A log PSS morphism with applications to Lagrangian embeddings

Abstract

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.

Symplectic cohomology rings of affine varieties in the topological limit

“…Achieving both of the above properties (and in particular the second property) entails a substantial refinement of our earlier work [GP,§3]: roughly speaking, to achieve the above integral filtration, orbits contributing to SC * (X) which wind a large number of times around D are made to to occur arbitrarily close (depending on the amount of winding) to D, in order to to compensate for action errors (between the usual symplectic action and the relevant divisorial winding number) that would otherwise magnify with winding. This integral filtration can also be thought of as a limit of the action filtrations on (symplectic cohomologies of) a family of Liouville domains which exhaust X, see Remark 2.20.…”

“…, k}. By the discussion in [GP,p. 15] the time it takes for p to flow by −Z to Σ is the time it takes for p to flow by −Z vert to Σ ∩ U I , where Z vert , the vertical component of Z with respect to the symplectic fibration π I : U I → D I is given by:…”

“…(4) There is a unique point s = s 0 with q (s 0 ) = 0 and q(s 0 ) = 0. (Compare [M2,proof of Theorem 5.16] and [GP,§3.1…”

“…Proof. This is the content of [GP,Lemma 3.7] and uses the "nice" property of θ and the ω-regularization of D (specifically Lemma 2.4); see also [M2,Theorem 5.16] for a similar discussion in the case of concave boundaries.…”

Floer Theory of Higher Rank Quiver 3-folds

“…The tripod spheres which enter into the proof of Theorem 1.1 exactly correspond to the simplest spectral networks after saddle connections, see [GMN13a, Figure 3]. The possible embedded graded Lagrangians in Y are constrained by results of [GP16], and one only obtains connect sums of copies of S 1 × S 2 and 3-tori. It would be interesting to construct unobstructed immersed special Lagrangian representatives for more general spectral networks.…”