Holomorphic symplectic geometry

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