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Hodge integrals, partition matrices, and the \(\lambda_g\) conjecture. (English) Zbl 1058.14046

Let \(M_{g,n}\) be the moduli stack of complex nonsingular curves of genus \(g\) with \(n\) distinct marked points, and denote by \(\overline M_{g,n}\) its Deligne-Mumford compactification via stable curves. It is well-known that \(\overline M_{g,n}\) is a nonsingular orbifold of dimension \(3g- 3+n\), and that the Hodge bundle \(\mathbb{E}\) on \(M_{g,n}\) extends to the compactification \(\overline M_{g,n}\). Denote the \(\lambda_g\) the determinant bundle of \(\mathbb{E}\), which is called the Hodge line bundle on \(\overline M_{g,n}\).
The main result of the paper under review is a formula for integrating tautological classes on \(\overline M_{g,n}\) against the Hogde line bundle \(\lambda_g\). More precisely, the authors derive a closed formula for integrals of the cotangent line classes against \(\lambda_g\) in the form \[ \int_{\overline M_{g,n}} \psi^{\alpha_1}_1\cdots \psi^{\alpha_n}_n \lambda_g= {2g+ n-3\choose \alpha_1\cdots\alpha_n}\, \int_{\overline M_{g,1}} \psi^{2g- 2}\lambda_g, \] where the integrals on the righthand side had been calculated by the same authors in a previous paper [C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139, No. 1, 173–199 (2000; Zbl 0960.14031)]. The study of integration against the Hodge line bundle is motivated by predictions in Gromov-Witten theory, on the one hand, and by the enumerative geometry (tautological ring) of the moduli space \(M^c_g\subset \overline{M}_g\) of stable curves of compact type. The above integrals are computed via relations obtained from virtual localization in Gromov-Witten theory.

MSC:

14H10 Families, moduli of curves (algebraic)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)

Citations:

Zbl 0960.14031
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