## The second cohomology with symplectic coefficients of the moduli space of smooth projective curves (1998)

Venue: | Compositio Math |

Citations: | 5 - 0 self |

### BibTeX

@ARTICLE{Kabanov98thesecond,

author = {Alexandre I. Kabanov},

title = {The second cohomology with symplectic coefficients of the moduli space of smooth projective curves},

journal = {Compositio Math},

year = {1998},

pages = {163--186}

}

### OpenURL

### Abstract

Abstract. Each finite dimensional irreducible rational representation V of the symplectic group Sp 2g(Q) determines a generically defined local system V over the moduli space Mg of genus g smooth projective curves. We study H 2 (Mg; V) and the mixed Hodge structure on it. Specifically, we prove that if g ≥ 6, then the natural map IH 2 ( ˜ Mg; V) → H 2 (Mg; V) is an isomorphism where ˜ Mg is the Satake compactification of Mg. Using the work of Saito we conclude that the mixed Hodge structure on H 2 (Mg; V) is pure of weight 2 + r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H 2 (Mg; V) for 3 ≤ g < 6. Results of this article can be applied in the study of relations in the Torelli group Tg. The moduli space Mg of smooth projective curves of genus g is a quasiprojective variety over C. Its points correspond to isomorphism classes of smooth projective complex curves of genus g. It has only finite quotient

### Citations

459 |
and Mapping class groups
- Birman, Braids
- 1974
(Show Context)
Citation Context ...irst morphism M 1 g → Mg is called the “universal curve”[10, p. 218]. Its fiber over a point [C] ∈ Mg is C/AutC. On the level of the mapping class groups there is a corresponding short exact sequence =-=[4]-=- 1 → π1(S) → Γ 1 g → Γg → 1. The morphism Mg,1 → M 1 g “forgets” the tangent vector, but remembers its base point. When g ≥ 2 it is the frame bundle of the relative holomorphic tangent bundle to the u... |

374 |
The irreducibility of the space of curves of given genus
- Deligne, Mumford
(Show Context)
Citation Context ...oduli spaces of curves. We start with the Deligne–Mumford compactification of Ms g . A stable curve is a reduced connected curve which has only nodes as singularities, and a finite automorphism group =-=[8]-=-. The Deligne–Mumford compactification M s g of Ms g is the moduli space of stable projective curves. It is a normal projective variety in which Ms g is a Zariski open subset [8], [31, Th. 5.1]. are c... |

257 | Geometric Invariant Theory - Mumford - 1965 |

231 |
Projectivity of the moduli space of stable curves
- Knudsen
- 1983
(Show Context)
Citation Context ...pen subset of Mg whose complement has codimension g − 2. There are natural surjective morphisms between different moduli spaces which correspond to forgetting marked points and marked tangent vectors =-=[25]-=-. We will consider the morphisms M 1 g → Mg and Mg,1 → M 1 g. The first morphism M 1 g → Mg is called the “universal curve”[10, p. 218]. Its fiber over a point [C] ∈ Mg is C/AutC. On the level of the ... |

129 |
Intersection homology
- Goresky, MacPherson
- 1983
(Show Context)
Citation Context ...the main theorem of this article. The proof consists of a sequence of lemmas and propositions. We assume that the reader is familiar with intersection cohomology, and suggest the references [3], [5], =-=[12]-=-. Notation. For the rest of the paper we omit R • from the notation for the derived functors. For example, if f : X → Y is a continuous map between topological spaces, then f∗ = R • f∗. As we mentione... |

129 |
Stability of the homology of the mapping class groups of orientable surfaces
- Harer
- 1985
(Show Context)
Citation Context ...+· · ·+gag. This is the smallest integer r such that V (λ) ⊆ H1(S) ⊗r . (A good reference is [11].) Then V(λ) can be realized uniquely as a variation of Hodge structure of weight |λ|. Harer proved in =-=[17]-=- that the cohomology H k (Mg; Z) stabilizes when g ≥ 3k, and Ivanov later improved the range of stability [21]. [22]. He showed that H k (Mg; Z) stabilizes when g ≥ 2k + 2. In [22] Ivanov also proved ... |

129 | The red book of varieties and schemes - Mumford - 1988 |

90 |
The second homology group of the mapping class group of an oriented surface
- Harer
- 1983
(Show Context)
Citation Context ...D COHOMOLOGY OF THE MODULI SPACE 3 Looijenga’s result also provides very specific information about the MHS on H k (Mg; V(λ)). Combined with computations of H k (Mg; Q) in low dimensions due to Harer =-=[16]-=-, [19], [20], it implies that H k (Mg; V(λ)) is pure of weight k + |λ| when k ≤ 4 and g is in the stability range. In particular, H 2 (Mg; V(λ)) is pure of weight 2+|λ| when g ≥ 6+2|λ|. Recently, Pika... |

74 | Sheaf theory - Bredon - 1967 |

65 | Infinitesimal presentations of the Torelli group
- Hain
- 1997
(Show Context)
Citation Context ...eld theory. By a result of Johnson [23], Tg is finitely generated when g ≥ 3. Thus, tg is also finitely generated when g ≥ 3. It is not known for any g ≥ 3 whether Tg is finitely presented or not. In =-=[15]-=- Hain gives an explicit presentation of tg for g ≥ 3. More specifically, he proves that for each choice of x0 ∈ Mg there is a canonical MHS on tg which is compatible with the bracket. Thus, where Gr W... |

65 |
The structure of the Torelli group. I. A finite set of generators for
- Johnson
- 1983
(Show Context)
Citation Context ...g; V) is an isomorphism when k = 1 for all g ≥ 3, and when k = 2 for all g ≥ 6. The group H 1 (Mg; V) is easily computed when g ≥ 3 for all symplectic local systems V using Johnson’s fundamental work =-=[23]-=- (cf. [14]). Let X be an algebraic variety. From Saito’s work [37], [38] we know that H • (X; V) has natural mixed Hodge structure (MHS) if V → X is an admissible polarized variation of Hodge structur... |

38 | Torelli groups and geometry of moduli spaces of curves, in Current topics in complex algebraic geometry
- Hain
(Show Context)
Citation Context ...n isomorphism when k = 1 for all g ≥ 3, and when k = 2 for all g ≥ 6. The group H 1 (Mg; V) is easily computed when g ≥ 3 for all symplectic local systems V using Johnson’s fundamental work [23] (cf. =-=[14]-=-). Let X be an algebraic variety. From Saito’s work [37], [38] we know that H • (X; V) has natural mixed Hodge structure (MHS) if V → X is an admissible polarized variation of Hodge structure, and IH ... |

35 |
On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen
- Ivanov
- 1991
(Show Context)
Citation Context ... realized uniquely as a variation of Hodge structure of weight |λ|. Harer proved in [17] that the cohomology H k (Mg; Z) stabilizes when g ≥ 3k, and Ivanov later improved the range of stability [21]. =-=[22]-=-. He showed that H k (Mg; Z) stabilizes when g ≥ 2k + 2. In [22] Ivanov also proved that H k (Mg,1; V(λ)) is independent of g when g ≥ 2k+2+|λ|. (The space Mg,1 is the moduli space of curves with a ma... |

34 | Smooth Deligne-Mumford compactifications by means of Prym level structures - Looijenga - 1994 |

31 |
Théorie de Hodge I, Actes Congrès Intern
- Deligne
- 1971
(Show Context)
Citation Context ...kHQ/Wk−1HQ by GrW k H. We shall say that an integer m is a weight of a mixed Hodge structure H if GrW m H ̸= 0. We use abbreviations: MHS for mixed Hodge structure, and MHM for mixed Hodge module. In =-=[7]-=- Deligne proved that the rational cohomology of every quasi-projective variety possesses a natural MHS. In [38] Saito proved that the cohomology and intersection cohomology of an algebraic variety wit... |

29 |
Stabilization of the homology of Teichmüller modular groups, Original: Algebra i Analiz 1
- Ivanov
- 1989
(Show Context)
Citation Context ...can be realized uniquely as a variation of Hodge structure of weight |λ|. Harer proved in [17] that the cohomology H k (Mg; Z) stabilizes when g ≥ 3k, and Ivanov later improved the range of stability =-=[21]-=-. [22]. He showed that H k (Mg; Z) stabilizes when g ≥ 2k + 2. In [22] Ivanov also proved that H k (Mg,1; V(λ)) is independent of g when g ≥ 2k+2+|λ|. (The space Mg,1 is the moduli space of curves wit... |

27 |
Representation theory: a first course, GTM 129
- Fulton, Harris
- 1991
(Show Context)
Citation Context ...t λ. Fix fundamental weights λ1,λ2,... ,λg of Sp 2g. If λ = a1λ1 + a2λ2 + · · · + agλg, define |λ| = a1+2a2+· · ·+gag. This is the smallest integer r such that V (λ) ⊆ H1(S) ⊗r . (A good reference is =-=[11]-=-.) Then V(λ) can be realized uniquely as a variation of Hodge structure of weight |λ|. Harer proved in [17] that the cohomology H k (Mg; Z) stabilizes when g ≥ 3k, and Ivanov later improved the range ... |

23 | The projectivity of the moduli space of stable curves III : The line bundles on Mg,n, and a proof of the projectivity of M g,n in characteristic 0 - Knudsen - 1983 |

18 |
The third homology group of the moduli space of curves
- Harer
- 1991
(Show Context)
Citation Context ...MOLOGY OF THE MODULI SPACE 3 Looijenga’s result also provides very specific information about the MHS on H k (Mg; V(λ)). Combined with computations of H k (Mg; Q) in low dimensions due to Harer [16], =-=[19]-=-, [20], it implies that H k (Mg; V(λ)) is pure of weight k + |λ| when k ≤ 4 and g is in the stability range. In particular, H 2 (Mg; V(λ)) is pure of weight 2+|λ| when g ≥ 6+2|λ|. Recently, Pikaart pr... |

18 | The local Torelli problem for algebraic curves in Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry - Oort, Steenbrink - 1979 |

15 | Mixed Hodge structures on the intersection cohomology of links - Durfee, Saito - 1990 |

14 | Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map
- Looijenga
- 1996
(Show Context)
Citation Context ...lizes when g ≥ 2k + 2. In [22] Ivanov also proved that H k (Mg,1; V(λ)) is independent of g when g ≥ 2k+2+|λ|. (The space Mg,1 is the moduli space of curves with a marked non-zero tangent vector.) In =-=[28]-=- Looijenga calculated the stable cohomology groups of Mg with symplectic coefficients as a module over stable cohomology groups of Mg with trivial coefficients. In particular, this implies that H k (M... |

13 | Moduli theory and classification theory of algebraic varieties - Popp - 1977 |

10 | de Jong, Moduli of curves with non-abelian level structure - Pikaart, J - 1995 |

9 | Progress in the theory of complex algebraic curves - Eisenbud, Harris - 1989 |

8 |
Completions of mapping class groups and the cycle C − C − , Mapping class groups and moduli spaces of Riemann surfaces
- Hain
- 1993
(Show Context)
Citation Context ...was the motivation for this article. The Torelli group Tg is the kernel of the surjective homomorphism Γg → Sp 2g(Z). One can consider the Malcev Lie algebra tg associated to Tg. (For definitions see =-=[13]-=-). This Lie algebra is an analogue of the Lie algebra associated to the pure braid group on m strings, which is important in the study of Vassiliev invariants and conformal field theory. By a result o... |

7 |
A study of variation of mixed Hodge structure, Publ
- Kashiwara
- 1986
(Show Context)
Citation Context ...tion cohomology of an algebraic variety with coefficients in an admissible variation of MHS carry MHSs. The definition of an admissible variation of MHS is given for curves in [40], and in general in =-=[24]-=- (also see [37, 2.1]). There is a strong belief that when both MHSs of Deligne and Saito exist they are the same. Let V be an irreducible symplectic local system over Mg determined by highest weight λ... |

7 | Stability of projective varieties, L’Enseign - Mumford - 1977 |

6 |
The fourth homology group of the moduli space of curves, Duke University preprint
- Harer
(Show Context)
Citation Context ... OF THE MODULI SPACE 3 Looijenga’s result also provides very specific information about the MHS on H k (Mg; V(λ)). Combined with computations of H k (Mg; Q) in low dimensions due to Harer [16], [19], =-=[20]-=-, it implies that H k (Mg; V(λ)) is pure of weight k + |λ| when k ≤ 4 and g is in the stability range. In particular, H 2 (Mg; V(λ)) is pure of weight 2+|λ| when g ≥ 6+2|λ|. Recently, Pikaart proved i... |

5 |
An orbifold partition of M n g , in The moduli space of curves
- Pikaart
- 1995
(Show Context)
Citation Context ...it implies that H k (Mg; V(λ)) is pure of weight k + |λ| when k ≤ 4 and g is in the stability range. In particular, H 2 (Mg; V(λ)) is pure of weight 2+|λ| when g ≥ 6+2|λ|. Recently, Pikaart proved in =-=[34]-=- that the stable cohomology H k (Mg; Q) is pure of weight k. Combined with Looijenga’s computations, this shows that H k (Mg; V(λ)) is pure of weight k + |λ| whenever g ≥ 2k + 2 + 2|λ|. Unlike the sta... |

4 |
Die Singularitäten der Modulmannigfaltigkeit Mg(n) der stabilen Kurven vom Geschlecht g ≥ 2 mit n-Teilungspunktstruktur, Crelle 343
- Mostafa
- 1983
(Show Context)
Citation Context ...] [8, p. 106], [29, Bem. 1], [35, Rem. 2.3.7]. This is a projective variety according to [32, Th. 4, III.8], and there is a finite morphism Mg[l] → Mg determined by forgetting a level l structure. In =-=[29]-=- Mostafa proves that Mg[l] is not smooth, at least when g ≥ 3. However, in this article we are interested in particular strata of the boundary of Mg[l]. The irreducible component ∆1 of the boundary of... |

1 |
Geometry of Algebraic Curves II, preliminary manuscript
- Arbarello, Cornalba, et al.
- 1994
(Show Context)
Citation Context ... S • ) ∼ = H 0 (X ◦ ; H 3 j ! S • ). Therefore to prove the theorem it suffices to show that φ from the exact sequence above is the zero morphism. The distinguished triangle j ! S• ��������� �� j∗S • =-=[1]-=- ����������� j∗i∗VsTHE SECOND COHOMOLOGY OF THE MODULI SPACE 13 implies that H 3 j ! S • ∼ = H 2 j ∗ i∗V. Then the morphism φ composed with this isomorphism can be factored as H 2 (Mg; V) → H 2 (X ◦ ;... |

1 |
On the theory of theta-functions, the moduli of abelian varieties and the moduli of curves
- Baily
- 1962
(Show Context)
Citation Context ...hen s = 0. The boundary Mg − Mg is the union of irreducible divisors The singularities of M s g [g/2] ⋃ i=0 where each divisor ∆i has the following property. When i = 0 there is birational morphism M =-=[2]-=- g−1 → ∆0; when 1 ≤ i < g − i there is birational morphism M 1 i × M 1 g−i → ∆i; and when i = g − i there is a birational ∆i, morphism from the Z/2Z-quotient of M 1 i × M1 i to ∆i. Definition 2.1. (cf... |

1 |
The cohomology of the moduli space of curves, CIME notes, Theory of moduli
- Harer
- 1988
(Show Context)
Citation Context ...2 i , h∗(w1 i ) = w2 i , and [f2 ◦ h] = [f1] where the homotopy is required to preserve the marked points and tangent vectors. The space of equivalence classes T s g,r is called the Teichmüller space =-=[18]-=-, [19, p. 26]. It is known that T s g,r is a contractible complex manifold of dimension 3g − 3 + s + 2r when 2g − 2 + s + 2r > 0. The mapping class group Γs g,r is defined to be Diff+ (S)/Diff + 0 (S)... |