18 2.3 Functorial properties of Bott-Chern cohomology 20 2.4 Bott-Chern characteristic classes of holomorphic vector bundles 21 Chapter 3 The derived category Dbcoh (X) 24 3.1 Definition of the derived category Dbcoh (X) 24 3.2 Derived pullbacks 25 3.3 Derived tensor products 25 3.4 Derived direct images 26 Chapter 4 Preliminaries on linear algebra and differential geometry 27 4.1 Filtered vector space and exterior algebras 27 4.2 Supercommutators and supertraces 30 4.3 Morphisms and cones 31 4.4 Generalized Hermitian metrics 32 4.5 Clifford algebras 34 4.6 A simple remark of differential geometry 35 Chapter 5 The antiholomorphic superconnections of Block 37 5.1 The antiholomorphic superconnections 37 5.2 A conjugation result 39 5.3 Antiholomorphic superconnections and coherent sheaves 43 5.4 The determinant line bundle 45 5.5 The case whereHE = 0 46 5.6 Superconnections, morphisms, and cones 48 5.7 Antiholomorphic superconnections and pullbacks 49 5.8 Antiholomorphic superconnections and tensor products 50 Chapter 6 An equivalence of categories 51 6.1 Some properties of Dbcoh (X) 51 6.2 Three categories 53 6.3 The functor FX 54 6.4 The functor FX 63 6.5 The functor FX as an equivalence of categories 63 6.6 Compatibility with pullbacks 65 6.7 Compatibility with tensor products 66 6.8 Compatibility with direct images 67 Chapter 7 Antiholomorphic superconnections and generalized metrics 68 7.1 The adjoint of an antiholomorphic superconnection 68 7.2 Curvature 71 Chapter 8 Generalized metrics and Chern character forms 73 8.1 The Chern character forms 74 8.2 A trivial example 77 8.3 The Chern character of pullbacks 78 8.4 Chern character and tensor products 79 8.5 The case where HE is locally free 79 8.6 The Chern character forms and the scaling of the metric 80 8.7 The Chern character of a cone 82 8.8 The Chern character on Dbcoh (X) 83 8.9 The Chern character on K (X) 84 8.10 Spectral truncations 86 Chapter 9 The case of embeddings 91 9.1 Embeddings, direct images, and transversality 91 9.2 Deformation to the normal cone 94 9.3 A Riemann-Roch-Grothendieck theorem for embeddings 95 9.4 The uniqueness of the Chern character 99 Chapter 10 Submersions and elliptic superconnection forms 100 10.1 A Riemann-Roch-Grothendieck theorem for submersions 100 10.2 Replacing F by E 101 10.3 The adjoint of Ap∗E0′′ 102 10.4 Some connections on TX 105 10.5 A Lichnerowicz formula for AD,2 108 10.6 A Lichnerowicz formula for Ap∗E0,2 110 10.7 The elliptic superconnection forms 111 Chapter 11 Elliptic superconnection forms and direct images 113 11.1 Spectral truncations in infinite dimensions 113 11.2 The existence theorem 115 11.3 The Chern character of the direct image 116 11.4 The case where HR p∗E is locally free 118 Chapter 12 A proof of Theorem 10.1.1 when ∂–X∂XωX = 0 120 Chapter 13 The hypoelliptic superconnections 127 13.1 The total space of TX 128 13.2 A Hermitian metric on TX 128 13.3 The antiholomorphic superconnection EM 130 13.4 A metric description of AEM′′ 131 13.5 The adjoint of A′′ 132 13.6 The antiholomorphic superconnection AEM′′ 133 13.7 The superconnection AZ 133 13.8 Two Hermitian forms 135 13.9 Self-adjointness of AZ, BZ 137 13.10 A formula for the curvature of AZ 138 13.11 The hypoelliptic curvature 141 13.12 Scaling the metric gTX 142 13.13 Hypoelliptic and elliptic superconnections 143 Chapter 14 The hypoelliptic superconnection forms 146 14.1 Construction of the hypoelliptic superconnection forms 146 14.2 The limit as b → 0 of the hypoelliptic superconnection forms 147 Chapter 15 The hypoelliptic superconnection forms when ∂–X∂XωX = 0 149 15.1 Finite and infinite-dimensional traces 149 15.2 The time parameter 152 15.3 The case when ∂–X∂XωX = 0 155 Chapter 16 Exotic superconnections and Riemann-Roch-Grothendieck 162 16.1 A deformation of the fundamental 2-form ωX 163 16.2 A formula for A2Z,θ 163 16.3 The superconnection AY,θ 164 16.4 The scaling identities 165 16.5 The uniform estimates 166 16.6 The exotic superconnection forms of vector bundles 167 16.7 The limit as t → 0 of ch (A′′Y ,ωX/t, gD, gdTX/t3, θt) 168 16.8 A proof of Theorem 10.1.1 172 References 173 Subject index 176 Index of notation 178