For every ruled surface over a positive-genus base surface, the space of symplectic forms in any class of positive self-intersection has infinitely many connected components.
Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventiones mathematicae 82 (1985), no
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Infinite connected components of the space of symplectic forms on ruled surfaces
For every ruled surface over a positive-genus base surface, the space of symplectic forms in any class of positive self-intersection has infinitely many connected components.