In this paper we introduce a refined multiplicity for rational tropical curves in arbitrary dimension, which generalizes the refined multiplicity introduced by F. Block and L. Göttsche in [3]. We then prove an invariance statement for the count of rational tropical curves in several enumerative problems using this new refined multiplicity. This leads to the definition of Block-Göttsche polynomials in any dimension.

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.

In this paper we introduce a refined multiplicity for rational tropical curves in arbitrary dimension, which generalizes the refined multiplicity introduced by Block and Göttsche (Compositio Mathematica 152(1): 115–151, 2016). We then prove an invariance statement for the count of rational tropical curves in several enumerative problems using this new refined multiplicity. This leads to the definition of Block–Göttsche polynomials in any dimension.

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus g curves of fixed degree passing through g points. We compute the multiplicity provided by a correspondence theorem due to T. Nishinou and show that it is possible to refine this multiplicity in the style of Block-Göttsche to get tropical refined invariants.

In 2015, Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through a set consisting of real points and pairs of complex conjugated points chosen generically on the toric boundary of the surface. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, we address the computation of the refined count when the pairs of complex conjugated points are chosen purely imaginary and belonging to the same component of the toric boundary. Despite the non-genericity, we relate this refined count for purely imaginary values to the refined invariant of Mikhalkin for generic values. That allows us to extend the relation between these classical refined invariants and the tropical refined invariants from Block–Göttsche.