Disk counting on toric varieties via tropical curves (Q2851602)

The paper gives a ``tropical'' solution to the problem of enumeration of holomorphic disks in toric varieties with Lagrangian boundary conditions, which is related to the Lagrangian intersection Floer theory. More precisely, the enumerative problem is to count stable maps of a holomorphic disk with marked points in the interior and one marked point on the boundary into an \(n\)-dimensional projective toric variety \(X\) subject to the following conditions: (i) the maps have a given degree, i.e., prescribed intersection numbers with the toric divisors, (ii) the boundary of the disk is mapped into a given Lagrangian torus fiber of the moment map, (iii) interior marked points are mapped to given generic subvarieties (or points) of \(X\), and the boundary marked point is mapped to a prescribed point orbit in the Lagrangian torus, (iv) the degree and the incidence conditions are adjusted in order to define finitely many maps. The main result states that the number in question equals the number of certain rational tropical curves in \({\mathbb R}^n\), which have one ``stop'', match a collection of tropical incidence conditions (in particular, a ``stop'' at a given point), and are counted with specific multiplicities.NEWLINENEWLINE A tropical rational curve with a ``stop'' is defined similarly to a usual tropical curve: it is a proper map into \({\mathbb R}^n\) of a tree with one univalent vertex so that the map is \({\mathbb Z}\)-linear on each edge of the tree and satisfy a balancing condition at each multivalent vertex. The proof basically follows the lines of the author's previous paper, joint with \textit{B. Siebert} [Duke Math. J. 135, No. 1, 1--51 (2006; Zbl 1105.14073)]. The enumerative numbers computed in the paper look like open genus zero Gromov-Witten invariants. However, they remain invariant only under small variation of incidence conditions and are well-defined only in a neighborhood of a maximal degeneration limit. The author provides an example when the counted numbers do coincide with certain genus zero Gromov-Witten invariants.