Wild Harmonic Bundles and Wild Pure Twistor D-Modules

Description

Contents 1. Introduction 1.1. Contents of this monograph 1.2. Asymptotic behaviour of wild harmonic bundles 1.3. Application to meromorphic flat bundles 1.4. Application to holonomic 𝐷-modules and wild pure twistor 𝐷-modules 1.5. Acknowledgement Part I. Good Meromorphic 𝜚-Flat Bundles 2. Good Formal Property of Meromorphic 𝜚-Flat Bundle 2.1. Good set of irregular values and truncations 2.2. Good lattice in the formal case 2.3. Good lattice of meromorphic 𝜚-flat bundle 2.4. Decompositions 2.5. Good filtered 𝜚-flat bundle 2.6. Good lattice at the level 𝑚 2.7. Good Deligne-Malgrange lattice and Deligne-Malgrange lattice 2.8. Family of filtered λ-flat bundles and KMS-structure 3. Stokes Structure of Good 𝜚-Meromorphic Flat Bundle 3.1. Preliminaries 3.2. Good meromorphic 𝜚-flat bundle 3.3. Good meromorphic 𝜚-flat bundle at the level 𝑚 3.4. Some notation 3.5. Preliminary in the smooth divisor case 3.6. Proof of the statements in Section 3.3 3.7. Proof of the statements in Section 3.2 4. Full Stokes Data and Riemann-Hilbert-Birkhoff Correspondence 4.1. Full pre-Stokes data 4.2. Full Stokes data 4.3. Riemann-Hilbert-Birkhoff correspondence 4.4. Extension of Stokes data 4.5. Deformation 5. 𝐿^2-Cohomology of Filtered λ-Flat Bundle on Curves 5.1. Local quasi-isomorphisms for fixed λ 5.2. Proof for fixed λ 5.3. Local quasi-isomorphisms in family 5.4. Proof in the family case 6. Meromorphic Variation of Twistor Structure 6.1. Variation of polarized pure twistor structure 6.2. Good meromorphic prolongment of variation of twistor structure Part II. Prolongation of Wild Harmonic Bundle 7. Prolongments 𝒫ℰ^λ for Unramifiedly Good Wild Harmonic Bundles 7.1. Definition of wild harmonic bundles 7.2. Simpson’s main estimate and acceptability of the associated bundles 7.3. Proof of the estimates 7.4. The associated good filtered λ-flat bundle 7.5. Comparison of the irregular decompositions in the case 𝑚 < 0_ℓ 7.6. Small deformation of 𝒫ℰ^λ in the smooth divisor case 7.7. Boundedness of some section 8. Some Basic Results in the Curve Case 8.1. Norm estimate for holomorphic sections of 𝒫_𝒸 ℰ^λ 8.2. Comparison of the data at 𝑂 8.3. A characterization of the lattices for generic λ 8.4. Harmonic forms 9. Associated Family of Meromorphic λ-Flat Bundles 9.1. Filtered bundle ((𝒫_∗)^(λ_0)) ℰ 9.2. Family of meromorphic flat λ-connections on ((𝒫_∗)^(λ_0)) ℰ 9.3. Estimate of the norms of partially flat sections 9.4. Locally uniform comparison of the irregular decompositions 10. Smooth divisor case 10.1. Acceptability of (𝒫^(λ_0)) ℎ 10.2. Locally uniform comparison of the irregular decompositions 10.3. Norm estimate for a fixed λ 10.4. Norm estimate in family 11. Prolongation and reduction of variations of polarized pure twistor structures 11.1. Canonical prolongation 11.2. Reduction and uniqueness 11.3. Preliminary for convergence 11.4. One step reduction 11.5. Full reduction in the smooth divisor case 11.6. End of Proof of Theorem 11.2.2 11.7. Norm estimate 11.8. Regular meromorphic variation of twistor structure on a disc (Appendix) 12. Prolongation as ℛ-triple 12.1. ℛ-module associated to unramifiedly good wild harmonic bundle 12.2. Review of some results in the tame case 12.3. The case where Irr(𝜃) consists of only one element 12.4. Specialization of the associated ℛ_𝑋-module 12.5. Construction of the ℛ-triple (𝔈,𝔈,ℭ) 12.6. A characterization of the prolongment in the one dimensional case 12.7. Specialization of the associated ℛ-triples 12.8. Prolongation of ramified wild harmonic bundle on curve Part III. Kobayashi-Hitchin correspondence 13. Preliminaries 13.1. Preliminaries for μ_𝐿-polystable filtered flat sheaves 13.2. Mehta-Ramanathan type theorem 13.3. Auxiliary metrics 13.4. Harmonic bundles on curves 13.5. Some characterizations of wildness of harmonic bundle 13.6. The filtered flat bundle associated to wild harmonic bundle 13.7. Perturbation 14. Construction of an initial metric and preliminary correspondence 14.1. Around a crossing point 14.2. Around smooth point 14.3. Some formulas for the parabolic Chern character 14.4. Preliminary correspondence 14.5. Bogomolov-Gieseker inequality 15. Preliminaries for the resolution of turning points 15.1. Resolution for a tuple of ideals 15.2. Separation of the ramification and the polar part of a Higgs field 15.3. Resolution for generically good Higgs field 16. Kobayashi-Hitchin correspondence and some applications 16.1. Kobayashi-Hitchin correspondence for good filtered flat bundles 16.2. Applications to algebraic meromorphic flat bundles 16.3. Proof of Theorems 16.2.1 and 16.2.4 16.4. Minor refinement of the result in Section 16.2 Part IV. Application to Wild Pure Twistor 𝐷-Modules 17. Wild pure twistor 𝐷-modules 17.1. Review of wild pure twistor 𝐷-modules due to Sabbah 17.2. Wild harmonic bundles and wild pure twistor 𝐷-modules on curves 17.3. Gysin map for wild pure twistor 𝐷-modules 18. The Hard Lefschetz Theorem 18.1. Statement 18.2. Step 1 18.3. Step 2 18.4. Step 3 19. Correspondences 19.1. Wild harmonic bundles and wild pure twistor 𝐷-modules 19.2. Prolongation to polarized wild pure twistor 𝐷-modules 19.3. Wildness and uniqueness 19.4. Application to algebraic semisimple holonomic 𝐷-modules Part V. Appendix 20. Preliminaries from analysis on multi-sectors 20.1. Hukuhara-Malmquist type theorem 20.2. Estimates of some integrals on a sector 20.3. Some Estimates on a multi-sector 21. Acceptable bundles 21.1. Some general results on vector bundles on Kähler manifolds 21.2. Twist of the metric of an acceptable bundle 21.3. Prolongation to filtered bundle 21.4. One dimensional case 21.5. Extension of holomorphic sections I 21.6. Extension of holomorphic sections II 21.7. Proof of Theorems 21.3.1 and 21.3.2 21.8. Small deformation 21.9. Complement 22. Review on ℛ-modules, ℛ-triples and variants 22.1. Filtered rings 22.2. ℛ-modules 22.3. Specialization of ℛ-modules 22.4. ℛ_𝑋 (∗𝑡)-modules 22.5. Formal ℛ-modules 22.6. Preliminaries for 𝐷-modules 22.7. Complement 22.8. The sheaves 𝔇𝔟_(𝑋×𝑇/𝑇) and (𝔇𝔟_(𝑋×𝑇/𝑇))^mod 22.9. ℛ-triples 22.10. Specialization of ℛ-triples 22.11. ℛ (∗𝑡)-triples 22.12. Comparison Bibliography Index