International Congress of Mathematicians

244 P. BiranTheorem E'. Let (M,oj) be a closed Kahler manifold with [OJ] £ H 2 (M;Q) andn 2 (M) = 0. Then for every e > 0 there exists a Lagrangian CW-complex A f _ c(M,(jj) with the following property: every symplectic embedding tp : B 2n (e) —¥ (M,oj)must satisfy Image (ip) n A f _ ^ 0.By a Lagrangian CW-complex we mean a subspace A f _ c M which topologicallyis a CW-complex and the interior of each of its cells is a smoothly embeddeddisc of M on which u vanishes.2.4. Methods for studying intersectionsLagrangian intersections. The first systematic study of Lagrangian intersectionswas based on the theory of generating function [12, 26] (an equivalent theory wasindependently developed in contact geometry [13]). Gromov's theory of pseudoholomorphiccurves [22] gave rise to an alternative approach which culminated inwhat is now called Floer theory. Each of these theories has its own advantage. Floertheory works in larger generality and seems to have a richer algebraic structure, onthe other hand the theory of generating functions leads in some cases to sharperresults (see [20]).Since Floer theory will appear in the sequel, let us outline a few facts aboutit (the reader is referred to the works of Floer [16] and of Oh [29, 30] for details).Let (M,(jj) be a symplectic manifold and F 0 ,Fi c (M,oj) two Lagrangiansubmanifolds. In "ideal" situations Floer theory assigns to this data an invariantHF(L 0 ,Li). This is a Z2-vector space obtained through an infinite dimensionalversion of Morse-Novikov homology performed on the space of paths connectingF 0 to L\. The result of this theory is a chain complex CF(L 0 ,Li) whose underlyingvector space is generated by the intersection points F 0 fl L\ (one perturbsF 0 ,Fi so their intersection becomes transverse). The homology of this complexHF(L 0 ,Li) is called the Floer homology of the pair (F 0 ,Fi). The most importantfeature of FF(F 0 ,Fi) is its invariance under Hamiltonian isotopies: if L' 0 ,L'iare Hamiltonianly isotopie to F 0 ,Fi respectively, then HF(L' 0 ,L'i) = FF(F 0 ,Fi).From this point of view HF(L 0 , L\) can be regarded as a quantitative obstructionfor Hamiltonianly separating F 0 from L\. Indeed, the rank of HF(L 0 , L\) is a lowerbound on the number of intersection points of any pair of transversally intersectingLagrangians L' 0 , L\ in the Hamiltonian deformation classes of F 0 , L\ respectively.Let us explain the "ideal situations" in which Floer homology is defined. Firstof all there are restrictions on M : due to analytic difficulties manifolds are requiredto be either closed or to have symplectically convex ends (e.g. C", cotangent bundlesor any Stein manifold). More serious restrictions are posed on the Lagrangians. Forsimplicity we describe them only for the case when L\ is Hamiltonianly isotopieto F 0 . From now on we shall write L = L 0 and F' = L\. In Floer's originalsetting [16] the theory was defined under the assumption that the homomorphismA u : n 2 (M,L) —¥ R, defined by D >-¥ f D uj, vanishes. The reason for this comesfrom the construction of the differential of the Floer complex: the main obstructionfor defining a meaningful differential turns out to be existence of holomorphic discswith boundary on F or F'. These discs appear as a source of non-compactness ofthe space of solutions of the PDEs involved in the construction. Since holomorphic