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2015 | OriginalPaper | Buchkapitel

Enumerative Aspects of the Gross-Siebert Program

verfasst von: Michel van Garrel, D. Peter Overholser, Helge Ruddat

Erschienen in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Verlag: Springer New York

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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.
Vorheriges Kapitel Donaldson–Thomas Invariants and Wall-Crossing Formulas
Nächstes Kapitel Introduction to Modular Forms
Fußnoten
1
If we strove for maximal generality, we would assume π to be flat and locally finitely presented.
 
Literatur
1.
2.
3.
Auroux, D.: Mirror symmetry and T-duality in the complement of an anticanonical divisor. J. Gökova Geometry Topology 1, 51–91
4.
Barannikov, S.: Semi-infinite hodge structures and mirror symmetry for projective spaces. arXiv preprint math/0010157 (2000)
5.
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) MathSciNetMATH
6.
Boehm, J., Bringmann, K., Buchholz, A., Markwig, H.: Tropical mirror symmetry for elliptic curves. arXiv preprint arXiv:1309.5893 (2013)
7.
8.
Fukaya, K.: Multivalued morse theory, asymptotic analysis and mirror symmetry. Graphs Patterns Math. Theor. Phys. 73, 205–278 (2005) MathSciNetCrossRefMATH
9.
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) MathSciNetCrossRefMATH
10.
Fulton, W.: Introduction to toric varieties, vol. 131. Princeton University Press, Princeton (1993) MATH
11.
Gathmann, A.: Tropical algebraic geometry. Jahresbericht der DMV 108(1), 3–32
12.
Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145(01), 173–195 (2009) MathSciNetCrossRefMATH
13.
14.
15.
16.
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)
17.
Gross, M.: Mirror symmetry and the Strominger-Yau-Zaslow conjecture. In: Current Developments in Mathematics. Intl. Press, Boston
18.
Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turk. J. Math. 27, 33–60 (2003) MathSciNetMATH
19.
20.
21.
22.
Gross, M., Siebert, B., et al.: Mirror symmetry via logarithmic degeneration data I. J. Differ. Geom. 72(2), 169–338 (2006)
23.
Gross, M., Wilson, P.M.H., et al. Large complex structure limits of K3 surfaces. J. Differ. Geom. 55(3), 475–546 (2000)
24.
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) MathSciNetMATH
25.
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)
26.
27.
Itenberg, I., Kharlamov, V., Shustin, E.: Welschinger invariant and enumeration of real rational curves. Int. Math. Res. Not. 2003(49), 2639–2653 (2003) MathSciNetCrossRefMATH
28.
29.
Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic Analysis, Geometry, and Number Theory, Baltimore, 1988. Johns Hopkins University Press, Baltimore (1989)
30.
31.
32.
Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: The Unity of Mathematics, pp. 321–385. Springer, New York (2006)
33.
Kontsevich, M., Soibelman, Y.: Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry. arXiv preprint arXiv:1303.3253 (2013)
34.
Li, J.: Stable morphisms to singular schemes and relative stable morphisms. J. Differ. Geom. 57(3), 509–578 (2001) MathSciNetMATH
35.
Manin, IU I.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, vol. 47. American Mathematical Society, Providence (1999) MATH
36.
37.
Mikhalkin, G.: Amoebas of algebraic varieties and tropical geometry. In: Different Faces of Geometry, pp. 257–300. Springer, New York (2004)
38.
39.
Milne, J.: Étale Cohomology. Volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton (1980)
40.
Nishinou, T.: Disc counting on toric varieties via tropical curves. arXiv preprint math/0610660 (2006)
41.
42.
Ruddat, H.: Log Hodge groups on a toric Calabi-Yau degeneration. Mirror Symmetry Trop. Geom. Contemp. Math. 527, 113–164 (2008) MathSciNetCrossRefMATH
43.
Ruddat, H., Siebert, B.: Canonical coordinates in toric degenerations (2014)
44.
Shustin, E.: A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces. arXiv preprint math/0406099 (2004)
45.
Steenbrink, J.: Limits of Hodge structures. Invent. Math. 31(3), 229–257 (1975/1976)
46.
47.
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) MathSciNetCrossRefMATH
Metadaten
Titel
Enumerative Aspects of the Gross-Siebert Program
verfasst von
Michel van Garrel
D. Peter Overholser
Helge Ruddat
Copyright-Jahr
2015
Verlag
Springer New York
Buch
Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Print ISBN: 978-1-4939-2829-3

Electronic ISBN: 978-1-4939-2830-9

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-1-4939-2830-9_11

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