×

zbMATH — the first resource for mathematics

Tropical descendant Gromov-Witten invariants. (English) Zbl 1171.14039
The authors introduce the tropical version of rational descending Gromov-Witten invariants. These tropical invariants can be computed with a certain lattice path count, similar to G. Mikhalkin’s one [J. Am. Math. Soc. 18, 313–377 (2005; Zbl 1092.14068)], and satisfy a certain WDVV equation (thus being equal to their conventional counterparts under some mild restrictions). This provides a new way to compute descending Gromov-Witten invariants. A similar tropical analogue of descending Gromov-Witten invariants was studied in [M. Gross, Mirror symmetry for \(\mathbb P^2\) and tropical geometry, preprint arXiv:0903.1378 (2009)].

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H99 Curves in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Allermann, L., Rau, J.: First steps in tropical intersection theory. Preprint. math. AG/0709.3705 · Zbl 1193.14074
[2] Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic Geometry, Proceedings of the Summer Research Institute Santa Cruz 1995. Proc. Symp. Pure Math., vol. 62, part 2, pp. 45–96 (1997) · Zbl 0898.14018
[3] Gross, M.: Mirror Symmetry for \({\mathbb{P}^2}\) and Tropical Geometry. Preprint. math.AG/0903. 1378
[4] Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compositio Mathematica (to appear). math.AG/0708.2268 · Zbl 1169.51021
[5] Gathmann A., Markwig H.: Kontsevich’s formula and the WDVV equations in tropical geometry. Adv. Math. 217, 537–560 (2008) math.AG/0509628 · Zbl 1131.14057 · doi:10.1016/j.aim.2007.08.004
[6] Kock, J.: Notes on Psi classes. http://mat.uab.es/\(\sim\)kock/GW/notes/psi-notes.pdf
[7] Kerber, M., Markwig, H.: Intersecting Psi-classes on tropical M 0,n . Preprint. math.AG/0709.3953 · Zbl 1205.14070
[8] Mann, B.: An equivalent condition for the reducibility of tropical curves. Preprint, University of Michigan, Ann Arbor (in preparation)
[9] Markwig, H.: The enumeration of plane tropical curves. PhD thesis, TU Kaiserslautern (2006)
[10] Mikhalkin G.: Enumerative tropical geometry in \({\mathbb{R}^2}\) . J. Am. Math. Soc. 18, 313–377 (2005) math.AG/0312530 · Zbl 1092.14068 · doi:10.1090/S0894-0347-05-00477-7
[11] Mikhalkin, G.: Tropical geometry and its applications. In: Sanz-Sole, M., et al. (eds.) Invited Lectures, vol. II. Proceedings of the ICM Madrid, pp. 827–852 (2006). math.AG/0601041 · Zbl 1103.14034
[12] Mikhalkin, G.: Moduli spaces of rational tropical curves. Preprint. math.AG/0704.0839 (2007) · Zbl 1203.14027
[13] Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry. Idempotent Mathematics and Mathematical Physics, Proceedings Vienna (2003). math/0306366 · Zbl 1093.14080
[14] Speyer, D., Sturmfels, B.: Tropical mathematics. Preprint. math.CO/0408099 (2004) · Zbl 1065.14071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.