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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.

##### MSC:
 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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##### References:
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