Holomorphic symplectic geometry

Holomorphic symplectic geometry

-

Documents
136 pages
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

Holomorphic symplectic geometry Arnaud Beauville Universite de Nice Lisbon, March 2011 Arnaud Beauville Holomorphic symplectic geometry

  • holomorphic symplectic

  • degenerate ?x ?

  • ?? locally

  • pi ?

  • unlike riemannian

  • universite de nice

  • dpr ?


Sujets

Informations

Publié par
Ajouté le 19 juin 2012
Nombre de lectures 32
Langue English
Signaler un abus

Holomorphic symplectic geometry
Arnaud Beauville
Universite de Nice
Lisbon, March 2011
Arnaud Beauville Holomorphic symplectic geometryd’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
Arnaud Beauville Holomorphic symplectic geometryd’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
Arnaud Beauville Holomorphic symplectic geometry() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
Arnaud Beauville Holomorphic symplectic geometryThen (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Arnaud Beauville Holomorphic symplectic geometry(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
Arnaud Beauville Holomorphic symplectic geometry Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Arnaud Beauville Holomorphic symplectic geometryAll this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
Arnaud Beauville Holomorphic symplectic geometryglobal X compact, usually projective or Kahler.
I. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
Arnaud Beauville Holomorphic symplectic geometryI. Symplectic structure
De nition
A symplectic form on a manifold X is a 2-form ’ such that:
d’ = 0 and ’(x)2Alt(T (X )) non-degenerate 8x2 X .x
() locally ’ = dp ^dq +::: +dp ^dq (Darboux)1 1 r r
Then (X;’) is a symplectic manifold.
(In mechanics, typically q $ positions, p $ velocities)i i
Unlike Riemannian geometry, symplectic geometry is locally
trivial; the interesting problems are global.
All this makes sense with X complex manifold, ’ holomorphic.
global X compact, usually projective or Kahler.
Arnaud Beauville Holomorphic symplectic geometry