Duke Mathematical Journal

The symplectic arc algebra is formal

Mohammed Abouzaid and Ivan Smith

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one Hochschild cohomology class on a Floer A-algebra associated to the (k,k)-nilpotent slice Yk obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification Y¯k. The space Y¯k is obtained as the Hilbert scheme of a partial compactification of the A2k1-Milnor fiber. A sequel to this paper will prove formality of the symplectic cup and cap bimodules and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.

Article information

Duke Math. J., Volume 165, Number 6 (2016), 985-1060.

Received: 6 January 2014
Revised: 16 June 2015
First available in Project Euclid: 28 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

symplectic topology Khovanov homology Fukaya category nilpotent slice


Abouzaid, Mohammed; Smith, Ivan. The symplectic arc algebra is formal. Duke Math. J. 165 (2016), no. 6, 985--1060. doi:10.1215/00127094-3449459. https://projecteuclid.org/euclid.dmj/1453994024

Export citation


  • [1] M. Abouzaid, A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Etudes Sci. 112 (2010), 191–240.
  • [2] M. Abouzaid and I. Smith, Khovanov homology from Floer cohomology, preprint, arXiv:1504.01230v1 [math.SG].
  • [3] I. Biswas and G. Schumacher, “Generalized Petersson–Weil metric on the Douady space of embedded manifolds” in Complex Analysis and Algebraic Geometry, de Gruyter, Berlin, 2000, 109–115.
  • [4] S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves, I: The $\frak{sl}_{2}$ case, Duke Math. J. 142 (2008), 511–588.
  • [5] V. de Silva, Products in the symplectic Floer homology of Lagrangian intersections, D.Phil dissertation, Oxford University, 1998.
  • [6] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245–274.
  • [7] T. Ekedahl and R. Skjelnes, Recovering the good component of the Hilbert scheme, Ann. Math. 179 (2014), 805–841.
  • [8] A. Floer, H. Hofer, and D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), 251–292.
  • [9] J. Fogarty, Algebraic families on an algebraic surface, I & II, Amer. J. Math. 90 (1968), 511–521; 95 (1973), 660–687.
  • [10] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Vol. I, Stud. Adv. Math., Amer. Math. Soc., Providence, 2010.
  • [11] W. Fulton, Intersection Theory, Springer, New York.
  • [12] S. Ganatra, Symplectic cohomology and duality for the wrapped Fukaya category, preprint, arXiv:1304.7312v1 [math.SG].
  • [13] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359–426.
  • [14] M. Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002), 665–741.
  • [15] M. Khovanov, Crossingless matchings and the cohomology of $(n,n)$-Springer varieties, Commun. Contemp. Math. 6 (2004), 561–577.
  • [16] M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), 203–271.
  • [17] P. Kronheimer and T. Mrowka, Khovanov homology is an unknot-detector, Publ. Math. Inst. Hautes Etudes Sci. 113 (2001), 97–208.
  • [18] R. Lipshitz, A cylindrical reformulation of Heegaard Floer theory, Geom. Topol. 10 (2006), 955–1096.
  • [19] C. Manolescu, Nilpotent slices, Hilbert schemes, and the Jones polynomial, Duke Math. J. 132 (2006), 311–369.
  • [20] S. Mau, K. Wehrheim, and C. Woodward, $A_{\infty}$-functors for Lagrangian correspondences, preprint, available at www.math.rutgers.edu/~ctw/papers.html.
  • [21] D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ. 52, Amer. Math. Soc., Providence, 2004.
  • [22] I. Mirkovic and M. Vybornov, Quiver varieties and Beilinson–Drinfeld Grassmannians of type $A$, preprint, arXiv:0712.4160v2 [math.AG].
  • [23] H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, 1999.
  • [24] P. Ozsváth and Z. Szabó, On the Heegaard Floer homology of branched double covers, Adv. Math. 194 (2005), 1–33.
  • [25] T. Perutz, “Hamiltonian handleslides for Heegaard Floer homology” in Proceedings of Gökova Geometry-Topology Conference 2007, Gökova Geometry/Topology Conference (GGT), Gökova, 2008, 15–35.
  • [26] J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), 419–447.
  • [27] R. Rezazadegan, Seidel–Smith cohomology for tangles, Selecta Math. 15 (2009), 487–518.
  • [28] R. Rezazadegan, Pseudoholomorphic quilts and Khovanov homology, preprint, arXiv:0912.0669v3 [math.SG].
  • [29] P. Seidel, “Lectures on four-dimensional Dehn twists” in Symplectic 4-Manifolds and Algebraic Surfaces, Lecture Notes in Math. 1938, Springer, Berlin, 2008, 231–267.
  • [30] P. Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000), 103–146.
  • [31] P. Seidel, “Fukaya categories and deformations” in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, 351–360.
  • [32] P. Seidel, Fukaya Categories and Picard–Lefschetz Theory, European Math. Soc. Publ. House, 2008.
  • [33] P. Seidel and I. Smith, A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J. 134 (2006), 453–514.
  • [34] P. Seidel and J. Solomon, Symplectic cohomology and q-intersection numbers, Geom. Funct. Anal. 22 (2012), 443–477.
  • [35] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37–108.
  • [36] I. Smith, Floer cohomology and pencils of quadrics, Invent. Math. 189 (2012), 149–250.
  • [37] I. Smith, Quiver algebras as Fukaya categories, Geom. Topol. 19 (2015), 2577–2617.
  • [38] J. Solomon, Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, preprint, arXiv:math/0606429v1 [math.SG].
  • [39] C. Stroppel, Parabolic category $\mathcal{O}$, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009), 954–992.
  • [40] J. Varouchas, Stabilité de la classe des variétés Kählériennes par certains morphismes propres, Invent. Math. 77 (1984), 117–127.
  • [41] J. Varouchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), 13–52.
  • [42] C. Voisin, “On the punctual Hilbert scheme of a symplectic four-fold” in Symposium in Honor of C. H. Clemens (Salt Lake City, Utah, 2000), Contemp. Math. 312, Amer. Math. Soc., Providence, 2002, 265–289.
  • [43] K. Wehrheim and C. Woodward, Quilted Floer cohomology, Geom. Topol. 14 (2010), 833–902.