Amibes de variétés algébriques et dénombrement de courbes

Itenberg, Ilia

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 921, pp. 335-361.

Les amibes des variétés algébriques dans ${\left({ℂ}^{*}\right)}^n$ sont les images de ces variétés par l’application des moments $\mathrm{Log}:{\left({ℂ}^{*}\right)}^n\to {ℝ}^n$, $\mathrm{Log}:(z_1,...,z_n)\mapsto (log|z_1|,...,log|z_n\left|\right)$. Des résultats obtenus par G. Mikhalkin montrent l’utilité des amibes pour l’étude des variétés algébriques réelles et complexes. Les amibes peuvent être déformées en des complexes polyédraux appelés variétés algébriques tropicales. Cette déformation permet, en particulier, de calculer les invariants de Gromov-Witten du plan projectif et d'autres surfaces toriques en dénombrant des courbes tropicales.

Amoebas of algebraic varieties in ${\left({ℂ}^{*}\right)}^n$ are the images of these varieties under the moment map $\mathrm{Log}:{\left({ℂ}^{*}\right)}^n\to {ℝ}^n$, $\mathrm{Log}:(z_1,...,z_n)\mapsto (log|z_1|,...,log|z_n\left|\right)$. G. Mikhalkin’s results show the usefulness of amoebas in the study of real and complex algebraic varieties. Amoebas can be deformed to certain polyhedral complexes which are called tropical algebraic varieties. This deformation gives a possibility to compute Gromov-Witten invariants of the projective plane and other toric surfaces by counting tropical curves.

[1] V. I. Arnol'D - Mathematical methods of classical mechanics, Nauka, Moscou, 1974, en russe ; traduction anglaise : Graduate Texts in Mathematics vol. 60, Springer-Verlag, New York, 1989. | Zbl

[2] M. F. Atiyah - “Angular momentum, convex polyhedra and algebraic geometry”, Proc. Edinburgh Math. Soc. (2) 26 (1983), p. 121-133. | MR | Zbl

[3] G. M. Bergman - “The logarithmic limit set of an algebraic variety”, Trans. Amer. Math. Soc. 157 (1971), p. 459-469. | MR | Zbl

[4] L. Caporaso & J. Harris - “Counting plane curves of any genus”, Invent. Math. 131 (1998), p. 345-392. | MR | Zbl

[5] A. Degtyarev & V. Kharlamov - “Topological properties of real algebraic varieties : Rokhlin's way”, Russian Math. Surveys 55 (2000), no. 4, p. 735-814. | MR | Zbl

[6] M. Forsberg, M. Passare & A. Tsikh - “Laurent determinants and arrangements of hyperplane amoebas”, Adv. in Math. 151 (2000), p. 45-70. | MR | Zbl

[7] W. Fulton - Introduction to Toric Varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, 1993. | MR | Zbl

[8] I. Gelfand, M. M. Kapranov & A. V. Zelevinsky - Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, 1994. | MR | Zbl

[9] A. Henriques - “An analogue of convexity for complements of amoebas of varieties of higher codimensions”, Prépublication, Berkeley, 2001. | Zbl

[10] D. Hilbert - “Mathematische Probleme”, Arch. Math. Phys. (3) 1 (1901), p. 213-237. | JFM

[11] I. Itenberg, V. Kharlamov & E. Shustin - “Welschinger invariant and enumeration of real rational curves”, Internat. Math. Res. Notices 49 (2003), p. 2639-2653. | MR | Zbl

[12] M. M. Kapranov - “Amoebas over non-Archimedian fields”, Prépublication, 2000.

[13] V. Kharlamov & S. Orevkov - “Asymptotic growth of the number of classes of real plane algebraic curves as the degree grows”, Zapiski Nauchn. Semin. POMI 266 (2000), p. 218-233, en russe ; traduction anglaise : J. of Math. Sciences 113 (2003), p. 666-674. | MR | Zbl

[14] -, “The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves”, Prépublication arXiv : math.AG/0301245, 2003. | Zbl

[15] M. Kontsevitch & Yu. Manin - “Gromov-Witten classes, quantum cohomology and enumerative geometry”, Comm. Math. Phys. 164 (1994), p. 525-562. | MR | Zbl

[16] M. Kontsevitch & Ya. Soibelman - “Homological mirror symmetry and torus fibrations”, Prépublication arXiv : math.SG/0011041, 2000. | MR | Zbl

[17] G. L. Litvinov & V. P. Maslov - “The correspondence principle for Idempotent Calculus and some computer applications”, in Idempotency (J. Gunawardena, 'ed.), Cambridge University Press, Cambridge, 1998, p. 420-443. | MR | Zbl

[18] G. L. Litvinov, V. P. Maslov & A. N. Sobolevskii - “Idempotent Mathematics and Interval Analysis”, Prépublication arXiv : math.SC/9911126, 1999. | MR | Zbl

[19] G. Mikhalkin - “Real algebraic curves, moment map and amoebas”, Ann. of Math. 151 (2000), p. 309-326. | MR | Zbl

[20] -, “Amoebas of algebraic varieties”, Prépublication arXiv : math.AG/0108225, 2001.

[21] -, “Decomposition into pairs-of-pants for complex algebraic hypersurfaces”, Prépublication arXiv : math.GT/0205011, 2002.

[22] -, “Counting curves via lattice paths in polygons”, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), p. 629-634. | MR | Zbl

[23] -, “Enumerative tropical algebraic geometry in ${ℝ}^2$”, Prépublication arXiv : math.AG/0312530, 2003.

[24] G. Mikhalkin & H. Rullård - “Amoebas of maximal area”, Internat. Math. Res. Notices 9 (2001), p. 441-451. | MR | Zbl

[25] G. Mikhalkin & O. Viro - “Amoebas of algebraic varieties”, en préparation.

[26] M. Passare & H. Rullård - “Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope”, Prépublication, Université de Stockholm, 2000. | Zbl

[27] J.-E. Pin - “Tropical semirings”, in Idempotency (Bristol 1994), Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998, p. 50-69. | MR | Zbl

[28] J.-J. Risler - “Construction d'hypersurfaces réelles [d'après Viro]”, in Sém. Bourbaki (1992/93), Astérisque, vol. 216, Société Mathématique de France, 1993, exp. no 763, p. 69-86. | Numdam | MR | Zbl

[29] L. Ronkin - “On zeroes of almost periodic functions generated by holomorphic functions in a multicircular domain”, in Complex Analysis in Modern Mathematics, Fazis, Moscou, 2000, p. 243-256. | Zbl

[30] H. Rullård - “Stratification des espaces de polynômes de Laurent et la structure de leurs amibes”, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), p. 355-358. | MR | Zbl

[31] -, “Polynomial amoebas and convexity”, Prépublication, Université de Stockholm, 2001.

[32] E. Shustin - “Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry”, Prépublication arXiv : math.AG/0211278, 2002. | MR | Zbl

[33] F. Sottile - “Enumerative Real Algebraic Geometry”, Prépublication arXiv : math.AG/0107179, 2001. | MR | Zbl

[34] B. Sturmfels - “On the Newton polytope of the resultant”, J. Algebraic Combin. 3 (1994), p. 207-236. | MR | Zbl

[35] -, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 2002. | MR | Zbl

[36] O. Viro - “Dequantization of real algebraic geometry on a logarithmic paper”, in Proceedings of the European Congress of Mathematicians (2000). | Zbl

[37] -, “Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves”, in Proc. Leningrad Int. Topological Conf., Leningrad, 1982, Nauka, Leningrad, 1983, en russe, p. 149-197. | Zbl

[38] -, “Gluing of plane real algebraic curves and construction of curves of degrees $6$ and $7$”, Lect. Notes in Math., vol. 1060, Springer, Berlin etc., 1984, p. 187-200. | MR | Zbl

[39] -, “Progress in the topology of real algebraic varieties over the last six years”, Russian Math. Surveys 41 (1986), no. 3, p. 55-82. | Zbl

[40] R. Walker - Algebraic curves, Princeton Univ. Press, Princeton, N.J., 1950. | MR | Zbl

[41] J.-Y. Welschinger - “Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry”, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), p. 341-344. | MR | Zbl

[42] -, “Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry”, Prépublication arXiv : math.AG/0303145, 2003. | Zbl

[43] G. Wilson - “Hilbert's sixteenth problem”, Topology 17 (1978), no. 1, p. 53-73. | MR | Zbl