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Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY Simon Salamon Politecnico di Torino

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Colloque Paul GauduchonPalaiseau, 20/05/05 Contents Dirac operators Model geometries 4-forms and spinors Types of Q structures Q symplectic manifolds

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Colloque Paul GauduchonPalaiseau, 20/05/05 4-FORMS AND SPINORS 4-FORMS AND SPINORS

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Colloque Paul GauduchonPalaiseau, 20/05/05 4-forms in dimension 8 Possible dimensions include

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Colloque Paul GauduchonPalaiseau, 20/05/05 A simple example

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Colloque Paul GauduchonPalaiseau, 20/05/05 A complex variant

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Colloque Paul GauduchonPalaiseau, 20/05/05 A complex variant

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Colloque Paul GauduchonPalaiseau, 20/05/05 The quaternionic 4-form

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Colloque Paul GauduchonPalaiseau, 20/05/05 Set of OQSs Symmetric spaces 3-forms 8 = forms 8 = 3 + 5

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Colloque Paul GauduchonPalaiseau, 20/05/05 Triality for Sp(2)Sp(1)

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Colloque Paul GauduchonPalaiseau, 20/05/05 Clifford multiplication X determines 8 = 3 + 5

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Colloque Paul GauduchonPalaiseau, 20/05/05 TYPES OF QUATERNIONIC STRUCTURES TYPES OF QUATERNIONIC STRUCTURES

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Colloque Paul GauduchonPalaiseau, 20/05/05 Reduction of structure The 4-form determines the metric and Levi-Civita connection on the bundle with fibre The 4-form determines the metric and Levi-Civita connection on the bundle with fibre

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Colloque Paul GauduchonPalaiseau, 20/05/05 Intrinsic torsion

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Colloque Paul GauduchonPalaiseau, 20/05/05 Q symplectic manifolds

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Colloque Paul GauduchonPalaiseau, 20/05/05 Quaternionic manifolds Nijenhuis = 0

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Colloque Paul GauduchonPalaiseau, 20/05/05 M 8 has an integrable twistor space I,J,K can be chosen with I complex Quaternionic manifolds

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Colloque Paul GauduchonPalaiseau, 20/05/05 DIRAC OPERATORS

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Colloque Paul GauduchonPalaiseau, 20/05/05 Rigidity principle G acts trivially on M Wolf space

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Colloque Paul GauduchonPalaiseau, 20/05/05 The tautological section An Sp(2)Sp(1) structure determines or

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Colloque Paul GauduchonPalaiseau, 20/05/05 Proposition [Witt] The tautological section

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Colloque Paul GauduchonPalaiseau, 20/05/05 Killing spinors M QK, X an infinitesimal isometry

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Colloque Paul GauduchonPalaiseau, 20/05/05 Killing spinors

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Colloque Paul GauduchonPalaiseau, 20/05/05 MODEL GEOMETRIES

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Colloque Paul GauduchonPalaiseau, 20/05/05 M is QK ( ) M is Einstein ( ) M 8 is symmetric Quaternion-Kahler manifolds

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Colloque Paul GauduchonPalaiseau, 20/05/05 Wolf spaces M 8 QK symmetric

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Colloque Paul GauduchonPalaiseau, 20/05/05 1. Projection Links with HK and G 2 holonomy

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Colloque Paul GauduchonPalaiseau, 20/05/05 Complex coadjoint orbits Any nilpotent orbit N has both QK and HK metrics The hunt for potentials: [Biquard-Gauduchon, Swann]

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Colloque Paul GauduchonPalaiseau, 20/05/05 2. The case SL(3,C) 8 = 3 + 5

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Colloque Paul GauduchonPalaiseau, 20/05/05 2. The case SL(3,C) M 8 parametrizes a subset of OQSs

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Colloque Paul GauduchonPalaiseau, 20/05/05 QUATERNIONIC SYMPLECTIC MANIFOLDS

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Colloque Paul GauduchonPalaiseau, 20/05/05 Q contact structures On hypersurfaces and asymptotic boundaries of QK manifolds with non-degenerate Levi form

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Colloque Paul GauduchonPalaiseau, 20/05/05 An extra integrability condition is needed for n=1 and allows one to extend QCSs on S 7 [Duchemin] Without the integrability condition, extension to a Q symplectic metric is nonetheless possible Q contact structures

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Colloque Paul GauduchonPalaiseau, 20/05/05 3. The case SO(5,C) Fibration based on the reduction

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Colloque Paul GauduchonPalaiseau, 20/05/05 3. The case SO(5,C) Total space is both Kahler and QK:

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Colloque Paul GauduchonPalaiseau, 20/05/05 3. The case SO(5,C) X 6 has a subspace of 3-forms

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Colloque Paul GauduchonPalaiseau, 20/05/05 T 2 product examples Ingredients: symplectic with closed primitive 3-forms giving closed 4-form Ingredients: symplectic with closed primitive 3-forms giving closed 4-form

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Colloque Paul GauduchonPalaiseau, 20/05/05 Compact nilmanifold examples have 3 transverse simple closed 3-forms, with reduction T 2 product examples Applications to SL/CY geometry [Giovannini, Matessi]

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Colloque Paul GauduchonPalaiseau, 20/05/05 8-DIMENSIONAL QUATERNIONIC GEOMETRY 8-DIMENSIONAL QUATERNIONIC GEOMETRY

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