On the Fukaya category of a Fano hypersurface in projective space

Nick Sheridan

doi:10.1007/s10240-016-0082-8

Nick Sheridan ¹

¹ Department of Mathematics, Princeton University, Fine Hall Washington Road 08544-1000 Princeton NJ USA

Publications Mathématiques de l'IHÉS, Volume 124 (2016), pp. 165-317

Abstract

This paper is about the Fukaya category of a Fano hypersurface $X \subset {CP}^{n}$ . Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed–open string maps, weak proper Calabi–Yau structure, Abouzaid’s split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurface $X$ : we construct a configuration of monotone Lagrangian spheres in $X$ , and compute the associated disc potential. The result coincides with the Hori–Vafa superpotential for the mirror of $X$ (up to a constant shift in the Fano index 1 case). As a consequence, we give a proof of Kontsevich’s homological mirror symmetry conjecture for $X$ . We also explain how to extract non-trivial information about Gromov–Witten invariants of $X$ from its Fukaya category.

Received: 2014-11-25
Accepted: 2016-02-06
Online First: 2016-02-15
Published online: 2016-11-07

MR Zbl

DOI: 10.1007/s10240-016-0082-8

Keywords: Modulus Space, Maslov Index, Quantum Cohomology, Hochschild Cohomology, Homological Mirror Symmetry

Author's affiliations:

Nick Sheridan ¹

¹ Department of Mathematics, Princeton University, Fine Hall Washington Road 08544-1000 Princeton NJ USA

@article{PMIHES_2016__124__165_0,
     author = {Nick Sheridan},
     title = {On the {Fukaya} category of a {Fano} hypersurface in projective space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {165--317},
     year = {2016},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {124},
     doi = {10.1007/s10240-016-0082-8},
     mrnumber = {3578916},
     zbl = {1453.53079},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-016-0082-8/}
}

TY  - JOUR
AU  - Nick Sheridan
TI  - On the Fukaya category of a Fano hypersurface in projective space
JO  - Publications Mathématiques de l'IHÉS
PY  - 2016
SP  - 165
EP  - 317
VL  - 124
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-016-0082-8/
DO  - 10.1007/s10240-016-0082-8
LA  - en
ID  - PMIHES_2016__124__165_0
ER  -

%0 Journal Article
%A Nick Sheridan
%T On the Fukaya category of a Fano hypersurface in projective space
%J Publications Mathématiques de l'IHÉS
%D 2016
%P 165-317
%V 124
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-016-0082-8/
%R 10.1007/s10240-016-0082-8
%G en
%F PMIHES_2016__124__165_0

Nick Sheridan. On the Fukaya category of a Fano hypersurface in projective space. Publications Mathématiques de l'IHÉS, Volume 124 (2016), pp. 165-317. doi: 10.1007/s10240-016-0082-8

References
Cited by

[1.] M. Abouzaid A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes Études Sci., Volume 112 (2010), pp. 191-240 | DOI | MR | Zbl | Numdam

[2.] M. Abouzaid, K. Fukaya, Y. G. Oh, H. Ohta and K. Ono, Quantum cohomology and split generation in Lagrangian Floer theory, in preparation.

[3.] P. Albers A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not., Volume 2008 (2008), p. 56 | MR | Zbl

[4.] D. Auroux Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol., Volume 1 (2007), pp. 51-91 | MR | Zbl

[5.] D. Auroux A beginner’s introduction to Fukaya categories, Contact and symplectic topology (2014), pp. 85-136 | MR | Zbl | DOI

[6.] A. Beauville Quantum cohomology of complete intersections, Mat. Fiz. Anal. Geom., Volume 2 (1995), pp. 384-398 | MR | Zbl

[7.] P. Biran; O. Cornea Lagrangian topology and enumerative geometry, Geom. Topol., Volume 16 (2012), pp. 963-1052 | MR | DOI | Zbl

[8.] P. Biran; O. Cornea Lagrangian cobordism and Fukaya categories, Geom. Funct. Anal., Volume 24 (2014), pp. 1731-1830 | MR | DOI | Zbl

[9.] P. Biran and C. Membrez, The Lagrangian Cubic Equation, 2014, | arXiv | MR

[10.] R. Bott; L. Tu Differential forms in algebraic topology (1982) | Zbl | MR | DOI

[11.] R. Buchweitz, Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, 1986.

[12.] C. H. Cho Products of Floer cohomology of torus fibers in toric Fano manifolds, Commun. Math. Phys., Volume 260 (2005), pp. 613-640 | MR | DOI | Zbl

[13.] C. H. Cho Strong homotopy inner product of an $A_{\infty}$ -algebra, Int. Math. Res. Not., Volume 2008 (2008), p. 35 | Zbl | MR

[14.] C. H. Cho, H. Hong and S. C. Lau, Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $P_{a, b, c}^{1}$ , 2013, | arXiv

[15.] C. H. Cho; Y. G. Oh Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math., Volume 10 (2006), pp. 773-814 | MR | DOI | Zbl

[16.] M. Cohen; S. Montgomery Group-graded rings, smash products, and group actions, Trans. Am. Math. Soc., Volume 282 (1984), pp. 237-258 | MR | DOI | Zbl

[17.] B. Crauder; R. Miranda Quantum cohomology of rational surfaces, The moduli space of curves (1995), pp. 33-80 | MR | Zbl | DOI

[18.] V. Dolgushev, A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold, Ph.D. thesis, MIT, 2005. | MR

[19.] T. Dyckerhoff Compact generators in categories of matrix factorizations, Duke Math. J., Volume 159 (2011), pp. 223-274 | MR | DOI | Zbl

[20.] D. Eisenbud Commutative algebra with a view toward algebraic geometry (1995) | Zbl | MR

[21.] A. Floer; H. Hofer; D. Salamon Transversality in elliptic Morse theory for the symplectic action, Duke Math. J., Volume 80 (1995), pp. 251-292 | MR | DOI | Zbl

[22.] K. Fukaya; Y. G. Oh Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math., Volume 1 (1997), pp. 96-180 | MR | DOI | Zbl

[23.] K. Fukaya; Y. Oh; H. Ohta; K. Ono Lagrangian Floer theory on compact toric manifolds: survey, Surveys in differential geometry (2012), pp. 229-298 | MR | Zbl | DOI

[24.] K. Fukaya; Y. G. Oh; H. Ohta; K. Ono Lagrangian intersection Floer theory—anomaly and obstruction (2007) | Zbl | MR

[25.] K. Fukaya, Y. G. Oh, H. Ohta and K. Ono, Lagrangian surgery and metamorphosis of pseudo-holomorphic polygons, 2009. Preprint, available at https://www.math.kyotou.ac.jp/~fukaya/fukaya.html.

[26.] K. Fukaya; Y. G. Oh; H. Ohta; K. Ono Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J., Volume 151 (2010), pp. 23-175 | MR | DOI | Zbl

[27.] K. Fukaya; Y. G. Oh; H. Ohta; K. Ono Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Sel. Math. New Ser., Volume 17 (2011), pp. 609-711 | MR | DOI | Zbl

[28.] S. Ganatra, Symplectic Cohomology and Duality for the Wrapped Fukaya Category, Ph.D. thesis, MIT, 2012. | MR

[29.] S. Ganatra, T. Perutz and N. Sheridan, Mirror symmetry: from categories to curve counts, 2015, | arXiv

[30.] E. Getzler Cartan homotopy formulas and the Gauss–Manin connection in cyclic homology, Isr. Math. Conf. Proc., Volume 7 (1993), pp. 1-12 | MR | Zbl

[31.] E. Getzler Lie theory for nilpotent $L_{\infty}$ algebras, Ann. Math., Volume 170 (2009), pp. 271-301 | MR | DOI | Zbl

[32.] A. Givental Equivariant Gromov–Witten invariants, Int. Math. Res. Not., Volume 1996 (1996), pp. 613-663 | MR | DOI | Zbl

[33.] M. Gross Tropical geometry and mirror symmetry (2011) | Zbl | MR | DOI

[34.] G. Hochschild; B. Kostant; A. Rosenberg Differential forms on regular affine algebras, Trans. Am. Math. Soc., Volume 102 (1962), pp. 383-408 | MR | DOI | Zbl

[35.] K. Hori Linear models of supersymmetric $D$ -branes, Symplectic geometry and mirror symmetry (2001), pp. 111-186 | MR | Zbl | DOI

[36.] M. Jinzenji On Quantum Cohomology Rings for Hypersurfaces in ${CP}^{N - 1}$ , J. Math. Phys., Volume 38 (1997), pp. 6613-6638 | MR | DOI | Zbl

[37.] A. Kapustin and Y. Li, $D$ -branes in topological minimal models: the Landau–Ginzburg approach, J. High Energy Phys., 07 (2004), 26 pp. (electronic). doi:. | DOI | MR

[38.] A. Keating Lagrangian tori in four-dimensional Milnor fibres, Geom. Funct. Anal., Volume 25 (2015), pp. 1822-1901 | MR | DOI | Zbl

[39.] M. Kontsevich Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians (1994), pp. 120-139 | MR | Zbl

[40.] M. Kontsevich, Lectures at ENS Paris. Notes by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona (1998).

[41.] M. Kontsevich Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003), pp. 157-216 | MR | DOI | Zbl

[42.] M. Kontsevich; Y. Soibelman Notes on $A_{\infty}$ algebras, $A_{\infty}$ categories and non-commutative geometry. I, Homological Mirror Symmetry: New Developments and Perspectives (2008), pp. 153-219 | MR | Zbl

[43.] T. Lada; M. Markl Strongly homotopy Lie algebras, Commun. Algebra, Volume 23 (1995), pp. 2147-2161 | MR | DOI | Zbl

[44.] H. Lawson; M. Michelsohn Spin geometry (1989) | Zbl | MR

[45.] L. Lazzarini Relative frames on $J$ -holomorphic curves, J. Fixed Point Theory Appl., Volume 9 (2011), pp. 213-256 | MR | DOI | Zbl

[46.] J. L. Loday Cyclic homology (1998) | Zbl | MR | DOI

[47.] D. McDuff; D. Salamon J-holomorphic Curves and Symplectic Topology (2004) | Zbl | MR | DOI

[48.] Y. G. Oh Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I, Commun. Pure Appl. Math., Volume 46 (1993), pp. 949-993 | MR | DOI | Zbl

[49.] Y. G. Oh Addendum to ‘Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs, I’, Commun. Pure Appl. Math., Volume 48 (1995), pp. 1299-1302 | Zbl | MR | DOI

[50.] Y. G. Oh; D. Kwon Structure of the image of (pseudo)-holomorphic disks with totally real boundary conditions, Commun. Anal. Geom., Volume 8 (2000), pp. 31-82 | MR | DOI | Zbl

[51.] D. Orlov Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Proc. Steklov Inst. Math., Volume 246 (2004), pp. 227-248 | MR | Zbl

[52.] S. Piunikhin; D. Salamon; M. Schwarz Symplectic Floer–Donaldson theory and quantum cohomology, Contact and symplectic geometry (1996), pp. 171-200 | MR | Zbl

[53.] A. F. Ritter and I. Smith, The monotone wrapped Fukaya category and the open-closed string map, Sel. Math. New Ser., to appear. | MR

[54.] Y. Ruan; G. Tian A mathematical theory of quantum cohomology, J. Differ. Geom., Volume 42 (1995), pp. 259-367 | MR | Zbl | DOI

[55.] P. Seidel Graded Lagrangian submanifolds, Bull. Soc. Math. Fr., Volume 128 (1999), pp. 103-149 | MR | Zbl | Numdam | DOI

[56.] P. Seidel Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians (2002), pp. 351-360 | MR | Zbl

[57.] P. Seidel Homological mirror symmetry for the quartic surface, Mem. Am. Math. Soc. (2015) | MR | Zbl

[58.] P. Seidel A biased view of symplectic cohomology, Current Developments in Mathematics (2008), pp. 211-253 | MR | Zbl

[59.] P. Seidel $A_{\infty}$ subalgebras and natural transformations, Homol. Homotopy Appl., Volume 10 (2008), pp. 83-114 | MR | DOI | Zbl

[60.] P. Seidel, Fukaya categories and Picard–Lefschetz Theory, J. Eur. Math. Soc. (2008). | Zbl

[61.] P. Seidel Suspending Lefschetz fibrations, with an application to local mirror symmetry, Commun. Math. Phys., Volume 297 (2010), pp. 515-528 | MR | DOI | Zbl

[62.] P. Seidel Abstract analogues of flux as symplectic invariants, Mém. Soc. Math. Fr., Volume 137 (2014), pp. 1-135 | MR | Zbl | Numdam

[63.] P. Seidel Homological mirror symmetry for the genus two curve, J. Algebraic Geom., Volume 20 (2011), pp. 727-769 | MR | DOI | Zbl

[64.] P. Seidel, Fukaya $A_{\infty}$ -structures associated to Lefschetz fibrations II, 2014, | arXiv | MR | Zbl

[65.] P. Seidel; R. Thomas Braid group actions on derived categories of coherent sheaves, Duke Math. J., Volume 108 (2001), pp. 37-108 | MR | DOI | Zbl

[66.] N. Sheridan On the homological mirror symmetry conjecture for pairs of pants, J. Differ. Geom., Volume 89 (2011), pp. 271-367 | MR | Zbl | DOI

[67.] N. Sheridan Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space, Invent. Math., Volume 199 (2015), pp. 1-186 | MR | DOI | Zbl

[68.] I. Smith Floer cohomology and pencils of quadrics, Invent. Math., Volume 189 (2012), pp. 149-250 | MR | DOI | Zbl

[69.] T. Tradler Infinity-inner-products on A-infinity-algebras, J. Homotopy Relat. Struct., Volume 3 (2008), pp. 245-271 | MR | Zbl

[70.] C. Weibel An introduction to homological algebra (1994) | Zbl | MR | DOI

[71.] Y. Yoshino Cohen–Macaulay Modules over Cohen–Macaulay Rings (1990) | Zbl | MR | DOI

Cited by Sources: