We study matrix factorization and curved module categories for Landau-Ginzburg models (X, W ) with X a smooth variety, extending parts of the work of Dyckerhoff. Following Positselski, we equip these categories with model category structures. Using results of Rouquier and Orlov, we identify compact generators. Via Toën's derived Morita theory, we identify Hochschild cohomology with derived endomorphisms of the diagonal curved module; we compute the latter and get the expected result. Finally, we show that our categories are smooth, proper when the singular locus of W is proper, and Calabi-Yau when the total space X is Calabi-Yau.

We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification (M, D) of X. We exhibit a broad class of pairs (M, D) (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be described topologically. Sample applications include: (a) a complete topological description of the symplectic cohomology ring of the complement, in any projective M , of the union of sufficiently many generic ample divisors whose homology classes span a rank one subspace, (b) complete additive and partial multiplicative computations of degree zero symplectic cohomology rings of many log Calabi-Yau varieties, and (c) a proof in many cases that symplectic cohomology is finitely generated as a ring. A key technical ingredient in our results is a logarithmic version of the PSS morphism, introduced in our earlier work [GP].

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We discuss homological mirror symmetry for the conifold from the point of view of the Strominger-Yau-Zaslow conjecture.Theorem 1.4. There is an equivalenceof triangulated categories sending S 0 and S 1 to O E and O E (−1) respectively.This paper is organized as follows: We review the construction of the SYZ mirror for the smoothed conifold from [AAK] in Section 2. In Section 3, we discuss the construction of Lagrangian submanifolds in Y 0 from paths on the z-plane. In Section 4, we recall the definition of the SYZ transform from [AP01, LYZ00] and prove Theorems 1.1 and 1.3. In Section 5, we give an explicit description of the derived category of coherent sheaves on the resolved conifold. In Section 6, we study the wrapped Fukaya category of Y 0 and prove Theorem 1.2. In Section 7, we study A ∞ -operations on vanishing cycles in Y 0 and prove Theorem 1.4. In Section 8, we study Floer cohomology of immersed Lagrangian S 2 × S 1 . In Section 9, we discuss extension of the main results of this paper to more general small toric Calabi-Yau 3-folds.