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Gromov-Witten theory of Deligne-Mumford stacks. (English) Zbl 1193.14070
The paper under review is devoted to establishing rigorous foundations to the Gromov-Witten theory of smooth complex Deligne-Mumford stacks with a projective coarse moduli space.
The contents of the paper were announced earlier by the same authors [Contemp. Math. 310, 1–24 (2002; Zbl 1067.14055)] with the aim of giving an algebro-geometric counterpart to the symplectic version of Gromov-Witten theory of orbifolds previously developed by W. Chen and Y. Ruan [Contemp. Math. 310, 25–85 (2002; Zbl 1091.53058)]. The stack-theoretical framework presented here is expected to be useful also within the symplectic setting.
The classical Gromov-Witten theory of a smooth projective variety $$X$$ within the algebro-geometric setting relies on the Kontsevich moduli space of stable maps from $$n$$-pointed curves of given genus $$g$$ to $$X$$ with image class $$\beta\in H_2(X,\mathbb Z)$$. To work out the Gromov-Witten theory of an orbifold $$\mathcal X$$ this space is then replaced by a moduli space of maps from orbifold curves to $$\mathcal X$$. Such a space had already been constructed by the first and the third author [J. Am. Math. Soc. 15, No. 1, 27–75 (2002; Zbl 0991.14007)] and later generalized by M. Olsson [Duke Math. J. 134, No. 1, 139–164 (2006; Zbl 1114.14002)] and [J. Reine Angew. Math. 603, 55–112 (2007; Zbl 1137.14004)] and goes with the name of moduli stack of twisted stable maps, $$\mathcal K_{g,n}(\mathcal X,\beta)$$.
There are several technical issues that need to be worked out in order to develop a satisfactory Gromov-Witten theory for $$\mathcal X$$. The existence of the virtual fundamental class for $$\mathcal K_{g,n}(\mathcal X,\beta)$$ follows without major difficulties from the classical case while evaluation maps are shown to land not in $$\mathcal X$$ but in a new gadget called the rigidified cyclotomic inertia stack of $$\mathcal X$$, denoted by $$\overline{\mathcal I}_\mu(\mathcal X)$$, parametrizing gerbes banded by some $$\mu_r$$ together with a representable morphism to $$\mathcal X$$. Gromov-Witten classes are then naturally defined in $$A^*(\overline{\mathcal I}_\mu(\mathcal X))_{\mathbb Q}$$ and differ from the classical case also by a correction term describing the index of the gerbe in $$\overline{\mathcal I}_\mu(\mathcal X)$$.
Some basic properties of classical Gromov-Witten invariants are then shown to hold also in the orbifold case, being of particular relevance the proof of the WDVV equation, which is Theorem 6.2.1 in the paper.

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14A20 Generalizations (algebraic spaces, stacks) 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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