Duke Mathematical Journal

Toric degenerations of toric varieties and tropical curves

Takeo Nishinou and Bernd Siebert

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We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory

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Duke Math. J. Volume 135, Number 1 (2006), 1-51.

First available in Project Euclid: 26 September 2006

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Zentralblatt MATH identifier

Primary: 14N10: Enumerative problems (combinatorial problems)
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]


Nishinou, Takeo; Siebert, Bernd. Toric degenerations of toric varieties and tropical curves. Duke Mathematical Journal 135 (2006), no. 1, 1--51. doi:10.1215/S0012-7094-06-13511-1. http://projecteuclid.org/euclid.dmj/1159281063.

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  • K. Behrend and Y. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke. Math. J. 85 (1996), 1--60.
  • P. Candelas, X. C. De La Ossa, P. S. Green, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21--74.
  • K. Fukaya and Y.-G. Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997), 96--180.
  • W. Fulton and R. Pandharipande, ``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.
  • W. Fulton and B. Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), 335--353.
  • A. B. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 1996, no. 13, 613--663.
  • M. Gross and B. Siebert, ``Affine manifolds, log structures, and mirror symmetry'' in Proceedings of Gökova Geometry-Topology Conference 2002 (Gökova, Turkey), Turkish J. Math. 27, Sci. Tech. Res. Council Turkey, Ankara, 2003, 33--60.
  • —, Mirror symmetry via logarithmic degeneration data, I, J. Differential Geom. 72 (2006), 169--338.
  • A. Grothendieck and M. Reynaud, Revêtements étales et groups fondamental, Séminaire de Géométrie Algébrique de Bois-Marie (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
  • E. N. Ionel and T. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), 45--96.
  • F. Kato, Log smooth deformation theory, Tohoku Math. J. (2) 48 (1996), 317--354.
  • —, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), 215--232.
  • K. Kato, ``Logarithmic structures of Fontaine-Illusie'' in Algebraic Analysis, Geometry, and Number Theory (Baltimore, 1988), Johns Hopkins Univ. Press, Baltimore, 1989, 191--224.
  • Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), 395--409.
  • F. F. Knudsen, The projectivity of the moduli spaces of stable curves, II: The stacks $M_g,n$, Math. Scand. 52 (1983), 161--199.
  • M. Kontsevich and Y. Soibelman, ``Homological mirror symmetry and torus fibrations'' in Symplectic Geometry and Mirror Symmetry (Seoul, 2000), World Sci., River Edge, N.J., 2001, 203--263.
  • A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau $3$-folds, Invent. Math. 145 (2001), 151--218.
  • J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), 509--578.
  • —, A degeneration formula of,GW-invariants, J. Differential Geom. 60 (2002), 199--293.
  • B. H. Lian, K. Liu, and S.-T. Yau, Mirror principle, I, Asian J. Math. 1 (1997), 729--763.
  • G. Mikhalkin, Enumerative tropical algebraic geometry in $\RR^2$, J. Amer. Math. Soc. 18 (2005), 313--377.
  • S. Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), 355--441.
  • B. Siebert, Logarithmic Gromov-Witten invariants, unfinished manuscript, 2001.
  • G. Tian, ``Quantum cohomology and its associativity'' in Current Developments in Mathematics, 1995 (Cambridge, Mass.), Internat. Press, Cambridge, Mass., 1994, 361--401.
  • S. Wewers, ``Deformation of tame admissible covers of curves'' in Aspects of,Galois Theory (Gainesville, Fla., 1996), London Math. Soc. Lecture Note Ser. 256, Cambridge Univ. Press, Cambridge, 1999, 239--282.