Elliptic Calabi–Yau Threefolds over a del Pezzo Surface

  • Simon Rose
  • Noriko YuiEmail author
Part of the Progress in Mathematics book series (PM, volume 319)


We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of degree 8, these elliptic threefolds are Calabi–Yau threefolds. We will discuss especially the generating functions of Gromov–Witten and Gopakumar–Vafa invariants.


del Pezzo surface Calabi–Yau threefold Modular forms 

2000 Mathematics Subject Classification. Primary:

14N10 11F11 14H52 



S. Rose was supported by a Jerry Marsden Postdoctoral Fellowship for the Fields major thematic program on Calabi–Yau Varieties: Arithmetic, Geometry and Physics from July to December 2013. N. Yui was supported in part by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant.


  1. [AS]
    M. Alim, E. Scheidegger, Topological strings on elliptic fibrations. Commun. Number Theory Phys. 8 (2012). doi:10.4310/CNTP.2014.v8.n4.a4Google Scholar
  2. [BL]
    J. Bryan, N.C. Leung, The enumerative geometry of K3 surfaces and modular forms. J. Am. Math. Soc. 13 (2), 371–410 (electronic) (2000). MR1750955 (2001i:14071)Google Scholar
  3. [GV]
    R. Gopakumar, C. Vafa, M-theory and topological strings I and II. (1998). arXiv:hep-th19809187; hep-th19812127Google Scholar
  4. [GP]
    L. Göttsche, R. Pandharipande, The quantum cohomology of blow-ups of P 2 and enumerative geometry. J. Differ. Geom. 48 (1), 61–90 (1998). 1622601 (99d:14057)Google Scholar
  5. [G]
    L. Göttsche, A conjectural generating function for numbers of curves on surfaces. Commun. Math. Phys. 196 (3), 523–533 (1998). 1645204 (2000f:14085)Google Scholar
  6. [H]
    F. Hirzebruch, Classical algebraic geometry and Calabi-Yau manifolds, in Calabi Conference at University of Wisconsin (2003)Google Scholar
  7. [HK]
    K. Hulek, R. Kloosterman, Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces. Ann. Inst. Fourier (Grenoble) 61 (3), 1133–1179 (2011). 2918726Google Scholar
  8. [K]
    S. Katz, Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds, in Snowbird Lectures on String Geometry. Contemporary Mathematics, vol. 401 (American Mathematical Society, Providence, RI, 2006), pp. 43–52. 2222528 (2007b:14121)Google Scholar
  9. [KMPS]
    A. Klemm, D. Maulik, R. Pandharipande, E. Scheidegger, Noether-Lefschetz theory and the Yau-Zaslow conjecture. J. Am. Math. Soc. 23 (4), 1013–1040 (2010). 2669707 (2011j:14121)Google Scholar
  10. [KMV]
    A. Klemm, P. Mayr, C. Vafa, BPS states of exceptional non-critical strings. Nucl. Phys. B Proc. Suppl. 58, 177–194 (1997). Advanced Quantum Field Theory (La Londe les Maures, 1996). 1486340 (99a:81145)Google Scholar
  11. [KMW]
    A. Klemm, J. Manschot, T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds. Commun. Number Theory Phys. 6 (4), 849–917 (2012). 3068410Google Scholar
  12. [M]
    Y. Manin, Cubic Forms: Algebra, Geometry, Arithmetic. North-Holland Mathematical Library (Elsevier Science, Amsterdam, 1986)Google Scholar
  13. [MP]
    D. Maulik, R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, in A Celebration of Algebraic Geometry, ed. by B. Hassett, J. McKernan, J. Starr, R. Vakil. Clay Mathematics Proceedings, vol. 18 (American Mathematical Society, Providence, RI, 2013), pp. 469–508Google Scholar
  14. [YZ]
    S.-T. Yau, E. Zaslow, BPS states, string duality, and nodal curves on K3. Nucl. Phys. B 471 (3), 503–512 (1996). MR1398633 (97e:14066)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  2. 2.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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