A log PSS morphism with applications to Lagrangian embeddings

Abstract

Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of distinguished classes in symplectic cohomology of the complement $X = M \backslash \mathbf{D}$. 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 $(M, \mathbf{D})$ 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 [SS]) 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 a complex 3-dimensional conic bundle over $(\mathbb{C}^*)^2$ must be diffeomorphic to $T^3$ or a connect sum $\#^n S^1 \times S^2$.