The paper under review is to set up a deformation problem from a Fukaya category of an affine variety \(M \subset \mathbb{C}^N\) to a Fukaya category of its projective closure \(\overline{M}\subset\mathbb{C} P^N\) in the abstract algebra sense. The paper outlines some expectations and conjectures for which the details and techniques involved have not been carried out yet.
In section 1, the various symplectic Floer homologies are briefly reviewed. The Fukaya \(A_{\infty}\)-category for the exact symplectic manifold \(M\) with contact boundary \(\partial M = L\) is discussed. The interesting case is to consider the holomorphic maps \(u: (\Sigma_{g, n}, \partial \Sigma_{g, n}) \to (M; L_1, \cdots, L_n)\) of an \(n\)-punctured surface of genus \(g\) with boundary mapping into Lagrangians. The \(A_{\infty}\) structure on such a symplectic cohomology theory gives the so-called Fukaya category (in general, it is formulated to be a derived (triangulated) category with slight extensions).
The Fukaya category seems to be the right setup to provide Kontsevich’s homological mirror symmetry. Section 3 discusses an example of the Fukaya category of Lagrangian spheres in the symplectic Lefschetz pencils. Sections 4 and 5 under proper assumptions give Conjectures 4 and 5. Conjecture 4 says that the Hochschild cohomology of the Fukaya category is isomorphic to the symplectic cohomology of the symplectic manifold; and Conjecture 5 says that the Fukaya category can be identified with its full subcategory consisting of only Lagrangian submanifolds, up to quasi-isomorphism.
For the entire collection see [Zbl 0993.00022].