##
**Logarithmic series and Hodge integrals in the tautological ring. With an appendix by Don Zagier.**
*(English)*
Zbl 1090.14005

Let \(M_g\) denote the coarse moduli space of non-singular complex algebraic curves of genus \(g\geq 2\). One of the most useful compactifications of \(M_g\) is given by the moduli space \(\overline M_g\) of Deligne-Mumford stable curves, which was constructed by P. Deligne and M. Mumford [Publ. Math., Inst. Hautes Étud. Sci. 36, 75–100 (1969; Zbl 0181.48803)]. The intersection theory in this so-called Deligne-Mumford compactification \(\overline M_g\) of \(M_g\), that is, the study of the Chow ring \(A^*(\overline M_g)\) of \(\overline M_g\), has been a subject of central interest in algebraic geometry ever since, beginning with D. Mumford’s pioneering work [in: Arithmetic and Geometry, II, Prog. Math. 36, 271–328 (1983; Zbl 0554.14008)]. In the past two decades, many remarkable properties of these Chow rings have been discovered, ranging from the existence of certain “tautological classes” in them up to striking numerical results related to (and predicted by) quantum field theories in physics. E. Witten’s famous conjecture in topological gravity theory [in: Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243–310 (1991; Zbl 0757.53049)], M. Kontsevich’s spectacular proof of it
[Commun. Math. Phys. 147, 1–23 (1992; Zbl 0756.35081)], and the avalanche of recent work in quantum cohomology theory, on Gromov-Witten invariants, and on the mirror symmetry phenomenon [D. A. Cox and S. Katz, Mirror symmetry and algebraic geometry (1999; Zbl 0951.14026)] not only utilized the enumerative geometry on compactified moduli spaces as a basic framework, but also propelled its rapid development in a crucial way.

In this vein, the main goal of the paper under review is to describe a new perspective on the intersection theory of \(\overline M_g\) that combines advances from both the classical theory of degenerations in algebraic geometry and topological quantum field theory in mathematical physics.

The central object of study is the so-called tautological ring \(R^*(\overline M_g)\) of \(\overline M_g\), i.e., the subring of \(A^*(\overline M_g)\) generated by the so far known tautological classes, together with its analogues \(R^*(\overline M_{g,n})\) for the moduli spaces \(\overline M_{g,n}\) of \(n\)-pointed stable curves. The authors consider the moduli filtration \(\overline M_g\supset M^c_g\supset M_g\supset\{[X_g]\}\), where \(X_g\) is a fixed smooth curve of genus \(g\) and \(M^c_g\) denotes the moduli space of stable curves with compact generalized Jacobian. There is an associated sequence of successive quotient maps of rational Chow rings, \[ A^*(\overline M_g)\to A^*(M^c_g)\to A^*(M_g)\to A^*(\{[X_g]\})\cong \mathbb{Q}, \] and the authors basically develop a uniform approach to the study of all these rings, and their related tautological rings with respect \(n\)-punctures, respectively, by successive descent methods. In fact, they show that the corresponding tautological rings in this sequence of quotient maps exhibit several parallel structures given by certain Hodge integrals of tautological classes, and that those can be effectively computed by a successive cutting-down method.

Some special cases of these computational results can be found in the authors’ earlier paper [Invent. Math. 139, No. 1, 173–199 (2000; Zbl 0960.14031)], whereas the subsequent paper under review largely completes the overall picture. The three main theorems of the present paper establish concrete integral formulae on \(\overline M_g\) which are of greatest significance. Namely, these formulae settle some earlier conjectures in this context reprove some related results in a unified way, strengthen evidence for Faber’s conjectures on the structure of the tautological rings \(R^*(\overline M_g)\) as formulated in his previous paper [C. Faber, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)], and reveal some surprising new combinatorial identies involving Stirling numbers of the second kind and Bernoulli numbers. In an appendix to the present paper, entitled “Polynomials arising from the tautological ring”, D. Zagier gives purely combinatorial proofs of these geometrically derived new identities.

Without any doubt, the novel approach to the study of the tautological ring of the moduli spaces \(\overline M_g\), especially the amazing reduction of their enumerative geometry to polynomial combinatorics, has the character of a very promising new research program in this direction, including both algebraic geometry and quantum field theory.

In this vein, the main goal of the paper under review is to describe a new perspective on the intersection theory of \(\overline M_g\) that combines advances from both the classical theory of degenerations in algebraic geometry and topological quantum field theory in mathematical physics.

The central object of study is the so-called tautological ring \(R^*(\overline M_g)\) of \(\overline M_g\), i.e., the subring of \(A^*(\overline M_g)\) generated by the so far known tautological classes, together with its analogues \(R^*(\overline M_{g,n})\) for the moduli spaces \(\overline M_{g,n}\) of \(n\)-pointed stable curves. The authors consider the moduli filtration \(\overline M_g\supset M^c_g\supset M_g\supset\{[X_g]\}\), where \(X_g\) is a fixed smooth curve of genus \(g\) and \(M^c_g\) denotes the moduli space of stable curves with compact generalized Jacobian. There is an associated sequence of successive quotient maps of rational Chow rings, \[ A^*(\overline M_g)\to A^*(M^c_g)\to A^*(M_g)\to A^*(\{[X_g]\})\cong \mathbb{Q}, \] and the authors basically develop a uniform approach to the study of all these rings, and their related tautological rings with respect \(n\)-punctures, respectively, by successive descent methods. In fact, they show that the corresponding tautological rings in this sequence of quotient maps exhibit several parallel structures given by certain Hodge integrals of tautological classes, and that those can be effectively computed by a successive cutting-down method.

Some special cases of these computational results can be found in the authors’ earlier paper [Invent. Math. 139, No. 1, 173–199 (2000; Zbl 0960.14031)], whereas the subsequent paper under review largely completes the overall picture. The three main theorems of the present paper establish concrete integral formulae on \(\overline M_g\) which are of greatest significance. Namely, these formulae settle some earlier conjectures in this context reprove some related results in a unified way, strengthen evidence for Faber’s conjectures on the structure of the tautological rings \(R^*(\overline M_g)\) as formulated in his previous paper [C. Faber, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)], and reveal some surprising new combinatorial identies involving Stirling numbers of the second kind and Bernoulli numbers. In an appendix to the present paper, entitled “Polynomials arising from the tautological ring”, D. Zagier gives purely combinatorial proofs of these geometrically derived new identities.

Without any doubt, the novel approach to the study of the tautological ring of the moduli spaces \(\overline M_g\), especially the amazing reduction of their enumerative geometry to polynomial combinatorics, has the character of a very promising new research program in this direction, including both algebraic geometry and quantum field theory.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14H10 | Families, moduli of curves (algebraic) |

14C15 | (Equivariant) Chow groups and rings; motives |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

11B73 | Bell and Stirling numbers |