International Congress of Mathematicians

ICM 2002 • Vol. II • 241-255Geometry of Symplectic IntersectionsP. Biran*AbstractIn this paper we survey several intersection and non-intersection phenomenaappearing in the realm of symplectic topology. We discuss their implicationsand finally outline some new relations of the subject to algebraicgeometry.2000 Mathematics Subject Classification: 53D35, 53D40, 14D06, 14E25.Keywords and Phrases: Symplectic, Lagrangian, Algebraic variety.1. IntroductionSymplectic geometry exhibits a range of intersection phenomena that cannotbe predicted nor explained on the level of pure topology or differential geometry.The main players in this game are certain pairs of subspaces (e.g. Lagrangiansubmanifolds, domains, or a mixture of both) whose mutual intersections cannotbe removed (or reduced) via the group of Hamiltonian or symplectic diffeomorphisms.The very first examples of such phenomena were conjectures by Arnold inthe 1960's, and eventually established and further explored by Gromov, Floer andothers starting from the mid 1980s.The first part of the paper will survey several intersection phenomena and themathematical tools leading to their discovery. We shall not attempt to present themost general results and since the literature is vast the exposition will be far fromcomplete. Rather we shall concentrate on various intersection phenomena trying tounderstand their nature and whether there is any relations between them.The second part is dedicated to "non-intersections", namely to situationswhere the principles of symplectic intersections break down. In the case of Lagrangiansubmanifolds this absence of intersections is reflected in the vanishing ofa symplectic invariant called Floer homology. This vanishing when interpreted algebraicallyleads to restrictions on the topology of Lagrangian submanifolds. Asa byproduct we shall explain how these restrictions can be used to study someproblems in algebraic geometry concerning hyperplane sections and degenerations.* School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Email:biran@math.tau.ac.il