Orbifold techniques in degeneration formulas

By D. Abramovich, Barbara Fantechi. 2011

77 citations recorded by the source corpus.

We give an approach for relative and degenerate Gromov--Witten invariants, inspired by that of Jun Li but replacing predeformable maps by transversal maps to a twisted target. The main advantage is a significant simplification in the definition of the obstruction theory. We reprove in our language the degeneration formula, extending it to the orbifold case.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3875c176ea00231f57856f51d8b67a1c79e7e5b4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Stable logarithmic maps to Deligne-Faltings pairs I (2010, methodology): “ible covers was introduced even earlier by Harris and Mumford [HM82]. Recently, the idea of expanded degeneration was systematically studied using orbifold techniques by D. Abramovich and B. Fantechi [AF], and an elegant proof of degeneration formula was given there. On the other hand, the idea of a”
Logarithmic Gromov–Witten theory with expansions (2019, methodology): “Many applications of relative Gromov–Witten theory come via the degeneration formula [7, 19, 46].”
Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations (2012, methodology): “At the cost of reintroducing Li’s expansions, they were able to develop a theory in which predeformability could be replaced by a more innocuous transversality condition [6].”
Relative and orbifold Gromov-Witten invariants (2010, background): “eneralizes it to rational curves in arbitrary targets. GROMOV–WITTEN INVARIANTS (October 24, 2018) 5 1.5. Theintermediary. Our comparison goes by way of a third theory, due to Abramovich and Fantechi [AF11], that combines the advantages of both while avoiding some of the disadvantages. This “relativ”
Expanded Degenerations and Pairs (2011, methodology): “The morphism Ttw spl →T tw × Z>0 T tw is a gerbe banded by µ r , and thus has degree 1/r (2) The morphism Ttw spl →T tw 0 has degree 1/r. This is used to prove the orbifold degeneration formula of [AF11]. The second morphism is notagerbe but rathermore like the embedding of a scheme in an (r−1)-s”
Local Gromov-Witten invariants are log invariants (2017, methodology): “The proof we will give here follows the main idea of that argument which is to apply the degeneration formula for Gromov-Witten invariants [17,16,13,1,9,14] to a twist of the degeneration to the normal cone of the preimage of D in Tot(O(−D)).”