The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Moduli spaces
Natacha Cappelle
October 30, 2015
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1
The main theorem : explanations
Compact self-dual Riemannian 4-manifold with positive scalar
curvature
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
2
Idea of the proof
3
Infinitesimal proof
4
Local proof
5
Global proof
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
Structure
1
The main theorem : explanations
Compact self-dual Riemannian 4-manifold with positive scalar
curvature
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
2
Idea of the proof
3
Infinitesimal proof
4
Local proof
5
Global proof
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1. The main theorem : explanations
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.1 Compact self-dual Riemannian 4-manifold with positive
scalar curvature
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.1 Compact self-dual Riemannian 4-manifold with positive
scalar curvature
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.1 Compact self-dual Riemannian 4-manifold with positive
scalar curvature
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.2 Principal G-bundle and ’adjoint bundle’
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.2 Principal G-bundle and ’adjoint bundle’
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.2 Principal G-bundle and ’adjoint bundle’
[2] Kobayashi, Nomizu, Foundations of Differential Geometry Volume I, 1963 Interscience Publishers, London, p.50
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.2 Principal G-bundle and ’adjoint bundle’
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.3 The space of moduli of irreducible self-dual connections
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.3 The space of moduli of irreducible self-dual connections
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.3 The space of moduli of irreducible self-dual connections
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.3 The space of moduli of irreducible self-dual connections
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.4 The index theorem
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.4 The index theorem
More information about the index theorem :
Atiyah, Singer, The index of elliptic operators : I, II and III, 1968
published in Annals of mathematics. Second Series, Vol. 87, No.3,
pp. 484-604.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Compact self-dual Riemannian 4-manifold with positive scalar curva
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
1.5 The theorem
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Structure
1
The main theorem : explanations
Compact self-dual Riemannian 4-manifold with positive scalar
curvature
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
2
Idea of the proof
3
Infinitesimal proof
4
Local proof
5
Global proof
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
2. Idea of the proof
Ref. : Atiyah, Hitchin, Singer, Self-duality in four-dimensional Riemannian geometry, 1978 published in Proceedings
of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 362, No. 1711, pp. 425-461.
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Structure
1
The main theorem : explanations
Compact self-dual Riemannian 4-manifold with positive scalar
curvature
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
2
Idea of the proof
3
Infinitesimal proof
4
Local proof
5
Global proof
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Structure
1
The main theorem : explanations
Compact self-dual Riemannian 4-manifold with positive scalar
curvature
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
2
Idea of the proof
3
Infinitesimal proof
4
Local proof
5
Global proof
Natacha Cappelle
Moduli spaces
The main theorem : explanations
Idea of the proof
Infinitesimal proof
Local proof
Global proof
Structure
1
The main theorem : explanations
Compact self-dual Riemannian 4-manifold with positive scalar
curvature
Principal G-bundle and ’adjoint bundle’
The space of moduli of irreducible self-dual connections
The index theorem
The theorem
2
Idea of the proof
3
Infinitesimal proof
4
Local proof
5
Global proof
Natacha Cappelle
Moduli spaces