Toric degenerations of toric varieties and tropical curves

By T. Nishinou, Bernd S Siebert. Duke mathematical journal, 2004

257 citations recorded by the source corpus.

We show that the counting of rational curves on a complete toric variety that are in general position 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. This generalizes results of Mikhalkin obtained by different methods in the surface case to arbitrary dimensions.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8cb4533299e9be269504306db7678ef3b22607f1

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Citations in the corpus

Towards Mirror Symmetry for Varieties of General Type (2012, methodology): “egendre transform. This will show that the mirror symmetry of (0.1), (0.2) ﬁts into the general setup of the Gross-Siebert program. We use a standard method of building toric degenerations, see e.g., [NS06]. Let B⊆MR be a (non-compact) polyhedron with a polyhedral decomposition P and a strictly conv”
From real affine geometry to complex geometry (2007, background): “etry, at a much deeper level than the usual description in terms of periods. AFFINE AND COMPLEX GEOMETRY 7 Furthermore, our description of the B-model side is tropical in nature. It is well-known [Mk][NiSi] that one should expect a correspondence between tropical curves on B and families of holomorp”
Mirror Symmetry via Logarithmic Degeneration Data I (2003, background): “The results of [34] and [36] suggest that this should be possible by counting certain piecewise straight graphs on B (“tropical curves”).”
Tropical geometry and correspondence theorems via toric stacks (2010, methodology): “However, to the best of our understanding, Nishinou-Siebert’s approach must work in large positive characteristic thoughthis is not claimed in [12].”
Decomposition of degenerate Gromov–Witten invariants (2017, methodology): “o Σ(Y) with the correct topology. In particular, the domain curve is rational and should have Dρ·βunbounded edges parallel to a ray ρ∈ Σ(Y), where Dρ ⊂ Y is the corresponding divisor. The argument of [NS] essentially carries out an explicit analysis of possible logarithmic curves associated with eac”
A MODULI STACK OF TROPICAL CURVES (2017, background): “ted to give a ﬁrm algebraic geometric foundation to the many enumerative applications of tropical geometry, as exempliﬁed by Ranganathan’s proof [Ran17] of the Nishinou-Siebert correspondence theorem [NS06] for rational curves on toric varieties. A ﬁrst step in this direction has been undertaken by ”
Exploded Manifolds (2009, background): “For instance, in the case that X is a toric manifold, and we use the toric boundary divisors, work of Mikhalkin, in [11] and of Siebert and Nishinou in [12] translates to saying that genus zero relative Gromov-Witten invariants can be computed by considering tropical curves in ExplX.”
On the tropical Torelli map (2009, background): “For instance, this program has been carried out successfully for many problems of real and complex enumerative geometry, see for example [32,40,50,24,25,11,33,7,21].”
The tropical vertex (2009, methodology): “The only difference between [20] and the present”
A Tropical Toolkit (2006, background): “The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective not-necessarily-normal toric varieties.”
Quivers, curves, and the tropical vertex (2009, unknown): “A transition to holomorphic curve counts with relative constraints is made via [15].”