Tropically constructed Lagrangians in mirror quintic threefolds

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. The homology spheres are mirror dual to the holomorphic curves contributing to the Gromov-Witten (GW) invariants. In view of Joyce’s conjecture, these Lagrangians are expected to have special Lagrangian representatives and hence solve a special Lagrangian enumerative problem in Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2,875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and the Joyce’s weight of each of these Lagrangians equals the multiplicity of the corresponding tropical curve. As applications, we show that disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians and we check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. Dehn twists along these Lagrangians generate an abelian subgroup of the symplectic mapping class group with that rank.

Refined curve counting with tropical geometry

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

Toric Degenerations, Tropical Curve, and Gromov–Witten Invariants of Fano Manifolds

In this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A and some moduli space of rank two bundles on a genus two curve.

Linear Systems on Tropical Curves

International audience A tropical curve $\Gamma$ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system $|D|$ of a divisor $D$ on a tropical curve $\Gamma$ analogously to the classical counterpart. We investigate the structure of $|D|$ as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, $|D|$ defines a map from $\Gamma$ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as a polyhedral complex. Une courbe tropicale $\Gamma$ est un graphe métrique pouvant contenir des arêtes infinies, et une fonction rationnelle tropicale est une fonction continue linéaire par morceaux à pentes entières. Le système linéaire complet $|D|$ d'un diviseur $D$ sur une courbe tropicale $\Gamma$ est défini de façon analogue au cas classique. Nous étudions la structure de $|D|$ en tant que complexe cellulaire et montrons que les systèmes linéaires sont des quotients de modules tropicaux engendrés par un nombre fini de sommets du complexe. Etant donné un ensemble fini de générateurs, $|D|$ définit une application de $\Gamma$ vers un espace projectif tropical, dont l'image peut être modifiée en une courbe tropicale de degré égal à $\mathrm{deg}(D)$. L'enveloppe convexe tropicale de l'image réalise le système linéaire $|D|$ en tant que complexe polyédral.

The Tropicalj-Invariant

If (Q,A) is a marked polygon with one interior point, then a general polynomialfbelonging toK[x,y] with supportAdefines an elliptic curveCfon the toric surfaceXA. IfKhas a non-archimedean valuation intoRwe can tropicalizeCfto get a tropical curve Trop(Cf). If in the Newton subdivision induced byfis a triangulation and the interior point occurs as the vertex of a triangle, then Trop(Cf) will be a graph of genus one and we show that the lattice length of the cycle of that graph is the negative of the valuation of thej-invariant ofCf.