COMPARISON THEOREMS FOR GROMOV-WITTEN INVARIANTS OF SMOOTH PAIRS AND OF DEGENERATIONS

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.