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E-raamat: Mixed Twistor D-modules

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  • Sari: Lecture Notes in Mathematics 2125
  • Ilmumisaeg: 19-Aug-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319100883
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Mixed Twistor D-modulesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2125
  • Ilmumisaeg: 19-Aug-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319100883
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We introduce mixed twistor D-modules and establish their fundamental functorial properties. We also prove that they can be described as the gluing of admissible variations of mixed twistor structures. In a sense, mixed twistor D-modules can be regarded as a twistor version of M. Saito's mixed Hodge modules. Alternatively, they can be viewed as a mixed version of the pure twistor D-modules studied by C. Sabbah and the author. The theory of mixed twistor D-modules is one of the ultimate goals in the study suggested by Simpson's Meta Theorem, and it would form a foundation for the Hodge theory of holonomic D-modules which are not necessarily regular singular.
1 Introduction
1(16)
1.1 Mixed Hodge Modules
1(5)
1.2 From Hodge Toward Twistor
6(2)
1.3 Mixed Twistor D-Modules
8(9)
1.3.1 Pure Twistor D-Modules
8(1)
1.3.2 Mixed Twistor D-Modules
9(1)
1.3.3 Gluing Procedure
10(1)
1.3.4 Admissible Variation of Mixed Twistor Structure
10(1)
1.3.5 Duality and Real Structure
11(6)
Part I Gluing and Specialization of R-Triples
2 Preliminary
17(32)
2.1 R-Triples
19(20)
2.1.1 R-Modules
19(1)
2.1.2 Strict Specializability for R-Modules
20(3)
2.1.3 Some Sheaves
23(3)
2.1.4 R-Triple
26(2)
2.1.5 Integrable R-Triple
28(2)
2.1.6 Smooth R-triples and Some Functorial Properties
30(3)
2.1.7 Variation of Twistor Structure
33(2)
2.1.8 Tate Object
35(2)
2.1.9 Other Basic Examples of Smooth R-Triples of Rank One
37(2)
2.2 Deformation Associated to Nilpotent Morphisms
39(5)
2.2.1 Twistor Nilpotent Orbit in R-Triple
39(4)
2.2.2 Variant
43(1)
2.3 Beilinson Triples
44(5)
2.3.1 Triples on a Point
44(1)
2.3.2 The Associated Twistor Nilpotent Orbit
45(1)
2.3.3 Appendix
46(3)
3 Canonical Prolongations
49(22)
3.1 Canonical Prolongations of R(*t)-Modules
50(10)
3.1.1 Strictly Specializable R(*t)-Modules
50(1)
3.1.2 The R-Module M[ *t]
50(3)
3.1.3 The R-Module M[ !t]
53(4)
3.1.4 Characterization
57(1)
3.1.5 Morphisms
57(1)
3.1.6 Canonical Prolongations of R-Modules
58(2)
3.2 Canonical Prolongations of R-Triples
60(5)
3.2.1 Canonical Prolongations of R(*t)-Triples
60(3)
3.2.2 Morphisms
63(1)
3.2.3 Canonical Prolongations of R-Triples
63(1)
3.2.4 Compatibility of Canonical Prolongation with Push-Forward
64(1)
3.3 Canonical Prolongations Across Hypersurfaces
65(6)
3.3.1 Canonical Prolongations Across Holomorphic Functions
65(3)
3.3.2 Canonical Prolongations Across Hypersurfaces
68(3)
4 Gluing and Specialization of R-Triples
71(32)
4.1 Beilinson Functors for R-Modules
72(8)
4.1.1 The Functors Πa,b, Π!a,b, Π*a,b and Π*!a,b for R-Modules
72(2)
4.1.2 Another Description
74(1)
4.1.3 The Induced Morphism
75(1)
4.1.4 Compatibility with the Push-Forward
76(1)
4.1.5 The Functors ψ(a) and Ξ(a) for R(*t)-Modules
77(1)
4.1.6 Beilinson Functors for R-Modules
78(1)
4.1.7 Beilinson Functors Along General Holomorphic Functions
79(1)
4.2 Beilinson Functors for R-Triples
80(9)
4.2.1 Functors Πa,b, Π*a,b and Π!a,b for R(*t)-Triple
80(2)
4.2.2 Functors Π*!a,b, ψ(a) and Ξ(a)
82(2)
4.2.3 Vanishing Cycle Functor for R-Triple
84(1)
4.2.4 Gluing of R-Triples
85(1)
4.2.5 Dependence on the Function t
86(1)
4.2.6 Compatibility with Push-Forward
87(1)
4.2.7 Beilinson Functors Along General Holomorphic Functions
88(1)
4.3 Comparison of the Nearby Cycle Functors
89(7)
4.3.1 Statements
89(1)
4.3.2 Preliminary (1)
90(1)
4.3.3 Preliminary (2)
91(1)
4.3.4 Construction of Isomorphisms
92(3)
4.3.5 Hermitian Adjoint
95(1)
4.4 Admissible Specializability
96(7)
4.4.1 Filtered R-Modules
96(3)
4.4.2 Filtered R-Triples
99(1)
4.4.3 Admissible Specializability Along Hypersurfaces
100(3)
5 Gluing of Good-KMS Smooth R-Triples
103(40)
5.1 Good-KMS Smooth R-Modules
104(4)
5.1.1 Good-KMS Meromorphic Prolongment
104(1)
5.1.2 Induced Bundles on the Intersection of Divisors
104(1)
5.1.3 Hukuhara-Levelt-Turrittin Type Decomposition
105(1)
5.1.4 Specialization
106(1)
5.1.5 Reduction with Respect to Stokes Structure
107(1)
5.2 Compatibility of Filtrations
108(4)
5.2.1 Compatibility with Hukuhara-Levelt-Turrittin Type Decomposition
108(1)
5.2.2 Extension of Good-KMS Smooth R-Modules
109(1)
5.2.3 Compatibility with KMS Structure
110(1)
5.2.4 Curve Test
111(1)
5.3 Canonical Prolongations of Good-KMS Smooth R-Modules
112(7)
5.3.1 Goal
112(1)
5.3.2 Uniqueness and Lemma 5.3.2
113(1)
5.3.3 Local Construction
114(1)
5.3.4 Preliminary
115(1)
5.3.5 Some Filtrations
116(2)
5.3.6 Globalization
118(1)
5.3.7 Ramified Covering
119(1)
5.4 Strict Specializability Along Monomial Functions
119(7)
5.4.1 Statement
119(1)
5.4.2 Refinement
120(1)
5.4.3 Preliminary
121(1)
5.4.4 Regular and Pure Case
122(2)
5.4.5 Regular and Filtered Case
124(1)
5.4.6 Good Irregular Case with Unique Irregular Value
125(1)
5.4.7 End of the Proof of Proposition 5.4.3
126(1)
5.5 Good-KMS Smooth R-Triple
126(6)
5.5.1 Reduction with Respect to Stokes Structure
126(2)
5.5.2 Specialization
128(1)
5.5.3 Canonical Prolongations
129(1)
5.5.4 Variant of Beilinson Functors
130(1)
5.5.5 Growth Order and the Compatibility of Stokes Filtrations
131(1)
5.5.6 I-Good-KMS Smooth R-Triples
132(1)
5.6 Gluing of Good-KMS Smooth R-Triples on the Intersections
132(11)
5.6.1 A Category
133(1)
5.6.2 Construction of the Functor
134(4)
5.6.3 Some Properties
138(1)
5.6.4 Dependence on Coordinate Systems
138(5)
Part II Mixed Twistor D-Modules
6 Preliminary for Relative Monodromy Filtrations
143(26)
6.1 Relative Monodromy Filtrations
144(3)
6.1.1 Definition and Basic Properties
144(2)
6.1.2 Canonical Decomposition
146(1)
6.1.3 A Criterion
146(1)
6.1.4 Functoriality for Tensor Product and Duality
147(1)
6.2 Transfer of Filtrations
147(6)
6.2.1 Gluing Data
147(1)
6.2.2 Inheritance of Relative Monodromy Filtration
148(1)
6.2.3 Transfer of Filtration
149(2)
6.2.4 Special Case
151(1)
6.2.5 Duality and Tensor Product
151(2)
6.3 Pure and Mixed Objects
153(16)
6.3.1 Setting
153(3)
6.3.2 A Category LA(Λ)
156(1)
6.3.3 S-Decomposability and Strict Support
156(2)
6.3.4 A Category LA(Λ1, Λ2)
158(1)
6.3.5 Pure Objects in LA(Λ1, Λ2)
158(2)
6.3.6 Mixed Objects in LA(Λ1, Λ2)
160(1)
6.3.7 Some Functors
161(1)
6.3.8 Gluing
162(1)
6.3.9 Another Description of MLA(Λ1, Λ2)
163(2)
6.3.10 Commutativity of the Transfer
165(1)
6.3.11 Canonical Prolongations
166(3)
7 Mixed Twistor D-Modules
169(26)
7.1 Admissible Specializability of Pre-mixed Twistor D-Modules
170(21)
7.1.1 Pre-mixed Twistor D-Modules
170(1)
7.1.2 Push-Forward by Projective Morphisms
171(4)
7.1.3 Admissible Specializability for Pre-mixed Twistor D-Modules
175(5)
7.1.4 Admissible Specializability and Push-Forward
180(1)
7.1.5 Gluing Along a Coordinate Function
181(1)
7.1.6 Localization
182(5)
7.1.7 Integrable Case
187(2)
7.1.8 Restriction of KMS-Spectrum
189(2)
7.2 Mixed Twistor D-Modules
191(4)
7.2.1 Definition
191(1)
7.2.2 Some Basic Properties
192(2)
7.2.3 Integrable Case
194(1)
8 Infinitesimal Mixed Twistor Modules
195(26)
8.1 Preliminary
196(5)
8.1.1 Pure Twistor Structure
196(2)
8.1.2 Mixed Twistor Structure
198(1)
8.1.3 Reduction
199(1)
8.1.4 Some Conditions for the Existence of Relative Monodromy Filtration
200(1)
8.2 Polarizable Mixed Twistor Structure
201(5)
8.2.1 Statements
201(2)
8.2.2 Proof of Proposition 8.2.1
203(2)
8.2.3 Proof of Proposition 8.2.3
205(1)
8.2.4 Proof of Lemmas 8.2.4 and 8.2.5
206(1)
8.3 Infinitesimal Mixed Twistor Modules
206(8)
8.3.1 Definition
206(1)
8.3.2 Statements
207(1)
8.3.3 Canonical Filtrations
208(1)
8.3.4 Property M2.2
209(1)
8.3.5 Property M0
210(1)
8.3.6 Property M3
211(1)
8.3.7 Transfer for Pre-IMTM
211(1)
8.3.8 Existence of Relative Monodromy Filtration in a Special Case
212(1)
8.3.9 End of the Proof of Proposition 8.3.14
213(1)
8.4 Nearby Cycle Functor Along a Monomial Function
214(2)
8.4.1 Beilinson IMTM and Its Deformation
214(1)
8.4.2 Statement
214(1)
8.4.3 Variant
215(1)
8.4.4 Reformulation
215(1)
8.4.5 Proof
216(1)
8.5 Twistor Version of a Theorem of Kashiwara
216(1)
8.5.1 A Purity Theorem (Special Case)
217(1)
8.5.2 Proof of Proposition 8.5.1
217(1)
8.6 Integrable Case
217(4)
8.6.1 Integrable Mixed Twistor Structure
217(1)
8.6.2 Integrable Polarizable Mixed Twistor Structure
218(1)
8.6.3 Infinitesimal Mixed Twistor Module
218(3)
9 Admissible Mixed Twistor Structures and Their Variants
221(26)
9.1 Admissible Mixed Twistor Structure
222(7)
9.1.1 Mixed Twistor Structure on (X, D)
222(1)
9.1.2 Pre-admissibility
222(1)
9.1.3 Admissibility in the Smooth Divisor Case
223(1)
9.1.4 Admissibility in the Normal Crossing Case
224(1)
9.1.5 Category of Admissible MTS
225(2)
9.1.6 Some Operations
227(2)
9.1.7 Curve Test
229(1)
9.1.8 Tensor Products
229(1)
9.2 Admissible Polarizable Mixed Twistor Structure
229(6)
9.2.1 Definition
229(2)
9.2.2 Category of Admissible (w, Λ)-Polarizable Mixed Twistor Structure
231(1)
9.2.3 An Equivalent Condition
232(2)
9.2.4 Specialization
234(1)
9.2.5 Some Operations
234(1)
9.3 Admissible IMTM
235(6)
9.3.1 Definitions
235(2)
9.3.2 Category of Admissible IMTM
237(2)
9.3.3 Some Operations
239(1)
9.3.4 A Remark on Nearby Cycle Functors
240(1)
9.4 Specialization of Admissible Mixed Twistor Structure
241(3)
9.4.1 Statement
241(1)
9.4.2 Some Notation
241(1)
9.4.3 Proof of Proposition 9.4.1
242(2)
9.5 Integrable Case
244(3)
9.5.1 Admissible Mixed Twistor Structure
244(1)
9.5.2 Admissible Polarizable Mixed Twistor Structure
245(1)
9.5.3 Admissible IMTM
246(1)
10 Good Mixed Twistor D-Modules
247(24)
10.1 Good Gluing Data
248(8)
10.1.1 An Equivalence
248(2)
10.1.2 Canonical Prolongments
250(1)
10.1.3 Beilinson Functors
251(3)
10.1.4 Nearby Cycle Functors, Maximal Functors and Vanishing Cycle Functors
254(2)
10.1.5 Gluing Along a Monomial Function
256(1)
10.2 Good Pre-Mixed Twistor D-Module
256(7)
10.2.1 Weak Admissible Specializability
256(1)
10.2.2 Local Case
257(3)
10.2.3 Global Case
260(3)
10.2.4 Gluing
263(1)
10.3 Good Mixed Twistor D-Modules
263(5)
10.3.1 Statement
263(1)
10.3.2 Preliminary
264(1)
10.3.3 Localizability of Good Pre-mixed Twistor D-Modules
264(1)
10.3.4 Proof of Theorem 10.3.1 and Proposition 10.3.2
265(2)
10.3.5 Proof of Lemma 10.3.4
267(1)
10.4 Integrable Case
268(3)
11 Some Basic Property
271(26)
11.1 Expression as Gluing of Admissible Mixed Twistor Structure
272(7)
11.1.1 Cell
272(1)
11.1.2 Cell of Mixed Twistor D-Modules
273(1)
11.1.3 Expression as a Gluing
273(2)
11.1.4 Gluing
275(1)
11.1.5 Admissibility of Cells
276(3)
11.2 Localization
279(6)
11.2.1 Localization Along Functions
279(1)
11.2.2 Localization Along Hypersurfaces
280(2)
11.2.3 The Underlying D-Modules
282(1)
11.2.4 Independence from Compactification
283(2)
11.3 Twist by Admissible Twistor Structure and Beilinson Functors
285(4)
11.3.1 Smooth Case
285(1)
11.3.2 Admissible Case
285(2)
11.3.3 Beilinson Functors
287(2)
11.4 External Tensor Product
289(8)
11.4.1 Preliminary
289(2)
11.4.2 External Tensor Product of Mixed Twistor D-Modules
291(4)
11.4.3 Compatibility
295(2)
12 D-Triples and Their Functoriality
297(74)
12.1 D-Triples and Their Push-Forward
298(16)
12.1.1 D-triples and D-Complex-Triples
298(3)
12.1.2 The Push-Forward
301(8)
12.1.3 Hermitian Adjoint of D-Complex-Triples
309(1)
12.1.4 Comparison with the Naive Push-Forward
310(2)
12.1.5 Rules for Signature (Appendix)
312(2)
12.2 Some Basic Functors for Non-degenerate D-Triples
314(5)
12.2.1 Category of Non-degenerate D-Triples
314(1)
12.2.2 Localization
314(1)
12.2.3 Tensor Product with Smooth D-Triples
315(1)
12.2.4 Beilinson Functors for D-Triples
316(1)
12.2.5 External Tensor Product
317(2)
12.3 De Rham Functor
319(5)
12.3.1 CX-Complex-Triples
319(1)
12.3.2 De Rham Functor for DX-Complex-Triples
320(1)
12.3.3 Compatibility with the Shift
321(1)
12.3.4 Compatibility with the Hermitian Adjoint
321(1)
12.3.5 Compatibility with the Push-Forward
322(1)
12.3.6 Compatibility with the External Tensor Product
323(1)
12.4 Duality of D-Triples
324(15)
12.4.1 Duality for Non-degenerate D-Triples
324(3)
12.4.2 Duality of Complexes of Non-degenerate DX-Triples
327(1)
12.4.3 Compatibility of the Push-Forward and the Duality
328(1)
12.4.4 Compatibility with Other Functors
329(5)
12.4.5 Functor γ*
334(1)
12.4.6 Push-Forward and Duality of D-Modules (Appendix)
335(4)
12.5 Proof of Theorems 12.4.1 and 12.4.5
339(11)
12.5.1 Preliminary
339(1)
12.5.2 Push Forward and the Functor CX
339(3)
12.5.3 Pairing on the Push-Forward
342(3)
12.5.4 Duality and Push-Forward
345(2)
12.5.5 Canonical Prolongation of Good Meromorphic Flat Bundles
347(2)
12.5.6 Special Case
349(1)
12.5.7 End of the Proof
350(1)
12.6 Real Structure
350(21)
12.6.1 Real Structure of Non-degenerate D-Triple
350(1)
12.6.2 Descriptions of Real Perverse Sheaves
351(5)
12.6.3 The de Rham Functor
356(3)
12.6.4 Regular Case
359(2)
12.6.5 R-Betti Structure
361(3)
12.6.6 Basic Examples
364(7)
13 Duality and Real Structure of Mixed Twistor D-Modules
371(42)
13.1 Duality of R-Modules
372(8)
13.1.1 Duality
372(1)
13.1.2 Compatibility with Push-Forward
373(2)
13.1.3 Specialization Along Χλ
375(1)
13.1.4 Twist by Smooth R-Modules
376(1)
13.1.5 Duality of Smooth R-Modules
377(2)
13.1.6 Duality of Integrable RX-Modules
379(1)
13.2 Duality and Strict Specializability of R-Modules
380(6)
13.2.1 Statement
380(1)
13.2.2 Preliminary
381(1)
13.2.3 RX0[ t]-Modules
381(2)
13.2.4 Filtered Free Module
383(1)
13.2.5 A Filtered Free Resolution
384(1)
13.2.6 Proof of Proposition 13.2.1
385(1)
13.3 Duality of Mixed Twistor D-Modules
386(8)
13.3.1 Statements
386(3)
13.3.2 Relative Monodromy Filtrations
389(1)
13.3.3 Duality of Smooth R-Triples
389(1)
13.3.4 Duality of Canonical Prolongation as R-Triples
390(1)
13.3.5 Duality of Minimal Extensions in the Pure Case
390(2)
13.3.6 Duality of the Canonical Prolongations in MTM
392(1)
13.3.7 Local Construction of the Pairing DT
392(1)
13.3.8 End of the Proof of Theorem 13.3.1
393(1)
13.4 Real Structure of Mixed Twistor D-Modules
394(10)
13.4.1 Some Functors
394(2)
13.4.2 Real Structure of Mixed Twistor D-Modules
396(1)
13.4.3 R-Betti Structure of the Underlying D-Modules
397(1)
13.4.4 Real Structure in the Integrable Case
398(6)
13.5 Relation with Mixed Hodge Modules
404(9)
13.5.1 Some Compatibilities
407(1)
13.5.2 Polarization
408(5)
14 Algebraic Mixed Twistor D-Modules and Their Derived Category
413(52)
14.1 Algebraic Mixed Twistor D-Modules
414(13)
14.1.1 Definition
414(1)
14.1.2 Restriction of KMS-Spectrum
415(1)
14.1.3 Some Functors for Algebraic Mixed Twistor D-Modules
415(7)
14.1.4 Cech Resolutions
422(1)
14.1.5 The Underlying Rλ0-Modules
423(2)
14.1.6 Real Structure
425(1)
14.1.7 Algebraic Integrable Mixed Twistor D-Modules
425(2)
14.2 Derived Category of Algebraic Mixed Twistor D-Modules
427(4)
14.2.1 Some Exact Functors
427(2)
14.2.2 A Version of Kashiwara's Equivalence
429(1)
14.2.3 Enhancement
430(1)
14.3 Push-Forward and Pull Back
431(23)
14.3.1 Push-Forward of Algebraic Holonomic D-Modules
431(11)
14.3.2 Push-Forward of Algebraic Mixed Twistor D-Modules
442(6)
14.3.3 Pull Back of Algebraic Mixed Twistor D-Modules
448(4)
14.3.4 Base Change
452(1)
14.3.5 Tensor and Inner Homomorphism
453(1)
14.3.6 Enhancement
453(1)
14.3.7 Mixed Hodge Modules
454(1)
14.4 Algebraicity of the R-Modules in the Integrable Case
454(11)
14.4.1 Preliminary
454(2)
14.4.2 Statement
456(1)
14.4.3 Preliminary
457(1)
14.4.4 Extension of R-Modules with Good-KMS Structure
458(1)
14.4.5 The Extension of Admissible Mixed Twistor Structure
459(1)
14.4.6 Affine Case
460(3)
14.4.7 Proof of Theorem 14.4.8
463(2)
15 Good Systems of Ramified Irregular Values
465(14)
15.1 Good System of Ramified Irregular Values
466(5)
15.1.1 Good Set of irregular Values
466(1)
15.1.2 Good System of Ramified Irregular Values
467(1)
15.1.3 Specialization of Good Set of Ramified Irregular Values
468(1)
15.1.4 Resolution
468(3)
15.2 Resolution of Turning Points for Lagrangian Covers
471(8)
15.2.1 Lagrangian Cover
471(1)
15.2.2 Statement
472(1)
15.2.3 Separation of Ramification and Polar Part
473(2)
15.2.4 Separation of Cover
475(1)
15.2.5 Proof of Theorem 15.2.7
476(3)
References 479(4)
Index 483
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