Contents

Acknowledgement xii

Chapter 1. Introduction 1

1.1. Simpson's Meta-Theorem 1

1.2. The purposes in this paper 1

1.3. On the purpose (1) 1

1.4. On the purpose (2) 7

1.5. Some Remark 10

1.6. The outline of the paper 10

Part 1. Preliminary 23

Chapter 2. Preliminary 24

2.1. Notation 24

2.2. Prolongation by an increasing order 26

2.3. Preliminary for /xc-equivariant bundle 28

2.4. Some elementary preliminary for convexity 30

2.5. Some lemmas for functions on a disc 33

2.6. An elementary remark on some distributions 36

2.7. Preliminary from elementary linear algebra 39

2.8. Preliminary from complex differential geometry 41

2.9. Preliminary from functional analysis 52

2.10. An estimate of the norms of Higgs field and the conjugate 54

2.11. Convergency of the sequence of harmonic bundles 58

2.12. Higgs field and twisted map 60

Chapter 3. Preliminary for Mixed Twistor Structure 62

3.1.

P1-holomorphic

vector bundle over X x P

1

62

3.2. Equivariant

P1-holomorphic

bundle over X

xF1

65

3.3. Tate objects and 0(p, q) 69

3.4. Equivalence of some categories 74

3.5. Variation of P1-holomorphic bundles 78

3.6. The twistor nilpotent orbit 84

3.7. Split polarized mixed twistor structure and the nilpotent orbit 89

3.8. The induced tuple on the divisor 96

3.9. Translation of some results due to Kashiwara, Kawai and Saito 100

3.10. 7^-triple in dimension 0 and twistor structure 107

Chapter 4. Preliminary for Filtrations 114

4.1. Filtrations and decompositions on a vector space 114