It is widely believed that the right "cycles" for symplectic geometry are Lagrangian submanifolds of symplectic manifolds (see for instance Weinstein's 1981 survey). This can be given several different meanings, depending on the kind of symplectic geometry one is interested in. In one direction, the development of Floer cohomology for Lagrangian submanifolds, culminating in 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 the relation between the Fukaya category of an affine variety M is contained in C~N and that of its projective closure M-bar is contained in CP~N. This can be set up as a "deformation problem" in the abstract algebraic sense.
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