Estimated transversality, projective maps and invariants of symplectic manifolds.

As first observed by Donaldson, very positive line bundles over compact symplectic manifolds admit many approximately holomorphic sections, which can be used to perform geometric constructions imitating classical results of complex algebraic geometry. This technique has given rise to constructions of symplectic submanifolds, symplectic Lefschetz pencils, and maps to the complex projective plane. The monodromy of these maps gives rise to interesting new invariants of symplectic manifolds.

Bibliography:

S.K. Donaldson, Symplectic submanifolds and almost-complex geometry, J. Diff. Geom. 44 (1996), 666-705.

see also:
J.C. Sikorav, Construction de sous-variétés symplectiques, Séminaire Bourbaki 844 (1998), Astérisque 252, (1998), 231-253.

S.K. Donaldson, Lefschetz fibrations in symplectic Geometry, Documenta Math., Extra Volume ICM 1998, II, 309-314.

see also:
S.K. Donaldson, Lefschetz pencils on symplectic manifolds, preprint (1999).

D. Auroux, Symplectic maps to projective spaces and symplectic invariants, to appear in Proc. 7th Gökova Geometry-Topology Conference (2000), International Press, preprint math.GT/0007130.

see also:
D. Auroux, Symplectic 4-manifolds as branched coverings of CP², Inventiones Math. 139 (2000), 551-602.
D. Auroux and L. Katzarkov, Branched coverings of CP² and invariants of symplectic 4-manifolds, Inventiones Math. 142 (2000), 631-673.