We introduce a class of Liouville manifolds with boundary which we call Liouville sectors. We define the wrapped Fukaya category, symplectic cohomology, and the openclosed map for Liouville sectors, and we show that these invariants are covariantly functorial with respect to inclusions of Liouville sectors. From this foundational setup, a local-to-global principle for Abouzaid's generation criterion follows.

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We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed n-manifolds of dimension n > 2 can be realized as exact Lagrangian submanifolds of T * S n with possibly exotic Weinstein symplectic structures. These Weinstein structures on T * S n , infinitely many of which are distinct, are formed by a single handle attachment to the standard 2n-ball along the Legendrian boundaries of flexible Lagrangians. We also formulate a number of open problems.

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].

Let M be a smooth projective variety and boldD an ample normal crossings divisor. From topological data associated to the pair (M,D), we construct, under assumptions on Gromov–Witten invariants, a series of distinguished classes in symplectic cohomology of the complement X=M∖D. Under further ‘topological’ assumptions on the pair, these classes can be organized into a log(arithmic) PSS morphism, from a vector space which we term the logarithmic cohomology of (M,D) to symplectic cohomology. Turning to applications, we show that these methods and some knowledge of Gromov–Witten invariants can be used to produce dilations and quasi‐dilations (in the sense of Seidel–Solomon [Geom. Funct. Anal. 22 (2012) 443–477]) in examples such as conic bundles. In turn, the existence of such elements imposes strong restrictions on exact Lagrangian embeddings, especially in dimension 3. For instance, we prove that any exact Lagrangian in any complex three‐dimensional conic bundle must be diffeomorphic to a product S1×normalΣg or a connect sum #nS1×S2.