Floor diagrams and enumerative invariants of line bundles over an elliptic curve

Abstract

We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are CP 1 -bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block-Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invariants, and the quasi-modularity of their generating series.

Tropical curves in abelian surfaces II: enumeration of curves in linear systems

“…In the case of toric surfaces, I. Itenberg and G. Mikhalkin proved in [14] that the count of tropical curves with fixed genus and degree passing through the right number of points with refined multiplicity is invariant. These results were generalized in various settings, see for instance [21], [1], [10], [3], [4].…”

“…Getting out of the toric situation, it is still possible to prove correspondence theorems by generalizing methods from the toric case. For instance, in [3], the author adapts methods from [20] to compute Gromov-Witten invariants of complex manifolds that are line bundles over an elliptic curve. However, this requires to consider families of complex varieties while correspondence for toric varieties could be seen as happening in the same variety.…”

Tropical Mirror Symmetry in Dimension One

“…Since the first version of this paper appeared as preprint, other researchers have continued working on related topics. We would like to point out in particular a series of papers by Blomme who studies the enumerative geometry of line bundles over elliptic curves and generalized further to the enumerative geometry of abelian surfaces [7,8], and the papers on enumerative geometry of elliptic fibrations by Oberdieck and Pixton [34,35].…”

“…Leaky tropical covers show up as floor diagrams representing counts of tropical curves in toric surfaces resp. in P 1 -bundles over E (see, e.g., [2,7,8,10,18]). We introduce them here, since they can be treated in terms of Feynman integrals analogously to their balanced versions.…”

Tropical curves in abelian surfaces I: enumeration of curves passing through points

“…By induction, we can assume that there are no non-trivalent vertices left. As in [4], the only case where this is not possible is when V is a quadrivalent vertex with a loop on it: an edge whose ends coincide but has non-zero slope. Such an edge would realize a circle, and this cannot happen since TA does not contain any elliptic curve.…”