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Singular plane curves and symplectic 4-manifolds
 

Summary: Singular plane curves and
symplectic 4-manifolds
Denis AUROUX
Symplectic manifolds
A symplectic structure on a smooth manifold is a 2-form
such that d = 0 and · · · is a volume form.
Example: R2n
, 0 = dxi dyi.
(Darboux: every symplectic manifold is locally (R2n
, 0),
i.e. there are no local invariants).
Example: Riemann surfaces (, vol) are symplectic.
Example: Every Kšahler manifold is symplectic.
(includes all complex projective manifolds)
but the symplectic category is much larger.
(Gompf 1994: G finitely presented group, (X4
, ) compact
symplectic such that 1(X) = G).
Symplectic manifolds are not always complex, but they are
almost-complex, i.e. there exists J End(TX) such that

  

Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)

 

Collections: Mathematics