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: Jun Li's original approach, Bumsig 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.

42 pages. Some minor changes. To appear in Annales de l'Institut Fourier

References in corpus (4)

• Gromov Witten invariants of exploded manifolds
• The Evaluation Space of Logarithmic Stable Maps
• Gromov-Witten theory of banded gerbes over schemes
• Logarithmic stacks and minimality

Cited by in corpus (4)

• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• A generalization of the double ramification cycle via log-geometry
• On the value of thinking tropically to understand Ionel's GW invariants relative normal crossing divisors
• Derived log stacks after Olsson