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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.

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.

Reviewer: Margarida Melo (Coimbra)