This paper is a companion to the authors' forthcoming work extending Heegaard Floer theory from closed 3-manifolds to compact 3-manifolds with two boundary components via quilted Floer cohomology. We describe the first interesting case of this theory: the invariants of 3-manifolds bounding S 2 ⨿ T 2 , regarded as modules over the Fukaya category of the punctured 2-torus. We extract a short proof of exactness of the Dehn surgery triangle in Heegaard Floer homology. We show that A ∞ -structures on the graded algebra A formed by the cohomology of two basic objects in the Fukaya category of the punctured 2-torus are governed by just two parameters ðm 6 , m 8 Þ, extracted from the Hochschild cohomology of A. For the Fukaya category itself, m 6 ≠ 0.symplectic manifolds | 3-manifolds | Heegaard Floer theory | homological mirror symmetry | Hochschild cohomology T his article is an offshoot of the authors' forthcoming work on Lagrangian correspondences and invariants of three-manifolds with boundary. In that work we will combine a detailed geometric examination of the Lagrangian correspondences between symmetric products of Riemann surfaces, studied by the second author in refs. 1 and 2, with the A ∞ quilted Floer theory of Ma'uWehrheim-Woodward (3) and the functoriality principle of ref. 4 (see also ref. 5). By doing so, we will extend the package of Heegaard Floer cohomology invariants (6) from closed 3-manifolds to compact 3-manifolds with boundary. To be precise, we construct invariants for compact, oriented, connected 3-manifolds with precisely two boundary components, marked as 'incoming' and 'outgoing'. When these are both spherical, our invariants capture the Heegaard Floer cochains of the capped-off 3-manifold. We refer to Auroux's work (7) for the relationship of this theory to bordered Heegaard Floer theory (8).The format of our invariants is alarmingly abstract: they take the form of A ∞ -functors between A ∞ -categories associated with the boundary surfaces, satisfying a composition law under sewing of cobordisms. Enthusiasts for extended topological quantum field theory (TQFT) will approve of this formulation, but geometric topologists will want to know how to extract topological information from it.In this article, we examine the next-to-simplest case of the theory by applying it to manifolds Y 3 with incoming boundary of genus 0 (which we cap off to formȲ ) and outgoing boundary of genus 1. The relevant A ∞ -categories are certain versions of the Fukaya category of a symplectic 2-torus T with a distinguished point z. The simplest version of the invariant for Y is an A ∞ -moduleM Y over the Fukaya categorŷ F ðT 0 Þ of exact; embedded curves in T 0 ≔ T \fzg:This module evaluates on each object X (which is a circle X ⊂ T with an exactness constraint and certain decorations) to give a cochain complexM Y ðXÞ. This complex is quasi-isomorphic to the Heegaard Floer cochains d CF Ã ðȲ ∪ T U X Þ, where U X is the solid torus in which the circle X bounds a disc. The different objects X correspond to different Dehn ...
This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the formal disc Spec Z [[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y 2 + xy = x 3 over Spec Z, the central fibre of the Tate curve; and, over the 'punctured disc' Spec Z ((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We also prove that the wrapped Fukaya category of the punctured torus is derived-equivalent over Z to coherent sheaves on the central fiber of the Tate curve.1 Beware: the circumstances under which one expects to find such an X are more subtle than those claimed by our one-sentence précis of Kontsevich's conjecture.We shall need to say what we mean by a CY structure for a category over a commutative ring L. The categories in question are of form H 0 C, where C is an A ∞ -category, and this permits us to make an expedient (but not fully satisfactory) definition: Definition 2.1 A CY structure consists on the L-linear A ∞ -category C consists of cochain-level maps φ A,B : hom C (A, B) ≃ hom C (B, A[n]) ∨ such that the induced maps on cohomology [φ A,B ⊗ 1 F ] : Hom H 0 (C× L F) (A, B) ≃ Hom H 0 (C× L F) (B, A[n]) ∨
In the second of a pair of papers, we complete our geometric construction of "Lagrangian matching invariants" for smooth four-manifolds equipped with broken fibrations. We prove an index formula, a vanishing theorem for connected sums and an analogue of the Meng-Taubes formula. These results lend support to the conjecture that the invariants coincide with Seiberg-Witten invariants of the underlying fourmanifold, and are in particular independent of the broken fibration. 53D40, 57R57; 57R15
Abstract. The nth symmetric product of a Riemann surface carries a natural family of Kähler forms, arising from its interpretation as a moduli space of abelian vortices. We give a new proof of a formula of Manton-Nasir [10] for the cohomology classes of these forms. Further, we show how these ideas generalise to families of Riemann surfaces.These results help to clarify a conjecture of D. Salamon [13] on the relationship between Seiberg-Witten theory on 3-manifolds fibred over the circle and symplectic Floer homology.
We prove a structural result in mirror symmetry for projective Calabi-Yau (CY) manifolds. Let X be a connected symplectic CY manifold, whose Fukaya category F(X) is defined over some suitable Novikov field K; its mirror is assumed to be some smooth projective scheme Y over K with 'maximally unipotent monodromy'. Suppose that some split-generating subcategory of (a dg enhancement of) D b Coh(Y) embeds into F(X): we call this hypothesis 'core homological mirror symmetry'. We prove that the embedding extends to an equivalence of categories, D b Coh(Y) ∼ = D π (F(X)), using Abouzaid's split-generation criterion. Our results are not sensitive to the details of how the Fukaya category is set up. In work-in-preparation [PS], we establish the necessary foundational tools in the setting of the 'relative Fukaya category', which is defined using classical transversality theory.
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