LEADER 04031nam 22006375 450 001 9910763598003321 005 20251113202407.0 010 $a3-031-27234-X 024 7 $a10.1007/978-3-031-27234-9 035 $a(MiAaPQ)EBC30941351 035 $a(Au-PeEL)EBL30941351 035 $a(DE-He213)978-3-031-27234-9 035 $a(CKB)28846142700041 035 $a(EXLCZ)9928846142700041 100 $a20231113d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCoherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck /$fby Jean-Michel Bismut, Shu Shen, Zhaoting Wei 205 $a1st ed. 2023. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2023. 215 $a1 online resource (181 pages) 225 1 $aProgress in Mathematics,$x2296-505X ;$v347 311 08$aPrint version: Bismut, Jean-Michel Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck Cham : Springer International Publishing AG,c2023 9783031272332 327 $aIntroduction -- Bott-Chern Cohomology and Characteristic Classes -- The Derived Category ${\mathrm{D^{b}_{\mathrm{coh}}}}$ -- Preliminaries on Linear Algebra and Differential Geometry -- The Antiholomorphic Superconnections of Block -- An Equivalence of Categories -- Antiholomorphic Superconnections and Generalized Metrics -- Generalized Metrics and Chern Character Forms -- The Case of Embeddings -- Submersions and Elliptic Superconnections -- Elliptic Superconnection Forms and Direct Images -- A Proof of Theorem 10-1 when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$. -- The Hypoelliptic Superconnections -- The Hypoelliptic Superconnection Forms -- The Hypoelliptic Superconnection Forms when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$ -- Exotic Superconnections and Riemann-Roch-Grothendieck -- Subject Index -- Index of Notation -- Bibliography. 330 $aThis monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian. Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource for manyresearchers in geometry, analysis, and mathematical physics. . 410 0$aProgress in Mathematics,$x2296-505X ;$v347 606 $aAlgebra, Homological 606 $aK-theory 606 $aDifferential equations 606 $aGeometry, Differential 606 $aCategory Theory, Homological Algebra 606 $aK-Theory 606 $aDifferential Equations 606 $aDifferential Geometry 615 0$aAlgebra, Homological. 615 0$aK-theory. 615 0$aDifferential equations. 615 0$aGeometry, Differential. 615 14$aCategory Theory, Homological Algebra. 615 24$aK-Theory. 615 24$aDifferential Equations. 615 24$aDifferential Geometry. 676 $a516.183 700 $aBismut$b Jean-Michel$044924 701 $aShen$b Shu$01439118 701 $aWei$b Zhaoting$01439119 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910763598003321 996 $aCoherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck$93601319 997 $aUNINA