Gromov-Witten theory of Deligne-Mumford stacks. (English) Zbl 1193.14070

Abramovich, Dan; Graber, Tom; Vistoli, Angelo

Am. J. Math. 130, No. 5, 1337-1398 (2008).

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.

Reviewer: Margarida Melo (Coimbra)

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

Keywords:

Deligne-Mumford stack; Gromov-Witten invariants; twisted curves; cyclotomic inertia stack

Citations:

Zbl 1067.14055; Zbl 1091.53058; Zbl 0991.14007; Zbl 1114.14002; Zbl 1137.14004