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

By D. Abramovich, Steffen Marcus, Jonathan Wise. 2012

77 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f86ab1fe0224b95fcf441e608efb2a21dd9d970e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Local Gromov-Witten invariants are log invariants (2017, methodology): “Because we will eventually need to make use of the theory of logarithmic stable maps, we will state the version in that setting, although it is worth noting that thanks to the comparison theorems of [2], we could equally well use the better known formalism for degeneration in terms of relative invar”
Logarithmic Gromov–Witten theory with expansions (2019, unknown): “Important conceptual advances were made in [9].”