Geometry & Topology

Quantum periods for $3$–dimensional Fano manifolds

Tom Coates, Alessio Corti, Sergey Galkin, and Alexander Kasprzyk

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

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

Abstract

The quantum period of a variety X is a generating function for certain Gromov–Witten invariants of X which plays an important role in mirror symmetry. We compute the quantum periods of all 3–dimensional Fano manifolds. In particular we show that 3–dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors.

Our methods are likely to be of independent interest. We rework the Mori–Mukai classification of 3–dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient VG, where G is a product of groups of the form GLn() and V is a representation of G. When G = GL1()r, this expresses the Fano 3–fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3–fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the quantum Lefschetz hyperplane theorem of Coates and Givental and the abelian/non-abelian correspondence of Bertram, Ciocan-Fontanine, Kim and Sabbah.

Article information

Source
Geom. Topol., Volume 20, Number 1 (2016), 103-256.

Dates
Received: 12 February 2014
Revised: 2 April 2015
Accepted: 5 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858924

Digital Object Identifier
doi:10.2140/gt.2016.20.103

Mathematical Reviews number (MathSciNet)
MR3470714

Zentralblatt MATH identifier
1348.14105

Subjects
Primary: 14J45: Fano varieties 14J33: Mirror symmetry [See also 11G42, 53D37]
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Keywords
quantum cohomology quantum period Fano manifold mirror symmetry

Citation

Coates, Tom; Corti, Alessio; Galkin, Sergey; Kasprzyk, Alexander. Quantum periods for $3$–dimensional Fano manifolds. Geom. Topol. 20 (2016), no. 1, 103--256. doi:10.2140/gt.2016.20.103. https://projecteuclid.org/euclid.gt/1510858924


Export citation

References

  • M Akhtar, T Coates, S Galkin, A M Kasprzyk, Minkowski polynomials and mutations, SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012)
  • D Auroux, Mirror symmetry and $T\!$–duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol. GGT 1 (2007) 51–91
  • D Auroux, Special Lagrangian fibrations, wall-crossing, and mirror symmetry, from: “Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry”, (H-D Cao, S-T Yau, editors), Surv. Differ. Geom. 13, International Press, Somerville, MA (2009) 1–47
  • V V Batyrev, Toric degenerations of Fano varieties and constructing mirror manifolds, from: “The Fano conference”, (A Collino, A Conte, M Marchisio, editors), Univ. Torino, Turin (2004) 109–122
  • A Beauville, Quantum cohomology of complete intersections, Mat. Fiz. Anal. Geom. 2 (1995) 384–398
  • K Behrend, The product formula for Gromov–Witten invariants, J. Algebraic Geom. 8 (1999) 529–541
  • K Behrend, B Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997) 45–88
  • A Bertram, I Ciocan-Fontanine, B Kim, Gromov–Witten invariants for abelian and nonabelian quotients, J. Algebraic Geom. 17 (2008) 275–294
  • I Ciocan-Fontanine, B Kim, C Sabbah, The abelian/nonabelian correspondence and Frobenius manifolds, Invent. Math. 171 (2008) 301–343
  • G Ciolli, Computing the quantum cohomology of some Fano threefolds and its semisimplicity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 7 (2004) 511–517
  • T Coates, A Corti, S Galkin, V Golyshev, A Kasprzyk, Mirror symmetry and Fano manifolds, preprint (2012) To appear in the Proceedings of the 6th European Congress of Mathematics
  • T Coates, A Corti, S Galkin, A Kasprzyk, Table of Laurent polynomial mirrors of $3$–dimensional Fano manifolds (2016) online supplement Available at \setbox0\makeatletter\@url http://msp.org/gt/2016/20-1/gt-v20-n1-x03-laurent.pdf {\unhbox0
  • T Coates, A Corti, H Iritani, H-H Tseng, Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009) 377–438
  • T Coates, S Galkin, A Kasprzyk, $3d$ Minkowski period sequences, online resource Available at \setbox0\makeatletter\@url http://www.grdb.co.uk/forms/period3 {\unhbox0
  • T Coates, A Gholampour, H Iritani, Y Jiang, P Johnson, C Manolache, The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces, Math. Res. Lett. 19 (2012) 997–1005
  • T Coates, A Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. 165 (2007) 15–53
  • T Coates, Y-P Lee, A Corti, H-H Tseng, The quantum orbifold cohomology of weighted projective spaces, Acta Math. 202 (2009) 139–193
  • D A Cox, S Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs 68, Amer. Math. Soc. (1999)
  • B Fantechi, Stacks for everybody, from: “European Congress of Mathematics, I”, (C Casacuberta, R M Miró-Roig, J Verdera, S Xambó-Descamps, editors), Progr. Math. 201, Birkhäuser, Basel (2001) 349–359
  • S Galkin, Small toric degenerations of Fano threefolds, preprint (2007) Available at \setbox0\makeatletter\@url http://member.ipmu.jp/sergey.galkin/papers/std.pdf {\unhbox0
  • S Galkin, Toric degenerations of Fano manifolds, PhD thesis, Steklov Math. Institute (2008) In Russian Available at \setbox0\makeatletter\@url http:/www.mi.ras.ru/~galkin \unhbox0
  • A Gathmann, Gromov–Witten invariants of blow-ups, J. Algebraic Geom. 10 (2001) 399–432
  • A B Givental, Equivariant Gromov–Witten invariants, Internat. Math. Res. Notices (1996) 613–663
  • A Givental, A mirror theorem for toric complete intersections, from: “Topological field theory, primitive forms and related topics”, (M Kashiwara, A Matsuo, K Saito, I Satake, editors), Progr. Math. 160, Birkhäuser, Boston (1998) 141–175
  • V V Golyshev, Classification problems and mirror duality, from: “Surveys in geometry and number theory: reports on contemporary Russian mathematics”, (N Young, editor), London Math. Soc. Lecture Note Ser. 338, Cambridge Univ. Press (2007) 88–121
  • M Gross, Toric degenerations and Batyrev–Borisov duality, Math. Ann. 333 (2005) 645–688
  • M Gross, B Siebert, Affine manifolds, log structures, and mirror symmetry, Turkish J. Math. 27 (2003) 33–60
  • M Gross, B Siebert, Mirror symmetry via logarithmic degeneration data, I, J. Differential Geom. 72 (2006) 169–338
  • M Gross, B Siebert, Mirror symmetry via logarithmic degeneration data, II, J. Algebraic Geom. 19 (2010) 679–780
  • N P Gushel', Fano varieties of genus $8$, Uspekhi Mat. Nauk 38 (1983) 163–164 In Russian; translated in Russian Math. Surveys 38 (1983) 192–193
  • N P Gushel', Fano $3$–folds of genus $8$, Algebra i Analiz 4 (1992) 120–134 In Russian; translated in St. Petersburg Math. J. 4 (1993) 115–129
  • J Harris, D Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982) 23–88
  • J Hu, Gromov–Witten invariants of blow-ups along points and curves, Math. Z. 233 (2000) 709–739
  • J Hu, Gromov–Witten invariants of blow-ups along surfaces, Compositio Math. 125 (2001) 345–352
  • H Iritani, An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222 (2009) 1016–1079
  • V A Iskovskih, Fano threefolds, I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977) 516–562 In Russian; translated in Math. USSR-Izv. 11 (1977) 485–527
  • V A Iskovskih, Fano threefolds, II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978) 506–549 In Russian; translated in Math. USSR-Izv. 12 (1978) 469–506
  • V A Iskovskih, Anticanonical models of three-dimensional algebraic varieties, from: “Current problems in mathematics, XII”, (R V Gamkrelidze, editor), VINITI, Moscow (1979) 59–157 In Russian; translated in J. Soviet Math. 13 (1980) 745–814
  • L Katzarkov, M Kontsevich, T Pantev, Hodge theoretic aspects of mirror symmetry, from: “From Hodge theory to integrability and TQFT tt*-geometry”, (R Y Donagi, K Wendland, editors), Proc. Sympos. Pure Math. 78, Amer. Math. Soc. (2008) 87–174
  • M Kontsevich, Enumeration of rational curves via torus actions, from: “The moduli space of curves”, (R Dijkgraaf, C Faber, G van der Geer, editors), Progr. Math. 129, Birkhäuser, Boston (1995) 335–368
  • M Kontsevich, Homological algebra of mirror symmetry, from: “Proceedings of the International Congress of Mathematicians, I”, (S D Chatterji, editor), Birkhäuser, Basel (1995) 120–139
  • M Kontsevich, Y Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994) 525–562
  • M Kontsevich, Y Soibelman, Homological mirror symmetry and torus fibrations, from: “Symplectic geometry and mirror symmetry”, (K Fukaya, Y-G Oh, K Ono, G Tian, editors), World Sci. Publ., River Edge, NJ (2001) 203–263
  • M Kontsevich, Y Soibelman, Affine structures and non-Archimedean analytic spaces, from: “The unity of mathematics”, (P Etingof, V Retakh, I M Singer, editors), Progr. Math. 244, Birkhäuser, Boston (2006) 321–385
  • H-H Lai, Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles, Geom. Topol. 13 (2009) 1–48
  • R Lazarsfeld, Positivity in algebraic geometry, II: Positivity for vector bundles, and multiplier ideals, Ergeb. Math. Grenzgeb. 49, Springer, Berlin (2004)
  • J Li, G Tian, Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998) 119–174
  • C Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012) 201–245
  • K Matsuki, Weyl groups and birational transformations among minimal models, Mem. Amer. Math. Soc. 557, Amer. Math. Soc. (1995)
  • S Mori, S Mukai, Classification of Fano $3$–folds with $B\sb{2}\geq 2$, Manuscripta Math. 36 (1981/82) 147–162 Correction in Manuscripta Math. 110 (2003) 407
  • S Mori, S Mukai, On Fano $3$–folds with $B\sb{2}\geq 2$, from: “Algebraic varieties and analytic varieties”, (S Iitaka, editor), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam (1983) 101–129
  • S Mori, S Mukai, Classification of Fano $3$–folds with $B\sb 2\geq 2$, I, from: “Algebraic and topological theories”, (M Nagata, S Araki, A Hattori, editors), Kinokuniya, Tokyo (1986) 496–545
  • S Mori, S Mukai, Extremal rays and Fano $3$–folds, from: “The Fano Conference”, (A Collino, A Conte, M Marchisio, editors), Univ. Torino, Turin (2004) 37–50
  • S Mukai, Biregular classification of Fano $3$–folds and Fano manifolds of coindex $3$, Proc. Nat. Acad. Sci. USA 86 (1989) 3000–3002
  • S Mukai, Curves and symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992) 7–10
  • S Mukai, Fano $3$–folds, from: “Complex projective geometry”, (G Ellingsrud, C Peskine, G Sacchiero, S A Strømme, editors), London Math. Soc. Lecture Note Ser. 179, Cambridge Univ. Press (1992) 255–263
  • S Mukai, Curves and Grassmannians, from: “Algebraic geometry and related topics”, (J-H Yang, Y Namikawa, editors), Conf. Proc. Lecture Notes Algebraic Geom. 1, Int. Press, Cambridge, MA (1993) 19–40
  • S Mukai, Curves and symmetric spaces, I, Amer. J. Math. 117 (1995) 1627–1644
  • S Mukai, New developments in Fano manifold theory related to the vector bundle method and moduli problems, Sūgaku 47 (1995) 125–144 In Japanese; translated in Sugaku Expositions 15 (1995) 125–150
  • S Mukai, Curves and symmetric spaces, II, Ann. of Math. 172 (2010) 1539–1558
  • S Mukai, M Reid, H Takagi, Classification of indecomposable Gorenstein Fano $3$–folds, unpublished manuscript
  • P E Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics 51, Narosa, New Delhi (1978)
  • R Pandharipande, Rational curves on hypersurfaces (after A Givental), from: “Séminaire Bourbaki 1997/98 (Exposé 848)”, Astérisque 252, Soc. Math. France, Paris (1998) 307–340
  • V V Przhiyalkovskiĭ, Gromov–Witten invariants of Fano threefolds of genera $6$ and $8$, Mat. Sb. 198 (2007) 145–158 In Russian; translated in Sb. Math. 198 (2007) 443–446
  • V V Przhiyalkovskiĭ, Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties, Mat. Sb. 198 (2007) 107–122 In Russian; translated in Sb. Math. 198 (2007) 1325–1340
  • V Przyjalkowski, On Landau–Ginzburg models for Fano varieties, Commun. Number Theory Phys. 1 (2007) 713–728
  • M I Qureshi, Families of polarised varieties in weighted flag varieties, PhD thesis, University of Oxford (2011)
  • S Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973) 69–84
  • A Strangeway, A reconstruction theorem for quantum cohomology of Fano bundles on projective space, preprint (2013)
  • A Strominger, S-T Yau, E Zaslow, Mirror symmetry is $T\!$–duality, Nuclear Phys. B 479 (1996) 243–259
  • A N Tjurin, Geometry of moduli of vector bundles, Uspehi Mat. Nauk 29 (1974) 59–88 In Russian; translated in Russian Math. Surveys 29 (1974) 57–88
  • A Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989) 613–670

Supplemental materials

  • Table of Laurent polynomial mirrors for each of the $105$ deformation families of $3$--dimensional Fano manifolds.