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Relative maps and tautological classes. (English) Zbl 1084.14054
The system of tautological rings of the moduli spaces \(\overline M_{g,n}\) is defined exploiting the inductive structure these spaces possess, and the complicated network of natural morphisms which connects them. A possible definition of these rings requires that they form the minimal system of subalgebras of the rational Chow rings of \(\overline M_{g,n}\) which are invariant by the natural marking forgetting and gluing pushforwards. One shows that the usual \(\psi\), \(\kappa\) and \(\lambda\) classes lie in the tautological rings. Explicit constructions of non-tautological classes are obtained for instance by T. Graber and R. Pandharipande [Mich. Math. J. 51, No. 1, 93–109 (2003; Zbl 1079.14511)].
An application of the virtual localization theorem shows that pushforwards of tautological Gromov-Witten classes from the moduli space of absolute stable maps to homogeneous targets \(\overline M_{g,n}(X, \beta)\) also yield tautological classes on \(\overline M_{g,n}\). This result is extended in the paper under review. The authors study the Chow classes obtained by pushing forward relative Gromov-Witten classes on the moduli space of relative stable maps to \(\mathbb P^1\) via the natural morphism to \(\overline M_{g,n}\) which remembers only the domain curve and its markings. It is shown that all such classes are tautological. The proof exploits a system of relations, obtained via the localization theorem, which constrain the pushforwards of the relative Gromov-Witten classes. As a corollary, these pushforwards are recursively computed in terms of \(\psi\), \(\kappa\) and \(\lambda\) classes.
Moreover, the authors strengthen the Ionel vanishing theorem [E.-N. Jonel, Invent. Math. 148, No. 3, 627–658 (2002; Zbl 1056.14076)], proving that any polynomial in the \(\psi\) and \(\kappa\) classes on \(\overline M_{g,n}\) of total degree at least \(g\geq 1\) is a pushforward of tautological classes supported on the boundary. In turn, this result is applied to prove the socle and vanishing claims of the Gorenstein conjectures for the tautological rings of \(\overline M_{g,n}\) and the related moduli spaces of curves of compact type and or with rational tails [see also the approach of T. Graber and R. Vakil, Duke Math. J. 130, No. 1, 1–37 (2005; Zbl 1088.14007)]. Finally, the authors discuss applications to the reconstruction of arbitrary genus Gromov-Witten invariants from a restricted set of invariants containing less than \(g\) descendants and kappa classes.

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
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