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Quantum reconstruction for Fano bundles on projective space

Part of: Symplectic geometry, contact geometry Surfaces and higher-dimensional varieties Projective and enumerative geometry

Published online by Cambridge University Press:  11 January 2016

Andrew Strangeway
Andrew Strangeway*
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdoma.strangeway09@imperial.ac.uk
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Abstract

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We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov-Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

MSC classification

Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14J45: Fano varieties
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 218 , June 2015 , pp. 1 - 28
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Ancona, V. and Maggesi, M., “On the quantum cohomology of Fano bundles over projective spaces” in The Fano Conference (Turin, 2002), University of Turin, Turin, 2004, 81–98. MR 2112569.Google Scholar
[2] Ancona, V., Peternell, T., and Wiśniewski, J. A., Fano bundles and splitting theorems on projective spaces and quadrics, Pacific J. Math. 163 (1994), 17–42. MR 1256175.CrossRefGoogle Scholar
[3] Behrend, K., Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601–617. MR 1431140. DOI 10.1007/s002220050132.CrossRefGoogle Scholar
[4] Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 45–88. MR 1437495. DOI 10.1007/s002220050136.CrossRefGoogle Scholar
[5] Beukers, F., “Irrationality proofs using modular forms” in Journées arithmétiques de Besançon (Besangon, 1985), Asterisque 147-148, Soc. Math. France, Paris, 1987, 271–283, 345. MR 0891433.Google Scholar
[6] Ciocan-Fontanine, I., Kim, B., and Sabbah, C., The abelian/nonabelian correspondence and Frobenius manifolds, Invent. Math. 171 (2008), 301–343. MR 2367022. DOI 10.1007/s00222-007-0082-x.CrossRefGoogle Scholar
[7] Clemens, H., Kollár, J., and Mori, S., Higher-dimensional Complex Geometry, Astérisque 166, Soc. Math. France, Paris, 1989. MR 1004926.Google Scholar
[8] Coates, T., Corti, A., Galkin, S., Golyshev, V., and Kasprzyk, A., Mirror symmetry and Fano manifolds, to appear in Proceedings of the 6th European Congress of Mathematics, preprint, arXiv: 1212.1722v1 [math.AG].Google Scholar
[9] Coates, T., Corti, A., Galkin, S., Golyshev, V., and Kasprzyk, A., (Fano search website), http://www.fanosearch.net (accessed 9 September 2014).Google Scholar
[10] Coates, T. and Givental, A., Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53. MR 2276766. DOI 10.4007/annals.2007.165.15.CrossRefGoogle Scholar
[11] Fulton, W., Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998. MR 1644323. DOI 10.1007/978-1-4612-1700-8.Google Scholar
[12] Fulton, W. and Pandharipande, R., “Notes on stable maps and quantum cohomology” in Algebraic Geometry (Santa Cruz, Calif., 1995), Proc. Sympos. Pure Math. 62 (Part 2), Amer. Math. Soc., Providence, 1997, 45–96. MR 1492534.CrossRefGoogle Scholar
[13] Givental, A. B., Equivariant Gromov-Witten invariants, Int. Math. Res. Not. IMRN 13 (1996), 613–663. MR 1408320. DOI 10.1155/S1073792896000414.CrossRefGoogle Scholar
[14] Givental, A. B., “A mirror theorem for toric complete intersections” in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996) , Progr. Math. 160, Birkhäuser, Boston, 1998, 141–175. MR 1653024.Google Scholar
[15] Golyshev, V. V., “Classification problems and mirror duality” in Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics, London Math. Soc. Lecture Note Ser. 338, Cambridge University Press, Cambridge, 2007, 88–121. MR 2306141. DOI 10.1017/CBO9780511721472.004.CrossRefGoogle Scholar
[16] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, reprint of 1978 original, Wiley Classics Lib., Wiley, New York, 1994. MR 1288523. DOI 10.1002/9781118032527.CrossRefGoogle Scholar
[17] Guest, M. A., “Introduction to homological geometry, I” in Integrable Systems, Geometry, and Topology, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, 2006, 83–121. MR 2222513.Google Scholar
[18] Guest, M. A., “Introduction to homological geometry, II” in Integrable Systems, Geometry, and Topology, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, 2006, 123–150. MR 2222514.Google Scholar
[19] Kock, J. and Vainsencher, I., An Invitation to Quantum Cohomology: Kontsevich's Formula for Rational Plane Curves, Progr. Math. 249, Birkhauser, Boston, 2007. MR 2262630.Google Scholar
[20] Kollár, J., Holomorphic and pseudo-holomorphic curves on rationally connected varieties, Port. Math. 67 (2010), 155–179. MR 2662865. DOI 10.4171/PM/1863.CrossRefGoogle Scholar
[21] Kontsevich, M. and Manin, Yu., Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525–562. MR 1291244.CrossRefGoogle Scholar
[22] Li, J. and Tian, G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174. MR 1467172. DOI 10.1090/S0894-0347-98-00250-1.CrossRefGoogle Scholar
[23] Mori, S. and Mukai, S., “Extremal rays and Fano 3-folds” in The Fano Conference (Turin, 2002), University of Turin, Turin, 2004, 37–50. MR 2112566.Google Scholar
[24] Mori, S. and Mukai, S., The On-Line Encyclopedia of Integer Sequences, Apery numbers, http://oeis.org/A005259 (accessed 8 September 2014).Google Scholar
[25] Qin, Z. and Ruan, Y., Quantum cohomology of projective bundles over P n , Trans. Amer. Math. Soc. 350, no. 9 (1998), 3615–3638. MR 1422617. DOI 10.1090/S0002-9947-98-01968-0.CrossRefGoogle Scholar
[26] Siebert, B. and Tian, G., On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), 679–695. MR 1621570.CrossRefGoogle Scholar
[27] Zagier, D., “Integral solutions of Apéry-like recurrence equations” in Groups and Symmetries, CRM Proc Lecture Notes 47, Amer. Math. Soc., Providence, 2009, 349–366. MR 2500571.CrossRefGoogle Scholar