Summary: Singular plane curves and
A symplectic structure on a smooth manifold is a 2-form
such that d = 0 and · · · is a volume form.
, 0 = dxi dyi.
(Darboux: every symplectic manifold is locally (R2n
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