Related papers: Three dimensional tropical correspondence formula
We study the fix point components of the big torus action on the moduli space of stable maps into a smooth projective toric variety, and apply Graber and Pandharipande's localization formula for the virtual fundamental class to obtain an…
Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently…
We prove N.Takahashi's conjecture determining the contribution of each contact point in genus-$0$ maximal contact Gromov-Witten theory of $\mathbb{P}^2$ relative to a smooth cubic $E$. This is a new example of a question in Gromov-Witten…
We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the…
We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of…
We show that the standard generating functions for genus 0 two-point twisted Gromov-Witten invariants arising from concavex vector bundles over symplectic toric manifolds are explicit transforms of the corresponding one-point generating…
The well known formulas express the curvature and the torsion of a curve in $R^3$ in terms of euclidean invariants of its derivatives. We obtain expressions of this kind for all curvatures of curves in $R^n$. It follows that a curve in…
Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold $X$. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of $X$ defined using Gromov-Witten theory by Klemm-Pandharipande.…
We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation…
We study rational curves of degree two on a smooth sextic 4-fold and their counting invariant defined using Donaldson-Thomas theory of Calabi-Yau 4-folds. By comparing it with the corresponding Gromov-Witten invariant, we verify a…
We prove a comparison formula for curve-counting invariants in the setting of the McKay correspondence, related to the crepant resolution conjecture for Donaldson-Thomas invariants. The conjecture is concerned with comparing the invariants…
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. The purpose of this paper is to advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the…
We state and prove a topological recursion relation that expresses any genus-g Gromov-Witten invariant of a projective manifold with at least a (3g-1)-st power of a cotangent line class in terms of invariants with fewer cotangent line…
In this paper, we define two numbers. One comes from counting tropical curves with a stop and the other is the number of holomorphic discs in toric varieties with Lagrangian boundary condition. Both of these curves should satisfy some…
We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone…
Tyomkin's correspondence theorem states the equality of counts of rational curves of fixed homology class in a toric surface satisfying point and cross-ratio conditions with their tropical counterparts. Such correspondence theorems allow us…
We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two…
An algorithm to determine all the Gromov-Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro…
This paper provides an introduction to exploded manifolds. The category of exploded manifolds is an extension of the category of smooth manifolds with an excellent holomorphic curve theory. Each exploded manifold has a tropical part which…
We use a topological framework to study descendent Gromov-Witten theory in higher genus, non-toric settings. Two geometries are considered: surfaces of general type and the Enriques Calabi-Yau threefold. We conjecture closed formulas for…