Descendant log Gromov-Witten invariants for toric varieties and tropical curves

Abstract

Using degeneration techniques, we prove the correspondence of tropical curve counts and log Gromov-Witten invariants with general incidence and psi-class conditions in toric varieties for genus zero curves and all non-superabundant higher-genus situations. We also relate the log invariants to the ordinary ones, in particular explaining the appearance of negative multiplicities in the descendant correspondence result of Mark Gross.

Intersection theory on tropicalizations of toroidal embeddings

“…Here, 'ancestor' means that we use the ψ-classes pulled back from M 0,n via the forgetful morphism. We note that this statement has been proven independently by Mandel and Ruddat [43] with completely different methods. They also show that the ancestor descendants coincide with the usual descendants, so that our result, in fact, yields an equality of logarithmic and tropical descendant Gromov-Witten invariants.…”

“…Remark We defined the ‐classes on to be the analogs of the tropical ‐classes on of , that is, as pullbacks of ‐classes on . It has been shown in that these are equal to the classes on obtained by taking Chern classes of cotangent line bundles.…”

Local Gromov-Witten invariants are log invariants

“…We conjecture also the similar statement where further markings are added on both sides (with zero contact orders to the D i for the left hand side). As evidence for this conjecture, note that it allows computing the local invariant 1 d 3 of O P 1 (−1, −1) for degree d curves from the unique Hurwitz cover of P 1 with maximal branching at 0 and ∞ and cyclic order d automorphism, so the log invariant is 1 d , see also [MR16,Remark 4.17]. We also checked Conjecture 1.4 for P n with D the toric boundary in the case of a single insertion of ψ n−1 [pt]: the log invariant can easily be computed from [MR16, Theorem 1.1]+[MR19] and equals 1 for all n and d. We are grateful to Andrea Brini for computing for us the local invariant for this situation confirming the conjecture in this case.…”

Moduli of stable maps in genus one and logarithmic geometry, II

“…The realizability problem for tropical curves is a combinatorial shadow of the problem of characterizing the closure of the main component in the space of logarithmic maps. The difficulty of the problem has limited tropical enumerative techniques to low target dimensions [12,14,15,30] or to genus 0 curves [19,20,29,33,35].…”