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The degeneration formula for logarithmic expanded degenerations. (English) Zbl 1288.14039
A degeneration formula for Gromov-Witten invariants was first proven by Jun Li in algebraic settings. He developed the stack of expanded degenerations and the notion of predeformable maps to prove the degeneration formula. One difficulty in Jun Li’s approach is that predeformability is not an open condition in general. Jun Li studied the deformation theory of predeformable maps using log structures for the construction of the virtual fundamental class without using the technology of logarithmic cotangent complex which was developed later on by Olsson. Recently, Abramovich and Fantechi introduced the notion of transversal maps using the theory of root stacks. Transversality is an open condition and hence the construction of the perfect obstruction theory and virtual fundamental class is more natural in this approach. Abramovich and Fantechi then proved a degeneration formula using the transversal maps.
The paper under review proves a degeneration formula for Gromov-Witten invariants inspired by Kim’s theory of log stable maps. Interestingly, the degeneration formula of this paper is very similar to that of Abramovich and Fantechi and the author of the paper under review expects that the theory of transversal maps and the theory of log stable maps are equivalent.

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14D06 Fibrations, degenerations in algebraic geometry
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