The Gromov-Witten class and a perturbation theory in algebraic geometry (Q2731070)

The author introduces a new notion of quasi manifold structure on an algebraic stack, proves some important properties of this structure and constructs the fundamental class. These notions are introduced to prove that the moduli stack, \({\mathcal M}_{gn\beta}(V)\), classifying the stable maps from \(n\)-pointed prestable curves of genus \(g\) to an algebraic proper and smooth variety \(V\) (so that the image of the fundamental class of the curve is \(\beta\), a Néron-Severi class of dimension 1) has quasi manifold structure. This result and the properties of the quasi manifold structures allow the author to prove that the fundamental class of the moduli \({\mathcal M}_{gn\beta}(V)\) satisfy all the axioms of \textit{M. Kontsevich} and \textit{Yu. Manin} [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)] for the Gromov-Witten class. The author proves the deformation invariance and the Borel localization formula for the Gromov-Witten class.