Annales de l’institut Fourier (AIF)

Sabbah, Claude

Non-commutative Hodge structures

Faber, Carel

AN
N
A
L
E
S
D
E
L’INSTI
T
U
T
F
O
U
R
IE
R
ANNALES
DE
L’INSTITUT FOURIER
Carel FABER
A remark on a conjecture of Hain and Looijenga
Tome 61, no 7 (2011), p. 2745-2750.
<http://aif.cedram.org/item?id=AIF_2011__61_7_2745_0>
© Association des Annales de l’institut Fourier, 2011, tous droits
réservés.
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions
générales d’utilisation (http://aif.cedram.org/legal/). Toute re-
production en tout ou partie de cet article sous quelque forme que
ce soit pour tout usage autre que l’utilisation à fin strictement per-
sonnelle du copiste est constitutive d’une infraction pénale. Toute
copie ou impression de ce fichier doit contenir la présente mention
de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
61, 7 (2011) 2745-2750
A REMARK ON A CONJECTURE OF
HAIN AND LOOIJENGA
by Carel FABER
Abstract. — We show that the natural generalization of a conjecture of Hain
and Looijenga to the case of pointed curves holds for all g and n if and only if the
tautological rings of the moduli spaces of curves with rational tails and of stable
curves are Gorenstein.
Résumé. — Nous montrons que la généralisation naturelle d’une conjecture
de Hain et Looijenga au cas des courbes épointées tient pour tout g et n si et
seulement si les anneaux tautologiques des espaces des modules des courbes à
queues rationnelles et des courbes stables sont des anneaux de Gorenstein.
Let Mg,n (resp. Mg,n) be the moduli space of smooth (resp. stable) n-
pointed curves of genus g and let M ctg,n be the moduli space of pointed
curves of compact type, the complement of the boundary divisor ∆irr of
irreducible singular curves and their degenerations. LetMrtg,n be the moduli
space of pointed curves with rational tails; for g > 2, it is the inverse
image of Mg under the natural morphism Mg,n →Mg, while Mrt1,n = M ct1,n
andMrt0,n = M0,n by definition. Here, (g, n) is a pair of nonnegative integers
such that 2g− 2 +n > 0. There is a natural partial ordering of these pairs:
(h,m) 6 (g, n) if and only if h 6 g and 2h − 2 + m 6 2g − 2 + n, or, in
other words, if and only if there exists a stable n-pointed curve of genus g
whose dual graph contains a vertex of genus h with valency m.
We recall the definition of the tautological algebras R•(Mg,n) from [4]:
the system {R•(Mg,n)}(g,n) is defined as the set of smallest Q-subalgebras
of the rational Chow rings A•(Mg,n) that is closed under push-forward
via all maps forgetting markings and all standard gluing maps. The well-
known ψ-, κ-, and λ-classes are tautological. The system is also closed
under pull-back via the forgetting and gluing maps. The successive quo-
tients R•(M ctg,n), R•(Mrtg,n), and R•(Mg,n) are defined as the restrictions
Keywords: Moduli spaces of curves, tautological ring, Gorenstein ring.
Math. classification: 14H10, 13H10.
2746 Carel FABER
to the respective open subsets. (Observe that it is in general not known
whether the corresponding tautological localization sequences are exact in
the middle.)
The following results are known:
(a) R•(Mrtg,n) vanishes in degrees > g−2+n−δ0g and is 1-dimensional
in degree g − 2 + n− δ0g.
(b) R•(M ctg,n) vanishes in degrees > 2g − 3 + n and is 1-dimensional in
degree 2g − 3 + n.
(c) R•(Mg,n) (vanishes in degrees > 3g − 3 + n and) is 1-dimensional
in degree 3g − 3 + n.
Statement (a) was proved by Looijenga [10] and Faber [1], [3]. Statements
(b) and (c) were proved by Graber and Vakil [6], [7] and Faber and Pand-
haripande [4].
Recall the following three conjectures:
(A) R•(Mrtg,n) is Gorenstein with socle in degree g − 2 + n− δ0g.
(B) R•(M ctg,n) is Gorenstein with socle in degree 2g − 3 + n.
(C) R•(Mg,n) is Gorenstein with socle in degree 3g − 3 + n.
(For a graded Q-algebra R•, to be Gorenstein with socle in degreem means
that it vanishes in degrees > m, that Rm is isomorphic to Q, and that the
pairings Ri ×Rm−i → Rm are perfect.)
In the case g = 0, the three conjectures coincide and have been proved
by Keel [9]. Conjecture (A) in the case n = 0 is due to the author [1] and
is true for g 6 23. Hain and Looijenga [8] raised (C) as a question and (A),
(B), and (C) were formulated in [11] (see also [3], [2]).
Hain and Looijenga also introduce a compactly supported version of
the tautological algebra: they define R•c(Mg,n) as the set of elements
in R•(Mg,n) that restrict trivially to the Deligne-Mumford boundary (i.e.,
the pull-back via any standard map from a product of moduli spacesMgi,ni
onto the closure of a boundary stratum vanishes). It is a graded ideal
in R•(Mg,n) and a module over R•(Mg,n). They then formulate the follow-
ing conjecture in the case n = 0:
Conjecture 1 (Hain and Looijenga [8]). — The intersection pairings
Rk(Mg)×R3g−3−kc (Mg)→ R3g−3c (Mg) ∼= Q, k = 0, 1, 2, . . .
are perfect (Poincaré duality) and R•c(Mg) is a free R•(Mg)-module of rank
one.
Observe that λ1 ∈ R1c(M1,1) and λgλg−1 ∈ R2g−1c (Mg) for g > 1.
(The author’s proof of the nonvanishing of Rg−2+n(Mrtg,n) for g > 0 uses
ANNALES DE L’INSTITUT FOURIER
A REMARK ON A CONJECTURE OF HAIN AND LOOIJENGA 2747
this fact.) So this class is supposed to be a generator of the R•(Mg)-
module R•c(Mg) (the unique generator of degree 2g − 1 up to a scalar).
However, the pull-backs of these classes to Mg,n don’t lie in R•c(Mg,n)
for n > 2, since they don’t vanish on the boundary strata corresponding
to curves with rational tails. Let us therefore define R•c(Mrtg,n) as the set of
elements in R•(Mg,n) that restrict trivially to Mg,n \Mrtg,n. Consider the
following conjectures:
(D) The intersection pairings
Rk(Mrtg,n)×R3g−3+n−kc (Mrtg,n)→ R3g−3+nc (Mrtg,n) ∼= Q
are perfect for k > 0.
(E) In addition to (D), R•c(Mrtg,n) is a free R•(Mrtg,n)-module of rank
one.
Conjecture (E) appears to be the natural generalization of Conjecture 1 to
the case n > 0. For reasons that will become clear in a moment, we also
include the weaker statement (D). Observe that (E) implies that λgλg−1 is
a generator of R•c(Mrtg,n) for g > 0 (the unique one of degree 2g − 1 up to
a scalar), by (a) above.
Theorem 1. — Conjectures (A) and (C) are true for all (g, n) if and
only if Conjecture (E) is true for all (g, n). More precisely,
A(g,n) and C(g,n) ⇒ E(g,n) ⇒ A(g,n) and D(g,n)
and
{D(g′,n′)}(g′,n′)6(g,n) ⇒ {C(g′,n′)}(g′,n′)6(g,n) .
Proof. Suppose first that (C) is not true for all (g, n) and let a minimal
counterexample be given by 0 6= α ∈ R•(Mg,n), i.e., R•(Mg′,n′) is Goren-
stein for all (g′, n′) < (g, n) and deg(αβ) = 0 for all β ∈ R•(Mg,n). (We
write deg for the degree homomorphism on R0(Mg,n) and its extension by
zero to all of R•(Mg,n)). It follows that g > 0.
Let π denote the standard map Mg−1,n+2 → Mg,n onto the boundary
divisor ∆irr. Let γ ∈ R•(Mg−1,n+2) be arbitrary. Then
deg((π∗α)γ) = deg(π∗((π∗α)γ)) = deg(απ∗γ) = 0,
since π∗γ is tautological. Since R•(Mg−1,n+2) is Gorenstein, it follows
that π∗α = 0.
Next, let π denote one of the standard maps Mg1,n1 ×Mg2,n2 → Mg,n
onto a boundary component parametrizing reducible singular curves (g1 +
g2 = g and n1 + n2 = n+ 2). We have the push-forward map
π∗ : R•(Mg1,n1)⊗Q R•(Mg2,n2)→ R•(Mg,n)
TOME 61 (2011), FASCICULE 7
2748 Carel FABER
and the pull-back map in the other direction (cf. [5]). The tensor product
is Gorenstein, with perfect pairing given by
deg((β1 ⊗ β2)(γ1 ⊗ γ2)) = deg(β1γ1) deg(β2γ2).
Let γ1 resp. γ2 be arbitrary elements of R•(Mg1,n1) resp. R•(Mg2,n2). Then
deg((π∗α)(γ1 ⊗ γ2)) = deg(π∗((π∗α)(γ1 ⊗ γ2))) = deg(απ∗(γ1 ⊗ γ2)) = 0,
since π∗(γ1 ⊗ γ2) is tautological. Again, it follows that π∗α = 0.
Therefore, 0 6= α ∈ R•c(Mg,n) and a fortiori 0 6= α ∈ R•c(Mrtg,n). But it
pairs to zero with all β and this contradicts D(g,n). The implication in the
second display follows as an immediate consequence.
The next step is to prove the implication E(g,n) ⇒ A(g,n). As mentioned
above, if g > 0 and E(g,n) holds, then λgλg−1 generates R•c(Mrtg,n) freely.
Suppose that A(g,n) fails: let 0 6= α ∈ R•(Mrtg,n) be such that it pairs to zero
with all β ∈ R•(Mrtg,n), i.e., deg(αβλgλg−1) = 0 for all β (note that g > 0).
From D(g,n), it follows that αλgλg−1 = 0, but this contradicts E(g,n). This
proves the second implication in the first display.
To prove the first implication, we first show that A(g,n) and C(g,n) im-
ply D(g,n). Assume that D(g,n) fails; the perfect pairing may fail on ei-
ther side. Suppose first that 0 6= α ∈ R•c(Mrtg,n) pairs to zero with all
of R•(Mrtg,n). We know that π∗α = 0, for every standard map π associ-
ated to a stratum in Mg,n \Mrtg,n. This means that the product of α and
a Chow class pushed forward via such a map is zero (hence the pairing is
well-defined). Since α pairs to zero with all of R•(Mrtg,n), it gives a coun-
terexample to C(g,n). If instead 0 6= α ∈ R•(Mrtg,n) pairs to zero with all
of R•c(Mrtg,n), then it pairs to zero with all classes of the form βλgλg−1,
for β ∈ R•(Mrtg,n) (note that g > 0). In this case, α gives a counterexample
to A(g,n).
We conclude by showing that A(g,n) and C(g,n) imply E(g,n). We already
have D(g,n). If E(g,n) doesn’t hold, then g > 0 and certainly λgλg−1 fails to
be a basis for R•c(Mrtg,n), i.e., multiplication by λgλg−1 fails to be surjective
or injective. From A(g,n) and D(g,n), it follows that the surjectivity and in-
jectivity of this map are equivalent (recall from [5], Cor. 1, that R•(Mg,n) is
finite-dimensional). But if 0 6= α ∈ R•(Mrtg,n) and αλgλg−1 = 0,
then A(g,n) fails. 
There is an analogous result in the compact type case. Begin by defin-
ing R•c(M ctg,n) as the set of elements in R•(Mg,n) that pull back to zero via
the standard map Mg−1,n+2 → Mg,n onto ∆irr. Conjectures (D) and (E)
ANNALES DE L’INSTITUT FOURIER
A REMARK ON A CONJECTURE OF HAIN AND LOOIJENGA 2749
have obvious analogues (Dct) and (Ect). We have that
B(g,n) and C(g,n) ⇒ Ect(g,n) ⇒ B(g,n) and Dct(g,n)
and
{Dct(g′,n′)}(g′,n′)6(g,n) ⇒ {C(g′,n′)}(g′,n′)6(g,n) .
The proof proceeds entirely analogously; the class λg now plays the role
of λgλg−1 (it is no longer necessary to treat the case g = 0 separately).
Acknowledgements. The author thanks Eduard Looijenga, Rahul
Pandharipande, and Michael Shapiro for useful discussions. The author
is supported by the Göran Gustafsson Foundation for Research in Natural
Sciences and Medicine and grant 622-2003-1123 from the Swedish Research
Council.
Note added in the second version (November 2010). Tavakol [13] has
proved that the tautological ring of M ct1,n = Mrt1,n is Gorenstein with socle
in degree n− 1 (Conjectures (A) and (B) for g = 1). From Theorem 1, the
tautological rings R•(M1,n) are Gorenstein if and only if E(1,n) holds for
all n > 1, in other words, if and only if R•c(Mrt1,n) is generated by λ1 as an
R•(Mrt1,n)-module.
Note added in the third version (April 2012). Tavakol [14] has now also
proved Conjecture (A) for g = 2: the tautological ring ofMrt2,n is Gorenstein
with socle in degree n.
Note added in the fourth version (June 2012). Petersen [12] has proved
that the tautological ring of M1,n is Gorenstein with socle in degree n
(Conjecture (C) for g = 1).
BIBLIOGRAPHY
[1] C. Faber, “A conjectural description of the tautological ring of the moduli space
of curves”, in Moduli of curves and abelian varieties, Aspects Math., E33, Vieweg,
Braunschweig, 1999, p. 109-129.
[2] ———, “Hodge integrals, tautological classes and Gromov-Witten theory”, Pro-
ceedings of the Workshop “Algebraic Geometry and Integrable Systems related to
String Theory” (Kyoto, 2000), no. 1232, Sūrikaisekikenkyūsho Kōkyūroku, 2001,
p. 78-87.
[3] C. Faber & R. Pandharipande, “Logarithmic series and Hodge integrals in the
tautological ring”, with an appendix by Don Zagier. Dedicated to William Fulton
on the occasion of his 60th birthday. Michigan Math. J. 48 (2000), p. 215-252.
[4] ———, “Relative maps and tautological classes”, J. Eur. Math. Soc. (JEMS) 7
(2005), no. 1, p. 13-49.
[5] T. Graber & R. Pandharipande, “Constructions of nontautological classes on
moduli spaces of curves”, Michigan Math. J. 51 (2003), no. 1, p. 93-109.
TOME 61 (2011), FASCICULE 7
2750 Carel FABER
[6] T. Graber & R. Vakil, “On the tautological ring of Mg,n”, Turkish J. Math. 25
(2001), no. 1, p. 237-243.
[7] ———, “Relative virtual localization and vanishing of tautological classes on mod-
uli spaces of curves”, Duke Math. J. 130 (2005), no. 1, p. 1-37.
[8] R. Hain & E. Looijenga, “Mapping class groups and moduli spaces of curves”, in
Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Part 2,
Amer. Math. Soc., Providence, RI, 1997, p. 97-142.
[9] S. Keel, “Intersection theory of moduli space of stable n-pointed curves of genus
zero”, Trans. Amer. Math. Soc. 330 (1992), no. 2, p. 545-574.
[10] E. Looijenga, “On the tautological ring of Mg”, Invent. Math. 121 (1995), no. 2,
p. 411-419.
[11] R. Pandharipande, “Three questions in Gromov-Witten theory”, Proceedings of
the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed.
Press, Beijing, 2002, p. 503-512.
[12] D. Petersen, “The structure of the tautological ring in genus one”, Preprint,
arXiv:1205.1586.
[13] M. Tavakol, “The tautological ring of Mct1,n”, Ann. Inst. Fourier (Grenoble) 61.7
(2011).
[14] ———, “The tautological ring of the moduli space Mrt2,n”, Preprint,
arXiv:1101.5242.
Manuscrit reçu le 6 décembre 2010,
accepté le 25 avril 2012.
Carel FABER
Department of Mathematics,
KTH Royal Institute of Technology,
Lindstedtsvägen 25,
10044 Stockholm, Sweden.
faber@math.kth.se
ANNALES DE L’INSTITUT FOURIER


A remark on a conjecture of Hain and Looijenga

Barbot, Thierry

Béguin, François

Zeghib, Abdelghani

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Batista da Silva, Márcio Henrique

Simons Type Equation in $\mathbb{S}^{2}\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ and Applications

Granier, Anne

A Galois $D$-groupoid for $q$-difference equations

Schweitzer, Paul A.

Riemannian manifolds not quasi-isometric to leaves in codimension one foliations

Ludwig, Ursula

A proof of the stratified Morse inequalities for singular complex algebraic curves using  the Witten deformation

Not a member yet

Non-commutative Hodge structures

A remark on a conjecture of Hain and Looijenga

Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes Application to the Minkowski problem in the Minkowski space

Simons Type Equation in $\mathbb{S}^{2}\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ and Applications

A Galois $D$ -groupoid for $q$ -difference equations

Riemannian manifolds not quasi-isometric to leaves in codimension one foliations

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Compatible complex structures on twistor space

Positive sheaves of differentials coming from coarse moduli spaces

Global existence for coupled Klein-Gordon equations with different speeds

3,503

3,732