International Congress of Mathematicians

ICM 2002 • Vol. II • 351^360Fukaya Categories and DeformationsPaul Seidel*AbstractIt is widely believed that the right "cycles" for symplectic geometry areLagrangian submanifolds of symplectic manifolds (see for instance Weinstein's1981 survey). This can be given several different meanings, depending onthe kind of symplectic geometry one is interested in. In one direction, thedevelopment of Floer cohomology for Lagrangian submanifolds, culminatingin recent work of Fukaya, Oh, Ohta and Ono, has led to the definition of a"Fukaya category" associated to a symplectic manifold. I want to look at therelation between the Fukaya category of an affine variety M C C^ and thatof its projective closure M C CP N . This can be set up as a "deformationproblem" in the abstract algebraic sense.2000 Mathematics Subject Classification: 57R17, 57R56, 18E30.Soon after their first appearance [7], Fukaya categories were brought to theattention of a wider audience through the homological mirror conjecture [14]. Sincethen Fukaya and his collaborators have undertaken the vast project of laying downthe foundations, and as a result a fully general definition is available [9, 6]. The taskthat symplectic geometers are now facing is to make these categories into an effectivetool, which in particular means developing more ways of doing computations in andwith them.For concreteness, the discussion here is limited to projective varieties which areCalabi-Yau (most of it could be carried out in much greater generality, in particularthe integrability assumption on the complex structure plays no real role). Thefirst step will be to remove a hyperplane section from the variety. This makesthe symplectic form exact, which simplifies the pseudo-holomorphic map theoryconsiderably. Moreover, as far as Fukaya categories are concerned, the affine piececan be considered as a first approximation to the projective variety. This is a fairlyobviousidea, even though its proper formulation requires some algebraic formalismof deformation theory. A basic question is the finite-dimensionality of the relevant* Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau, France. E-mail:seidel@math.polytechnique.fr