Obstructions associated with tropical curves

Abstract

Given a complex variety 𝑋, suppose there is a degeneration 𝔛 of it to a union of toric varieties glued along toric divisors. Then, one can use tropical curves to study holomorphic curves on 𝑋 through degenerate curves on the central fiber 𝑋₀ of 𝔛. In such studies, the case corresponding to so-called non-superabundant tropical curves is relatively well understood. This is the case where the curve satisfies a version of transversality. On the other hand, in the case corresponding to superabundant tropical curves, where transversality fails, not much is known. The first important step for such a study is describing the obstruction to deforming degenerate curves on 𝑋₀ to generic fibers of 𝔛. In this paper, we present a general formula describing such an obstruction. In the study of superabundant tropical curves, allowing higher-valent vertices (i.e., those with valency greater than three) is essential, although there have been few studies of them in this context. Our formula covers such cases.