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Orbifold thechniques in degeneration formulas. (English) Zbl 1375.14182
A complex algebraic stack \(X\) has nodal (codimension one) singularities, if it is locally isomorphic in the f.p.p.f. topology to \(\{ xy=0\} \times \mathbb{A}^n\). In this case, the singular locus \(D\) is smooth, and \(X\) is said first-order smoothable, if the line bundle \(\mathcal{E}xt^1(\omega_X,\mathcal{O}_X)\) on \(D\) is trivial. Let \(W_0 = X_1 \sqcup_D X_2\) be a nodal, first order smoothable proper Deligne-Mumford stack, which is the union of two smooth components \(X_1\) and \(X_2\) meeting transversally along \(D\). In this paper the authors give a new definition of Gromov-Witten invariants for stacks of the form as \(W_0 = X_1 \sqcup_D X_2\), and of relative Gromov-Witten invariants for smooth Deligne-Mumford stacks. In the case of relative Gromov-Witten invariants the target space is a pair \((X,D)\), where \(X\) is a proper Deligne-Mumford smooth stack with projective coarse moduli space and \(D\) is a smooth divisor. A degeneration formula is also proved, which expresses the Gromov-Witten invariants of \(W_0 = X_1 \sqcup_D X_2\) in terms of the relative invariants of the pairs \((X_1, D)\) and \((X_2,D)\).
The approach “follows closely that of J. Li” [J. Differ. Geom. 57, No. 3, 509–578 (2001; Zbl 1076.14540); J. Differ. Geom. 60, No. 2, 199–293 (2002; Zbl 1063.14069)]. One of the differences is to replace predeformable maps (used in J. Li’s approach) with transversal maps, where auxiliary orbifold structures along the nodes of both source curves and target spaces are introduced. This has the advantage of simplifying both, the definition of the obstruction theory on the moduli space of maps (that are used to define the Gromov-Witten invariants), and the proofs of the properties of the Gromov-Witten invariants.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D22 Fine and coarse moduli spaces
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