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We construct A-infinity functors between Fukaya categories associated to monotone Lagrangian correspondences between compact symplectic manifolds. We then show that the composition of A-infinity functors for correspondences is homotopic to the functor for the composition, in the case that the composition is smooth and embedded.
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Given a Lagrangian V cong CP^n in a symplectic manifold (M,omega), there is an associated symplectomorphism phi_V of M. We define the notion of a CP^n-object in an A-infinity-category A and use this to construct algebraically an A-infinity-functor Phi_V and prove that it induces an autoequivalence of the derived category DA. We conjecture that Phi_V corresponds to the action of phi_V and prove this in the lowest dimension n=1. The construction is designed to be mirror to a construction of Huybrechts and Thomas.
Let k>2. We prove that the cotangent bundles of oriented homotopy (2k-1)-spheres S and S are symplectomorphic only if the classes defined by S and S agree up to sign in the quotient group of oriented homotopy spheres modulo those which bound parallelizable manifolds. We also show that if the connect sum of real projective space of dimension (4k-1) and a homotopy (4k-1)-sphere admits a Lagrangian embedding in complex projective space, then twice the homotopy sphere framed bounds. The proofs build on previous work of Abouzaid and the authors, in combination with a new cut-and-paste argument, which also gives rise to some interesting explicit exact Lagrangian embeddings into plumbings. As another application, we show that there are re-parameterizations of the zero-section in the cotangent bundle of a sphere which are not Hamiltonian isotopic (as maps, rather than as submanifolds) to the original zero-section.
We construct families of imaginary special Lagrangian cylinders near transverse Maslov index $0$ or $n$ intersection points of positive Lagrangian submanifolds in a general Calabi-Yau manifold. Hence, we obtain geodesics of open positive Lagrangian submanifolds near such intersection points. Moreover, this result is a first step toward the non-perturbative construction of geodesics of closed positive Lagrangian submanifolds. Also, we introduce a method for proving $C^{1,1}$ regularity of geodesics of positive Lagrangians at the non-smooth locus. This method is used to show that $C^{1,1}$ geodesics of positive Lagrangian spheres persist under small perturbations of endpoints, improving the regularity of a previous result of the authors. In particular, we obtain the first examples of $C^{1,1}$ solutions to the positive Lagrangian geodesic equation in arbitrary dimension that are not invariant under isometries. Along the way, we study geodesics of positive Lagrangian linear subspaces in a complex vector space, and prove an a priori existence result in the case of Maslov index $0$ or $n.$ Throughout the paper, the cylindrical transform introduced in previous work of the authors plays a key role.
Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known elementary building blocks for Lagrangian cobordisms that are smoothly the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is decomposable into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.
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 $#^n S^1 times S^2$.
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