Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

Abstract

We consider four approaches to relative Gromov-Witten theory and Gromov-Witten theory of degenerations: J. Li's original approach, B. Kim's logarithmic expansions, Abramovich-Fantechi's orbifold expansions, and a logarithmic theory without expansions due to Gross-Siebert and Abramovich-Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov-Witten invariants associated to all four of these theories are identical.

Superabundant Curves and the Artin Fan

“…The map f u thus naturally has the structure of a logarithmic map. Composing with V(τ ) → V(τ) and pushing forward the logarithmic structure as in [5,Appendix B], we obtain a logarithmic map to V(τ). Ranging over all vertices u and gluing, we obtain a map f : C 0 → X(∆).…”

Skeletons and Fans of Logarithmic Structures

“…The idea is that since an Artin fan A is logarithmicallyétale, a map f : C → A X from a curve to A X is logarithmically unobstructed. Precursors to this result for specific X were obtained in [ACFW13,ACW10,AMW14,CMW12]. In [ACFW13] an approach to Jun Li's expanded degenerations was provided using what in hindsight we might call the Artin fan of the affine line A = A A 1 .…”

“…Moreover, the virtual fundamental classes can be understood with machinery available off the shelf of any deformation theory emporium. This observation from [AMW14] made it possible to prove a number of comparison results, including those described below. There are analogous constructions…”

“…In [ACFW13] an approach to Jun Li's expanded degenerations was provided using what in hindsight we might call the Artin fan of the affine line A = A A 1 . The papers [ACW10,AMW14,CMW12] use this formalism to prove comparison results in relative Gromov-Witten theory. For logarithmically smooth X, the map X → A X was used in [AW13] to prove that logarithmic Gromov-Witten invariants are invariant under logarithmic blowings up; in [ACMW] it was used for general X to complete a proof of boundedness of the space of logarithmic stable maps.…”

Boundedness of the space of stable logarithmic maps

“…An Artin fan whose tautological morphism to Log (which is necessarilý etale) is representable will be said to have faithful monodromy. Logarithmic morphisms between Artin fans are always logarithmicallyétale (see [AMW12,Lemma A.7]). …”