LEADER 12553nam 22006615 450 001 9910548172003321 005 20251113181617.0 010 $a9783030945008$b(electronic bk.) 010 $z9783030944995 024 7 $a10.1007/978-3-030-94500-8 035 $a(MiAaPQ)EBC6896792 035 $a(Au-PeEL)EBL6896792 035 $a(CKB)21325477200041 035 $a(PPN)260825301 035 $a(OCoLC)1300781100 035 $a(DE-He213)978-3-030-94500-8 035 $a(EXLCZ)9921325477200041 100 $a20220223d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aPeriodic Monopoles and Difference Modules /$fby Takuro Mochizuki 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (336 pages) 225 1 $aLecture Notes in Mathematics,$x1617-9692 ;$v2300 311 08$aPrint version: Mochizuki, Takuro Periodic Monopoles and Difference Modules Cham : Springer International Publishing AG,c2022 9783030944995 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- Acknowledgements -- Contents -- 1 Introduction -- 1.1 Background and Motivation -- 1.2 Monopoles of GCK-Type -- 1.3 Previous Works on Monopoles and Algebraic Objects -- 1.3.1 SU(2)-Monopoles with Finite Energy on R3 -- 1.3.2 The Correspondence due to Charbonneau and Hurtubise -- 1.3.3 Remark -- 1.4 Review of the Kobayashi-Hitchin Correspondences for ?-Flat Bundles -- 1.4.1 Harmonic Bundles and Their Underlying ?-Flat Bundles -- 1.4.2 Kobayashi-Hitchin Correspondences in the Smooth Case -- 1.4.3 Tame Harmonic Bundles and Regular Filtered ?-Flat Bundles -- 1.4.4 Wild Harmonic Bundles and Good Filtered ?-Flat Bundles -- 1.5 Equivariant Instantons and the Underlying Holomorphic Objects -- 1.5.1 Instantons and the Underlying Holomorphic Bundles -- 1.5.2 Instantons and Harmonic Bundles -- 1.5.3 Instantons and Monopoles -- 1.5.4 Instantons and Monopoles as Harmonic Bundles of Infinite Rank -- 1.5.4.1 Instantons as Harmonic Bundles of Infinite Rank -- 1.5.4.2 The Underlying ?-Flat Bundles of Infinite Rank -- 1.5.4.3 Monopoles as Harmonic Bundles of Infinite Rank -- 1.6 Difference Modules with Parabolic Structure -- 1.6.1 Difference Modules -- 1.6.2 Parabolic Structure of Difference Modules at Finite Place -- 1.6.3 Good Parabolic Structure at ? -- 1.6.4 Parabolic Difference Modules -- 1.6.5 Degree and Stability Condition -- 1.6.6 Easy Examples of Stable Parabolic Difference Modules (1) -- 1.6.6.1 The Case Where (?) Is Even -- 1.6.6.2 The Case Where (?) Is Odd -- 1.6.7 Easy Examples of Stable Parabolic Difference Modules (2) -- 1.7 Kobayashi-Hitchin Correspondences for Periodic Monopoles -- 1.7.1 The Correspondence in the Case ?=0 -- 1.7.1.1 Mini-complex Structure -- 1.7.1.2 Mini-holomorphic Bundles Associated with Monopoles -- 1.7.1.3 Dirac Type Singularity -- 1.7.1.4 Meromorphic Extension and Filtered Extension at Infinity. 327 $a1.7.1.5 Kobayashi-Hitchin Correspondence in the Case ?=0 -- 1.7.1.6 OM0Z(H0?)-Modules and C(w)-Modules with an Automorphism -- 1.7.2 The Correspondences in the General Case -- 1.7.2.1 Preliminary Consideration -- 1.7.2.2 Mini-complex Structure Corresponding to the Twistor Parameter ? -- 1.7.2.3 Another Coordinate System and the Compactification of M? -- 1.7.2.4 Mini-holomorphic Bundles Associated with Monopoles -- 1.7.2.5 Meromorphic Extension and Filtered Extension at Infinity -- 1.7.2.6 Kobayashi-Hitchin Correspondence of Periodic Monopoles of GCK Type -- 1.7.2.7 Difference Modules and OM?Z (H??)-Modules -- 1.8 Asymptotic Behaviour of Periodic Monopoles of GCK-Type -- 1.8.1 Setting -- 1.8.2 Decomposition of Mini-holomorphic Bundles -- 1.8.3 The Induced Higgs Bundles -- 1.8.3.1 Preliminary (1) -- 1.8.3.2 Preliminary (2) -- 1.8.3.3 The Induced Higgs Bundles -- 1.8.4 Asymptotic Orthogonality -- 1.8.5 Curvature Decay -- 1.8.6 The Filtered Extension in the Case ?=0 -- 1.8.7 The Filtered Extension for General ? -- 1.8.7.1 Ramified Covering Space -- 1.8.7.2 Approximation -- 1.8.7.3 Formal Completion of Asymptotic Harmonic Bundles at Infinity -- 1.8.7.4 The Formal Structure of PhE? at Infinity -- 2 Preliminaries -- 2.1 Outline of This Chapter -- 2.2 Mini-Complex Structures on 3-Manifolds -- 2.2.1 Mini-Holomorphic Functions on RC -- 2.2.2 Mini-Complex Structure on Three-Dimensional Manifolds -- 2.2.3 Tangent Bundles -- 2.2.4 Cotangent Bundles -- 2.2.5 Meromorphic Functions -- 2.3 Mini-Holomorphic Bundles -- 2.3.1 Mini-Holomorphic Bundles -- 2.3.2 Metrics and the Induced Operators -- 2.3.3 Splittings -- 2.3.4 Scattering Maps -- 2.3.5 Dirac Type Singularity of Mini-Holomorphic Bundles -- 2.3.6 Kronheimer Resolution of Dirac Type Singularity -- 2.3.7 Precise Description of Dirac Type Singularities -- 2.3.8 Subbundles and Quotient Bundles. 327 $a2.3.9 Basic Functoriality -- 2.4 Monopoles -- 2.4.1 Monopoles and Mini-Holomorphic Bundles -- 2.4.2 Euclidean Monopoles -- 2.4.3 Dirac Type Singularity -- 2.4.3.1 Dirac Monopoles (Examples) -- 2.4.4 Basic Functoriality -- 2.5 Dimensional Reduction from 4D to 3D -- 2.5.1 Instantons Induced by Monopoles -- 2.5.2 Holomorphic Bundles and Mini-Holomorphic Bundles -- 2.6 Dimensional Reduction from 3D to 2D -- 2.6.1 Monopoles Induced by Harmonic Bundles -- 2.6.2 Mini-Holomorphic Bundles Induced by Holomorphic Bundles with a Higgs Field -- 2.6.3 Mini-Holomorphic Sections and Monodromy -- 2.6.4 Appendix: Monopoles as Harmonic Bundles of Infinite Rank -- 2.7 Twistor Families of Mini-Complex Structures on RC and (R/TZ)C -- 2.7.1 Preliminary -- 2.7.2 Spaces -- 2.7.3 Twistor Family of Complex Structures -- 2.7.4 Family of Mini-Complex Structures -- 2.7.5 The Mini-Complex Coordinate System (t0,?0) -- 2.7.6 The Mini-Complex Coordinate System (t1,?1) -- 2.7.7 Coordinate Change -- 2.7.8 Compactification -- 2.7.9 Mini-Holomorphic Bundles Associated with Monopoles -- 2.7.9.1 Compatibility with the Dimensional Reduction from 4D to 3D -- 2.8 OM?-Modules and ?-Connections -- 2.8.1 Dimensional Reduction from OM?-Modules to ?-Flat Bundles -- 2.8.1.1 Setting -- 2.8.1.2 Some Vector Fields and Forms -- 2.8.1.3 A General Equivalence -- 2.8.1.4 Mini-Holomorphic Bundles and Flat ?-Connections -- 2.8.1.5 ?-Flat Bundles of Infinite Rank -- 2.8.1.6 Remark -- 2.8.2 Comparison of Some Induced Operators -- 2.8.2.1 Comparison of Mini-Holomorphic Bundles Induced by Harmonic Bundles -- 2.8.3 OM?-Modules and ?-Connections -- 2.8.3.1 Setting -- 2.8.3.2 A General Equivalence -- 2.8.3.3 Mini-Holomorphic Bundles and Meromorphic Flat ?-Connections -- 2.8.3.4 Another Description of the Construction -- 2.9 Curvatures of Mini-Holomorphic Bundles with Metric on M?. 327 $a2.9.1 Contraction of Curvature and Analytic Degree -- 2.9.2 Chern-Weil Formula -- 2.9.3 Another Description of G(h) -- 2.9.4 Change of Metrics -- 2.9.5 Relation with ?-Connections -- 2.9.5.1 ?-Flat Bundles of Infinite Rank with a Harmonic Metric -- 2.9.5.2 Remark -- 2.9.6 Dimensional Reduction of Kronheimer -- 2.9.7 Appendix: Ambiguity of the Choice of a Splitting -- 2.10 Difference Modules and OM?Z(H??)-Modules -- 2.10.1 Difference Modules with Parabolic Structure at Finite Place -- 2.10.2 Construction of Difference Modules from OM?Z(H??)-Modules -- 2.10.3 Construction of OM?Z(H?)-Modules from Difference Modules -- 2.10.4 Appendix: Mellin Transform and Parabolic Structures at Finite Place -- 2.10.4.1 Mellin Transform -- 2.10.4.2 Algebraic Nahm Transform for Filtered ?-Flat Bundles (Special Case) -- 2.11 Filtered Prolongation of Acceptable Bundles -- 2.11.1 Filtered Bundles on a Neighbourhood of 0 in C -- 2.11.1.1 G-Equivariance -- 2.11.1.2 Subbundles, Quotient and Splitting -- 2.11.1.3 Basic Functoriality -- 2.11.1.4 Pull Back -- 2.11.1.5 Push-Forward -- 2.11.1.6 Descent -- 2.11.1.7 Some Examples -- 2.11.2 Acceptable Bundles on a Punctured Disc -- 2.11.2.1 Basic Functoriality -- 2.11.2.2 Pull Back and Descent -- 2.11.3 Global Case -- 2.11.3.1 Filtered Bundles -- 2.11.3.2 Acceptable Bundles -- 3 Formal Difference Modules and Good Parabolic Structure -- 3.1 Outline of This Chapter -- 3.2 Formal Difference Modules -- 3.2.1 Formal Difference Modules of Level ?1 -- 3.2.2 Formal Difference Modules of Pure Slope -- 3.2.3 Slope Decomposition of Formal Difference Modules -- 3.3 Good Filtered Bundles of Formal Difference Modules -- 3.3.1 Filtered Bundles over C((yq-1))-Modules -- 3.3.1.1 G-Equivariance -- 3.3.1.2 Submodules, Quotient Modules and Splittings -- 3.3.1.3 Basic Functoriality -- 3.3.1.4 Pull Back -- 3.3.1.5 Push-Forward -- 3.3.1.6 Descent. 327 $a3.3.2 Good Filtered Bundles over Formal Difference Modules -- 3.3.3 The Induced Endomorphisms on the Graded Pieces -- 3.4 Geometrization of Formal Difference Modules -- 3.4.1 Ringed Spaces -- 3.4.2 Some Formal Spaces -- 3.4.3 Difference Modules and OH?,q(H?,q)-Modules -- 3.4.4 Lattices and the Induced Local Systems -- 3.5 Filtered Bundles in the Formal Case -- 3.5.1 Pull Back and Descent of OH?,p(H?,p)-Modules -- 3.5.2 Filtered Bundles -- 3.5.2.1 Subbundles and Quotient Bundles -- 3.5.2.2 Basic Functoriality -- 3.5.2.3 Pull Back -- 3.5.2.4 Push-Forward -- 3.5.2.5 Descent -- 3.5.3 Basic Filtered Objects with Pure Slope -- 3.5.4 Good Filtered Bundles over OH?,q(H?,q)-Modules with Level ?1 -- 3.5.5 Good Filtered Bundles over OH?,q(H?,q)-Modules -- 3.5.5.1 An Equivalence -- 3.5.5.2 Some Properties -- 3.5.6 Global Lattices on the Covering Space -- 3.5.7 Local Lattices -- 3.5.8 Complement for Good Filtered Bundles with Level ?1 -- 3.6 Formal Difference Modules of Level ?1 and Formal ?-Connections -- 3.6.1 Formal ?-Connections -- 3.6.2 Some Sheaves of Algebras on H?,q -- 3.6.3 From Formal ?-Connections to Formal Difference Modules -- 3.6.4 Equivalence -- 3.6.4.1 Simpler Cases of Proposition 3.6.8 -- 3.6.5 Example 1 -- 3.6.5.1 -- 3.6.5.2 -- 3.6.6 Example 2 -- 3.6.6.1 -- 3.6.6.2 -- 3.6.7 Comparison of Good Filtered Bundles -- 3.6.8 Comparison of the Associated Graded Pieces -- 3.6.9 Some Functoriality -- 3.7 Appendix: Pull Back and Descent in the R-Direction -- 3.7.1 Examples -- 4 Filtered Bundles -- 4.1 Outline of This Chapter -- 4.2 Filtered Bundles in the Global Case -- 4.2.1 Subbundles and Quotient Bundles -- 4.2.2 Degree and Slope -- 4.2.3 Stability Condition -- 4.2.4 Good Filtered Bundles of Dirac Type and Parabolic Difference Modules -- 4.2.4.1 Polystable Parabolic Difference Modules -- 4.2.4.2 Equivalence -- 4.3 Filtered Bundles on Ramified Coverings. 327 $a4.3.1 The Case ?=0. 330 $aThis book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized Cherkis?Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of Kobayashi?Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity. The theory of periodic monopoles of GCK type has applications to Yang?Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given, including major generalizations of results due to Charbonneau, Cherkis, Hurtubise, Kapustin, and others, and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette, Donaldson, Hitchin and Simpson. This work will be of interest to graduatestudents and researchers in differential and algebraic geometry, as well as in mathematical physics. 410 0$aLecture Notes in Mathematics,$x1617-9692 ;$v2300 606 $aGeometry, Differential 606 $aMathematical physics 606 $aGeometry, Algebraic 606 $aDifferential Geometry 606 $aMathematical Physics 606 $aAlgebraic Geometry 615 0$aGeometry, Differential. 615 0$aMathematical physics. 615 0$aGeometry, Algebraic. 615 14$aDifferential Geometry. 615 24$aMathematical Physics. 615 24$aAlgebraic Geometry. 676 $a516.36 700 $aMochizuki$b Takuro$f1972-$0319920 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910548172003321 996 $aPeriodic Monopoles and Difference Modules$92789042 997 $aUNINA