Abstract
For the last decade, Mark Gross and Bernd Siebert have worked with a number of collaborators to push forward a program whose aim is an understanding of mirror symmetry. In this chapter, we’ll present certain elements of the “Gross-Siebert” program. We begin by sketching its main themes and goals. Next, we review the basic definitions and results of two main tools of the program, logarithmic and tropical geometry. These tools are then used to give tropical interpretations of certain enumerative invariants. We study in detail the tropical pencil of elliptic curves in a toric del Pezzo surface. We move on to a basic illustration of mirror symmetry, Gross’s tropical construction for \(\mathbb{P}^{2}\). On the A-model side, we present the proof of Siebert and Nishinou that tropical geometry invariants coincide with classical geometry invariants via toric degenerations. We then summarize Gross’s tropical B-model and the theorem that links the two constructions, emphasizing the common tropical structures underlying both.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
If we strove for maximal generality, we would assume π to be flat and locally finitely presented.
References
Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne-Faltings pairs II. Asian J. Math. 18(3), 465–488 (2014)
Allermann, L., Rau, J.: First steps in tropical intersection theory. Mathematische zeitschrift 264(3), 633–670 (2010)
Auroux, D.: Mirror symmetry and T-duality in the complement of an anticanonical divisor. J. Gökova Geometry Topology 1, 51–91
Barannikov, S.: Semi-infinite hodge structures and mirror symmetry for projective spaces. arXiv preprint math/0010157 (2000)
Batyrev, V.V., Borisov, L.A.: On Calabi-Yau complete intersections in toric varieties. In: Higher-Dimensional Complex Varieties, Trento, 1994, pp. 39–65 (1996)
Boehm, J., Bringmann, K., Buchholz, A., Markwig, H.: Tropical mirror symmetry for elliptic curves. arXiv preprint arXiv:1309.5893 (2013)
Chen, Q.: Stable logarithmic maps to Deligne-Faltings pairs I. Ann. Math. 180(2), 455–521 (2014)
Fukaya, K.: Multivalued morse theory, asymptotic analysis and mirror symmetry. Graphs Patterns Math. Theor. Phys. 73, 205–278 (2005)
Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory on compact toric manifolds II: bulk deformations. Sel. Math. 17(3), 609–711 (2011)
Fulton, W.: Introduction to toric varieties, vol. 131. Princeton University Press, Princeton (1993)
Gathmann, A.: Tropical algebraic geometry. Jahresbericht der DMV 108(1), 3–32
Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145(01), 173–195 (2009)
Givental, A.B.: Equivariant Gromov-Witten invariants. Int. Math. Res. Not. 1996(13), 613–663 (1996)
Gross, M.: Toric degenerations and Batyrev-Borisov duality. Math. Ann. 333(3), 645–688 (2005)
Gross, M.: Mirror symmetry for \(\mathbb{P}^{2}\) and tropical geometry. Adv. Math. 224(1), 169–245 (2010)
Gross, M.: Tropical Geometry and Mirror Symmetry, Volume 114 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC (2011)
Gross, M.: Mirror symmetry and the Strominger-Yau-Zaslow conjecture. In: Current Developments in Mathematics. Intl. Press, Boston
Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turk. J. Math. 27, 33–60 (2003)
Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data, II. J. Algebr. Geom. 19(4), 679–780 (2010)
Gross, M., Siebert, B.: From real affine geometry to complex geometry. Ann. Math. 174(3), 1301–1428 (2011)
Gross, M., Siebert, B.: Logarithmic Gromov-Witten invariants. J. Am. Math. Soc. 26(2), 451–510 (2013)
Gross, M., Siebert, B., et al.: Mirror symmetry via logarithmic degeneration data I. J. Differ. Geom. 72(2), 169–338 (2006)
Gross, M., Wilson, P.M.H., et al. Large complex structure limits of K3 surfaces. J. Differ. Geom. 55(3), 475–546 (2000)
Hitchin, N.: The moduli space of special Lagrangian submanifolds. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 25(3–4), 503–515 (1997)
Illusie, L.: Logarithmic spaces (according to K. Kato), volume 15 of perspectives in mathematics. In: Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991). Academic, San Diego (1994)
Iritani, H.: Quantum D-modules and generalized mirror transformations. Topology 47(4), 225–276 (2008)
Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariant and enumeration of real rational curves. Int. Math. Res. Not. 2003(49), 2639–2653 (2003)
Kato, F.: Log smooth deformation and moduli of log smooth curves. Int. J. Math 11(2), 215–232 (2000)
Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic Analysis, Geometry, and Number Theory, Baltimore, 1988. Johns Hopkins University Press, Baltimore (1989)
Konishi, Y., Minabe, S.: Local B-model and mixed Hodge structure. Adv. Theor. Math. Phys. 14(4), 1089–1145 (2010)
Kontsevich, M.: What is tropical mathematics? Oct 2013. http://www.fields.utoronto.ca/video-archive/2013/10/22-2221
Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: The Unity of Mathematics, pp. 321–385. Springer, New York (2006)
Kontsevich, M., Soibelman, Y.: Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry. arXiv preprint arXiv:1303.3253 (2013)
Li, J.: Stable morphisms to singular schemes and relative stable morphisms. J. Differ. Geom. 57(3), 509–578 (2001)
Manin, IU I.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, vol. 47. American Mathematical Society, Providence (1999)
Markwig, H., Rau, J.: Tropical descendant Gromov-Witten invariants. Manuscr. Math. 129(3), 293–335 (2009)
Mikhalkin, G.: Amoebas of algebraic varieties and tropical geometry. In: Different Faces of Geometry, pp. 257–300. Springer, New York (2004)
Mikhalkin, G.: Enumerative tropical algebraic geometry in \(\mathbb{R}^{2}\). J. Am. Math. Soc. 18(2), 313–377 (2005)
Milne, J.: Étale Cohomology. Volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton (1980)
Nishinou, T.: Disc counting on toric varieties via tropical curves. arXiv preprint math/0610660 (2006)
Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)
Ruddat, H.: Log Hodge groups on a toric Calabi-Yau degeneration. Mirror Symmetry Trop. Geom. Contemp. Math. 527, 113–164 (2008)
Ruddat, H., Siebert, B.: Canonical coordinates in toric degenerations (2014)
Shustin, E.: A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces. arXiv preprint math/0406099 (2004)
Steenbrink, J.: Limits of Hodge structures. Invent. Math. 31(3), 229–257 (1975/1976)
Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479(1), 243–259 (1996)
Welschinger, J.-Y.: Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry. C. R. Math. Acad. Sci. Paris 336(4), 341–344 (2003)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Science+Business Media New York
About this chapter
Cite this chapter
van Garrel, M., Overholser, D.P., Ruddat, H. (2015). Enumerative Aspects of the Gross-Siebert Program. In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2830-9_11
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2830-9_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2829-3
Online ISBN: 978-1-4939-2830-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)