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Topological recursive relations in \(H^{2g}(\mathcal M_{g,n})\). (English) Zbl 1056.14076

Summary: Let \(\overline{\mathcal M}_{g,n}\) be the Deligne-Mumford compactification of the moduli space \({\mathcal M}_{g,n}\) of genus \(g\) smooth curves with \(n\) (distinct) marked points. (In this paper we work in the category of analytic orbifolds.) Let \(L_i\to \overline{\mathcal M}_{g,n}\) be the relative cotangent bundle at the marked point \(x_i\); the fiber of \(L_i\) over \((\Sigma,x_1,\dots, x_n)\) is the cotangent space to \(\Sigma\) at \(x_i\). The first Chern class of this bundle is denoted \(\psi_i= c_1(L_i)\) and is sometimes called a (gravitational) descendant. If \(\pi: \overline{\mathcal M}_{g,n+1}\to \overline{\mathcal M}_{g,n}\) is the map that forgets the last marked point then \(\kappa_a= \pi_*(\psi^{a+1}_{n+1})\) is called a tautological class (or Mumford-Morita-Miller class); since \(\kappa_a\in H^{2a}(\overline{\mathcal M}_{g,n})\) we define its degree to be \(a\), while the degree of each \(\psi_i\) equals 1.
E. Looijenga [Invent. Math. 121, 411–419 (1995; Zbl 0851.14017)] proved that in the Chow group \({\mathcal A}^*({\mathcal C}^n_g)\) a product of descendant classes of degree at least \(g+n-1\) vanishes, where \({\mathcal C}^n_g\) is the moduli space of smooth genus \(g\) curves with \(n\) (not necessarily distinct) points. In particular, in \({\mathcal M}_{g,0}\) any degree \(g-1\) monomial in tautological classes vanishes. However, with the above definition of tautological classes, this not true anymore in \({\mathcal M}_{g,n}\), for \(n\geq 1\) (for example in \({\mathcal M}_{2,1}\kappa_1= \psi_1\neq 0\)). In this paper, we obtain the following generalization of Looijenga’s result:
Theorem. When \(g\leq 2\), any product of degree at least \(g\) (or at least \(g-1\) when \(n= 0\)) of descendent or tautological classes vanishes when restricted to \(H^*({\mathcal M}_{g,n}, \mathbb{Q})\).
This generalizes a result of Looijenga and proves a version of Getzler’s conjecture. The method we use is the study of the relative Gromov-Witten invariants of \(\mathbb{P}^1\) relative to two points combined with the degeneration formulas previously given by E.-N. Ionel and T. H. Parker [Math. Res. Lett. 5, 563–576 (1998; Zbl 0943.83046)].

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
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