×

The family Floer functor is faithful. (English) Zbl 1371.53087

Applications of Fukaya categories to symplectic topology require an algebraic model for these categories; this involves finding a collection of Lagrangians which generate the category in the sense that the Fukaya category fully faithfully embeds in the category of modules over the corresponding \(A_\infty\) algebra. For closed symplectic manifolds, the known strategies for understanding such categories of modules rely on realising them, in an instance of homological mirror symmetry, as modules over the endomorphism algebra of (complexes of) coherent sheaves on an algebraic variety, or a non-commutative deformation thereof. Such descriptions are possible in a limited class of examples. The author is interested in the family Floer program which is to both give a more compelling proof of these equivalences, and extend the class of examples for which they can be proved. We shall call the symplectic side the \(A\)-side, and the algebro-geometric side the \(B\)-side. There are essentially only two previous results on family Floer cohomology, given by K. Fukaya and J. Tu.
The paper under review extends the author’s ICM address [“Family Floer cohomology and mirror symmetry”, Proc. Int. Congr. Math., 813–836 (2014)] by (1) constructing a map of morphism spaces from the \(A\)-side to the \(B\)-side, (2) constructing a map of morphism spaces from the \(B\)-side to \(A\)-side, (3) showing that the composition of these two maps is the identity on the \(A\)-side, leading to the main result of the paper under review, and (4) constructing an \(A_\infty\) functor. The formal result (which is the main theorem of the article) is the following:
Theorem: Let \(X\to Q\) be a Lagrangian torus fibration with \(\pi_2(X)=0\), and \(L\) and \(L'\) Lagrangians which are tautologically unobstructed. Given a sufficiently fine cover of \(Q\), we can associate to \(L\) and \(L'\) (twisted) sheaves \({\mathcal F}(L)\) and \({\mathcal F}(L')\) of perfect complexes (with respect to the induced cover of \(Y\)), as well as maps \[ CF^*(L, L')\mathop{\to}^{\mathcal C}\operatorname{Hom}({\mathcal F}(L),{\mathcal F}(L'))\mathop{\to}^{\mathcal P} CF^*(L,L') \] whose composition is homotopic to the identity up to sign. Given a finite collection of Lagrangians, the map \({\mathcal C}\) extends to a faithful \(A_\infty\) functor from the corresponding Fukaya category to the category of twisted sheaves of perfect complexes.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
14G22 Rigid analytic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abouzaid, M.: Family Floer cohomology and mirror symmetry. In: Proc. Int. Congress of Mathematicians (Seoul, 2014), 813-836 · Zbl 1373.53120
[2] Abouzaid, M.: A geometric criterion for generating the Fukaya category. Publ. Math. Inst. Hautes ´Etudes Sci. 112, 191-240 (2010)Zbl 1215.53078 MR 2737980 · Zbl 1215.53078
[3] Abouzaid, M.: Symplectic cohomology and Viterbo’s theorem. In: Free Loop Spaces in Geometry and Topology, Eur. Math. Soc., 271-485 (2015)Zbl 1326.55003 MR 3444367 · Zbl 1385.53078
[4] Abouzaid, M., Auroux, D., Katzarkov, L.: Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publ. Math. Inst. Hautes ´Etudes Sci. 123, 199-282 (2016)Zbl 06600165 MR 3502098 · Zbl 1368.14056
[5] Abouzaid, M., Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Homological mirror symmetry for compact toric manifolds. In preparation
[6] Adams, J. F.: On the cobar construction. Proc. Nat. Acad. Sci. U.S.A. 42, 409-412 (1956) Zbl 0071.16404 MR 0079266 · Zbl 0071.16404
[7] Boissonnat, J.-D., Dyer, R., Ghosh, A.: Delaunay triangulation of manifolds. Found. Comput. Math. (2017)(online) The family Floer functor is faithful2217 · Zbl 1395.57032
[8] Floer, A., Hofer, H., Salamon, D.: Transversality in elliptic Morse theory for the symplectic action. Duke Math. J. 80, 251-292 (1995)Zbl 0846.58025 MR 1360618 · Zbl 0846.58025
[9] Fukaya, K.: Cyclic symmetry and adic convergence in Lagrangian Floer theory. Kyoto J. Math. 50, 521-590 (2010)Zbl 1205.53090 MR 2723862 · Zbl 1205.53090
[10] Fukaya, K.: Floer homology for families. In progress · Zbl 1322.57025
[11] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory and mirror symmetry on compact toric manifolds. Ast´erisque 376 (2016)Zbl 1344.53001 MR 3460884 · Zbl 1344.53001
[12] Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory: anomaly and obstruction, Part I. AMS/IP Stud. Adv. Math. 46, Amer. Math. Soc. (2009)Zbl 1181.53002 MR 2553465 · Zbl 1181.53002
[13] Gross, M., Huybrechts, D., Joyce, D.: Calabi-Yau Manifolds and Related Geometries. Universitext, Springer, Berlin (2003)Zbl 1014.14019 MR 1963561 · Zbl 1001.00028
[14] Kirby, R. C., Taylor, L. R.: Pin Structures on Low-Dimensional Manifolds. In: Geometry of Low-Dimensional Manifolds, 2 (Durham, 1989), London Math. Soc. Lecture Note Ser. 151, Cambridge Univ. Press, Cambridge, 177-242 (1990)Zbl 0754.57020 MR 1171915 · Zbl 0754.57020
[15] Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, NJ, 203-263 (2001)Zbl 1072.14046 MR 1882331 · Zbl 1072.14046
[16] McDuff, D., Salamon, D.: Introduction to Symplectic Topology. 2nd ed., Oxford Math. Monogr., Clarendon Press, New York (1998)Zbl 1066.53137 MR 1698616 · Zbl 1066.53137
[17] McLaughlin, D. A.: Local formulae for Stiefel-Whitney classes. Manuscripta Math. 89, 1-13 (1996)Zbl 0845.55008 MR 1368532 · Zbl 0845.55008
[18] Phillips, A., Stone, D.: Lattice gauge fields, principal bundles and the calculation of topological charge. Comm. Math. Phys. 103, 599-636 (1986)Zbl 0597.53065 MR 0832541 · Zbl 0597.53065
[19] Seidel, P.: Fukaya Categories and Picard-Lefschetz Theory. Zurich Lectures Adv. Math., Eur. Math. Soc., Z¨urich (2008)Zbl 1159.53001 MR 2441780 · Zbl 1159.53001
[20] Seidel, P.: Homological mirror symmetry for the quartic surface. Mem. Amer. Math. Soc. 236, no. 1116, vi+129 pp. (2015)Zbl 1334.53091 MR 3364859 · Zbl 1334.53091
[21] Sheridan, N.: Homological mirror symmetry for Calabi-Yau hypersurfaces in projective space. Invent. Math. 199, 1-186 (2015)Zbl 1344.53073 MR 3294958 · Zbl 1344.53073
[22] Shtan’ko, M. A., Shtogrin, M. I.: Embedding cubic manifolds and complexes into a cubic lattice. Uspekhi Mat. Nauk 47, no. 1, 219-220 (1992) (in Russian)Zbl 0813.57021 MR 1171874 · Zbl 0813.57021
[23] Tate, J.: Rigid analytic spaces. Invent. Math. 12, 257-289 (1971)Zbl 0212.25601 MR 0306196 · Zbl 0212.25601
[24] Thurston, W. P.: Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc. 55, 467-468 (1976)Zbl 0324.53031 MR 0402764 · Zbl 0324.53031
[25] Tu, J.: On the reconstruction problem in mirror symmetry. Adv. Math. 256, 449-478 (2014) Zbl 1287.53077 MR 3177298 · Zbl 1287.53077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.