Siegel metric and curvature of the moduli space of curves

We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the curve X, in terms of the holomorphic sectional curvature of A_g and the second Gaussian map. Finally we extend the Kaehler form of the Siegel metric as a closed current on the Deligne-Mumford compatification of M_g and we determine its cohomology class as a multiple of the first Chern class of the Hodge bundle.


Introduction
During the last thirty years, some natural metrics on the moduli space of genus g curves M g have been extensively studied. Many of these metrics come from metrics on the Teichmüller space of which the moduli space of curves is the quotient by the mapping class group. One of these is the Weil-Petersson metric ω W P . It was introduced by Weil and is known to be Kähler, to have non-positive curvature operator and negative Ricci curvature, and to be geodesically convex. S. A. Wolpert showed that both its holomorphic sectional curvature and Ricci curvature have negative (genus dependent) upper bounds (but no lower bounds do exist). Moreover it is not complete ( [23]). The other canonical metrics, namely the Teichmüller metric (or the Kobayashi metric), the Caratheodory metric, the Kähler-Einstein metric, the induced Bergman metric, the McMullen metric, are complete. Recently Liu, Sun and Yau ( [10], [11]) showed their equivalence on M g and the equivalence with the Ricci metric and the perturbed Ricci metric introduced by them. The Kähler form of the WP metric has been extended by Masur ([12]) as a closed current on the Deligne-Mumford compactification M g of M g . Wolpert ([22]) determined its cohomology class in terms of the first Chern class of the Hodge bundle λ and the classes of the boundary.
Let A g be the moduli space of principally polarized abelian varieties of dimension g and let j : M g → A g be the period map sending a curve to its jacobian. It is an interesting and classical problem to understand the geometry of the image of M g in A g .
On A g there is a natural metric coming from the unique Sp(2g, R) invariant metric on the Siegel space H g ≃ Sp(2g, R)/U (g) of which A g is the quotient by Sp(2g, Z). The purpose of this paper is to study the metric on M g induced by this metric through the period map, which we call the Siegel metric. In [4] an explicit expression for the second fundamental form of the immersion j is given and it is proven that the second fundamental form lifts the second Gaussian map µ 2 : I 2 (K X ) → H 0 (X, 4K X ), as stated in an unpublished paper of Green-Griffiths (cf. [7]).
Here we use it to compute the curvature of the Siegel metric. In particular we give an explicit formula for the holomorphic sectional curvature of M g along the a Schiffer variation ξ P , for P a point on the curve X, in terms of the holomorphic sectional curvature of A g and the second Gaussian map µ 2 : I 2 (K X ) → H 0 (X, 4K X ).
Finally we give some properties of the holomorphic sectional curvature of M g , using results of [3]. In particular along a Schiffer variation ξ P the holomorphic sectional curvature H(ξ P ) of M g is strictly smaller than the holomorphic sectional curvature of A g unless P is either a Weierstrass point of a hyperelliptic curve or a ramification point of the g 1 3 on a trigonal curve. In these last cases H(ξ P ) = −1.
Furthermore we study the asymptotic behaviour of the Kähler form of the Siegel metric on M g showing that it extends as a closed current to M g , hence it defines a cohomology class in H 2 (M g , C) which we compute to be πλ.
In all what we have stated, we have considered M g as an orbifold. In the paper we make all computations using the covering of M g given by the moduli space of curves with level n ≥ 3 structures M (n) g and the moduli space A (n) g of principally polarized abelian varieties with level n structures. In fact M (n) g and A (n) g are smooth and by Local Torelli theorem proven in [18] we know that the period map j (n) : M is a two to one immersion outside the hyperelliptic locus and it is an injective immersion if we restrict to the hyperelliptic locus.
The paper is organized as follows: in Section 2 we define the Siegel metric and compute it on the tangent directions given by the Schiffer variations (Lemma 2.2). In Section 3 we give the expression of the curvature of the Siegel metric on A (n) g restricted to the Schiffer variations. Then, we show that the second fundamental form of the immersion of M is non zero at any non hyperelliptic curve and we exhibit a formula for the curvature of M (n) g (Thm.3.7). Finally we write the holomorphic sectional curvature of M (n) g along a Schiffer variation ξ P , using the second Gaussian map. In Section 4 we give some applications of results of [3] to the holomorphic sectional curvature of M (n) g . In Section 5 we study in particular the hyperelliptic locus HE g and we show that the second fundamental form of HE g in A (n) g is non zero at any point. In Section 6 we extend the Kähler form of the Siegel metric as a closed current on M g and we determine its cohomology class (6.1).
Acknowledgments. The authors thank Gilberto Bini and Pietro Pirola for several fruitful suggestions and discussions on the subject.

The Siegel metric
We introduce some notations. Let M g , resp. M (n) g be the moduli space of smooth genus g curves, resp. of smooth genus g curves with a fixed n-level structure. Denote by T g the Teichmuller space and by Γ g the mapping class group acting on T g with quotient M g . Let K(n) := ker(Γ g → Sp(2g, Z/nZ)) and recall that M (n) g is the quotient of T g by the action of K(n). Moreover, let K := ker(Γ g → Sp(2g, Z)) be the Torelli group and define the Torelli space T or g as the quotient of T g by the action of the Torelli group.
Let A g , resp. A (n) g be the moduli space of g-dimensional principally polarized Abelian varieties, resp. of g-dimensional principally polarized Abelian varieties with a n-level structure. Denote by H g := {Z ∈ M (g, C) | Z = t Z, ImZ > 0} the Siegel space so that A g is the quotient of H g by the action of Sp(2g, Z) and A (n) g is the quotient of H g by ker(Sp(2g, Z) → Sp(2g, Z/nZ)). Denote by j T or , j and j (n) the period maps which send a curve to its jacobian. We have the following diagram The Torelli theorem states that j is injective, while j tor and j (n) are two to one on the image and ramified over the hyperelliptic locus. In fact multiplication by −1 in H 1 (X, Z) = H 1 (JX, Z), where JX is the Jacobian of the curve X, is induced by an automorphism of abelian varieties but not by an automorphism of non hyperelliptic curves. Local Torelli Theorem says that outside the hyperelliptic locus and restricted to the hyperelliptic locus the period map is an immersion (cf. [18]). From now on we shall work on M (n) g and A (n) g , with n ≥ 3, since they are smooth, everything works in the same way on M g and A g but in the orbifold context.
We will now define the Siegel metric.
The Siegel space H g is a homogeneous space and it can be seen as the quotient Sp(2g, R)/U (g). We call the unique (up to scalar) invariant metric the Siegel metric.
OnLet F be the homogeneous vector bundle on H g associated to the standard g-dimensional representation of U (g, C). The Hodge metric h on F is the only (up to multiplication by scalars) invariant metric on the homogeneous bundle F . Moreover through the identification Hg ≃ S 2 F the Hodge metric on F defines the Siegel metric on H g . The Siegel metric on H g defines a metric on A (n) g and A g and, through the period map, an induced metric on M (n) g and M g outside the hyperelliptic locus, and on the hyperelliptic locus itself. We call all these metrics the Siegel metrics.
These metrics can be described in terms of polarized variation of Hodge structures. More precisely, on A (n) g we have the universal family φ : A → A (n) g , and the polarized variation of Hodge structures associated to the local system R 1 φ * Z. The associated Hodge bundle F 1 can be identified with is the sheaf of relative holomorphic one forms.
The polarization induces a Hermitian metric on R 1 φ * C and on F 1 , which we call the Hodge metric. In fact the pullback of F 1 on H g is the bundle F and the pullback of the metric is the Hodge metric on F . Hence the Siegel metric is induced by the Hodge metric through the identification we have the universal family ψ : C → M (n) g with induced relative dualizing sheaf K C|M (n) g . The local system R 1 ψ * Z coincides with the pullback of R 1 φ * Z through the period map: at a point [X] ∈ M (n) g , we have H 1 (X, Z) ∼ = H 1 (JX, Z). The non-degenerate Hermitian product on H 1 (X, C), defined by the polarization is the following: for any The Hodge bundle can be identified with ψ * (K C|M (n) g ), and the corresponding Hodge metric yields a metric on We finally observe that for the sake of simplicity we defined the Siegel metric on the fine moduli space M (n) g , but we also have a Siegel metric on M g viewed as an orbifold.
2.1. An explicit formula. We shall now give an explicit formula for the Siegel metric on M in terms of the basis of H 1 (T X ) given by Schiffer variations ξ P , for a set of 3g − 3 general points on X.
Now, we briefly recall the definition of ξ P . Consider the exact sequence Notice that H 0 (T X (P ) |P ) ∼ = C. If we denote the coboundary map by δ : H 0 (T X (P ) |P ) → H 1 (T X ), we have dim(Im(δ)) = 1. Any no zero element ξ P in Im(δ) is called a Schiffer variation. Let us a choose a local coordinate z in a neighborhood of P . Under the Dolbeault isomorphism H 1 (T X ) ∼ = H 0,1 (T X ), it is represented by the form where b P is a bump function around P . Notice that if we choose b P to be one in a neighborhood of P for this choice of local coordinate z, ξ P depends only on the choice of z. In what follows, we need to express ξ P in terms of a basis of ..,g is a basis of H 0,1 (X). This set can be viewed as the dual basis of {ω i }, where the non-degenerate pairing is given by Observe that we have Lemma 2.1. For a choice of a local coordinate z at P , we have Let C be a small circle around P such that b P ≡ 1 on C. Then the lemma follows by Stokes and Cauchy Theorems: The scalar product of the two Schiffer variations ξ P , ξ P ′ has the following form: Proof. Recall that on S 2 H 0,1 the scalar product is:

Curvature
We would like now to give a formula for the curvature of the Siegel metric on M (n) g . We will do the computation on the tangent vectors given by the ξ P 's. These depend on the choice of the local coordinates, but by linearity one can immediately derive the formulas at the tangent vectors , which are intrinsic. Recall that outside the hyperelliptic locus we have the sequence of tangent bundles: where I 2 := N * and m is the multiplication map. The Hermitian connection of the variation of Hodge structures R 1 ψ * C, the Gauss-Manin connection, defines a Hermitian connection on F 1 = ψ * K C|M (n) g , thus on F 1 * , as well as S 2 F 1 and S 2 F 1 * ≃ j (n) * T A (n) g , which we denote by ∇.
The exact sequence (5) defines a second fundamental form, Similarly the exact sequence (6) defines the second fundamental form ) and the second fundamental form σ. Namely, we have Let us now determine R (ξ P ), ξ P ′ (ξ R , ξ T ) in terms of the curvature form of the Hodge bundle.
Let z be a local coordinate in a neighborhood of S, and consider a local expression of ρ(Q)(ξ P ) ∈ H 0 (2K X ), ρ(Q)(ξ P ) = Ψ Q P (z)dz 2 . Lemma 3.5. Let Q ∈ I 2 (K X ), then ξ S (ρ Q (ξ P )) = 2πiΨ Q P (S). Proof. Recall that ξ S is represented by a form where z is a local coordinate in a neighborhood of S and b S is a bump function around S which is equal to one in a neighborhood of S. Let C be a small circle around S such that b S ≡ 1 on C. We have We want now to compute Ψ Q P (S). If P = S the form η P has the following local expression in a neighborhood of S: If P = S, the local expression of η P in a neighborhood of P is η P = (− 1 (z − z(P )) 2 + g(z))dz, and we have (cf. also [4], Thm.3.1) since Q ∈ I 2 (K X ), so i,j a ij f i (P )f j (P ) = 0, and i,j a ij f i (P )f ′ j (P ) = 0. Thus we have (15) Ψ Q P (P ) = where µ 2 (Q) is the second Gaussian map of X in Q. For the definition of the second Gaussian map see Section 4.
Proposition 3.6. Let ξ P be a Schiffer variation, and let {Q i } be an orthonormal basis of I 2 (K X ), denote by Ψ i P := Ψ Q i P . Then the following holds:

Proof. Fix an orthonormal basis
On the other hand, a basis of H 0 (2K X ) is given by the set {ξ * S }, where S runs in a set of 3g − 3 general points of X. This implies that Therefore, the following holds: Using lemma (3.5) we get   computed at the tangent vector ξ P is given by Proof. The proof immediately follows from (3.7), (15) and (2.2). By corollary (3.8) we see that the holomorphic sectional curvature of A given by the Schiffer variations ξ P is equal to −1, for all P ∈ X.
We shall now give another proof of this. We recall that the image of the sectional curvature of H g is the segment [−1, − 1 g ] and that the tangent directions V such that H(V ) = −1 correspond to the symmetric matrices of rank 1.
Let us now see as usual an element ξ ∈ H 1 (T X ) as a symmetric homomorphism H 0 (K X ) → H 0 (K X ) * through the exact sequence (12). Then the above observation shows that H(ξ) = −1 if and only if ξ has rank one. We therefore recall the characterisation of the elements ξ ∈ H 1 (T X ) such that ξ has rank 1. Moreover observe that the Schiffer variations are the points of the bicanonical curve φ 2K (X) ⊂ PH 1 (X, T X ). Then the statement follows as a corollary by the following result of Griffiths and by the theorem of Enriques-Babbage and Petri.
Define X ⊂ PH 1 (X, T X ), Theorem 3.9. ( [8]) Assume that g ≥ 3 and X is not hyperelliptic. Consider the image of the bicanonical map φ 2K (X) ⊂ PH 1 (X, T X ). Then φ 2K (X) ⊂ X with equality holding if and only if the canonical curve φ K (X) is cut out by quadrics.
Corollary 3.10. ( [8]) Assume that g ≥ 3 and X is not hyperelliptic. Then φ 2K (X) ⊂ X with equality holding if and only if the canonical curve φ K (X) is not trigonal, and it is not isomorphic to a plane quintic.

Second Gaussian map and holomorphic sectional curvature
We first recall the definition of the Gaussian maps (cf. [21]). Let X be a smooth projective curve, S := X × X, ∆ ⊂ S be the diagonal. Let L be a line bundle on X and L S := p * 1 (L) ⊗ p * 2 (L), where p i : S → X are the natural projections. Consider the restriction map µ n,L : H 0 (S, L S (−n∆)) → H 0 (∆, L S (−n∆) |∆ ).
Notice that since O(∆) |∆ ∼ = T X , we have In the case L = K X , I 2 (K X ) ⊂ H 0 (S, K S (−2∆)), so we can define the second Gaussian map as the restrictionμ 2,K|I 2 (K X ) .
Assume [X] ∈ M (n) g , with g ≥ 4, X non hyperelliptic. Then corollary (3.8) allows us to define a function F : X → R, given by the holomorphic sectional curvature evaluated along the tangent vectors given by the Schiffer variations: where {Q i } is an orthonormal basis of I 2 (K X ). Proposition 4.3. If g = 4, the set of points P ∈ X such that F (P ) = −1 is finite, which implies that F is non constant.
If g ≥ 5, X not hyperelliptic, nor trigonal, then F (P ) < −1 for all P ∈ X. If X is a trigonal curve of genus ≥ 4, F (P ) = H(ξ P ) = −1 for every P ∈ X which is a ramification point of the g 1 3 . Proof. Assume X has genus 4, then the dimension of I 2 is one and I 2 can be generated by a quadric Q of rank 4 which has norm 1. So ∀P ∈ X, F (P ) = −1− 1 64π 2 (α P,P ) 4 |µ 2 (Q)(P )| 2 , hence there is a finite number of points P such that µ 2 (Q)(P ) = 0, so in these points we have F (P ) = −1, while F (P ) < −1 elsewhere .
As regards the second statement, we observe that F (P ) = −1 if and only if µ 2 (Q i )(P ) = 0 for all i, where {Q i } is an orthonormal basis of I 2 . But then we must have µ 2 (Q)(P ) = 0 for all Q ∈ I 2 . So the proof follows by Theorem (4.2).
The last statement follows from (4.1).
Remark 4.4. The previous statements imply that for any curve X ∈ M (n) g , not hyperelliptic, nor trigonal, for every point P ∈ X the holomorphic sectional curvature of M (n) g , at X along the tangent directions given by ξ P is strictly smaller than the holomorphic sectional curvature of A

The hyperelliptic locus
We will now study the hyperelliptic locus HE g ⊂ M (n) g . Recall that by local Torelli, the restriction of the period map to HE g is an injective immersion (cf. [18]). Therefore we have the exact sequence and we denote by the associated second fundamental form and by ρ HE the second fundamental form of the dual exact sequence. At the point [X] ∈ HE g the dual exact sequence is 0 → I 2 → S 2 (H 0 (K X )) → H 0 (2K X ) + → 0, where H 0 (2K X ) + is the invariant part of H 0 (2K X ) under the hyperelliptic involution and I 2 is the vector space of the quadrics containing the rational normal curve, so that ρ HE : I 2 → Hom(T HEg, [X] , H 0 (2K X ) + ).
We recall that the set of Schiffer variations at the Weierstrass points P i generates T HEg,[X] .
So ρ HE (Q)(ξ P ) = 0 implies i,j a ij f i (P )ω j = 0, hence ∀j, i,j a ij f i (P ) = 0. Then Q ∈ ker(ρ HE ) if and only if i,j a ij f i (P ) = 0 for every Weierstrass point P ∈ X. Since there are 2g + 2 Weierstrass points, this implies that i,j a ij ω i = 0, hence Q = 0. Since σ HE (ξ P )(Q) = ρ HE (Q)(ξ P ) and ρ HE is injective, there must exist a Weierstrass point P ∈ X such that σ HE (ξ P ) = 0.
We also observe that with the same proof as in Lemma (3.5) and formula (15) one shows that ξ P (ρ HE (Q)(ξ P )) = µ 2 (Q)(P ) at a Weierstrass point P ∈ X.
Let us denote by H HE the holomorphic sectional curvature of T HEg , if [X] ∈ HE g and P ∈ X is a Weiestrass point, we have the same expression for H HE (ξ P ) as in (3.8), namely We recall now a result on the second Gaussian map proven in [3]. Proof. The proof immediately follows from (18) and from (5.2).

The class of the Siegel metric
Let M g (M g ). In [15] it is shown that the Hodge bundle extends to M g (M (n) g ) and its g-th exterior power is ample on M g (M (n) g ). We denote by λ both the first Chern class of the extension of the Hodge bundle on M g and on M (n) g . We will prove that the Kähler form of the Siegel metric on M g extends as a closed current to M g , hence it defines a cohomology class in H 2 (M g , C) which is a multiple of λ. Proof. On H g the Hodge metric is the only (up to multiplication by scalars) invariant metric on the homogeneous bundle F . Therefore we have an invariant metric on the line bundle Λ g F and thus its curvature is an invariant (1, 1) form β on H g .
On the other hand, the Siegel metric is the invariant metric obtained by the metric on S 2 F * induced by the Hodge metric and we denote byω its Kähler form.
Since both β andω are invariant (1, 1) forms and we are on the irreducible symmetric domain H g , there exists a constant c such thatω = cβ. This relation still holds on the corresponding forms on A (n) g which we denote in the same way.
g is constructed and it has the property that it is nonsingular and that D ∞ := A (n) g is a divisor with normal crossings.
In [14] it is shown that the Hodge bundle F 1 on A (n) g extends as a bundle on A (n) g , such that the Hodge metric has only logarithmic singularities at D ∞ .
Moreover in [14] (see also [5]), it is also proven that the extension of the second symmetric power is isomorphic to the sheaf of differential forms with logarithmic poles at D ∞ :  4)) that the extension of the Hodge metric has "good" singularities and that this implies that its first Chern class yields a closed current on A Therefore, since on A (n) g our Kähler formω = cβ, then alsoω can be extended as a closed (1,1) current on A (n) g , which we still callω and its cohomology class [ω] ∈ H 2 (A (n) g , C) is given by [ω] = cλ. In ( [16] (18.9), see also [17]) it is shown that the period map j (n) : M . In order to compute the constant c, we use the cycles introduced by Wolpert in ( [22]). In our case, since [ω] is a multiple of λ, it is sufficient to compute the value of [ω] on the 1-dimensional family given by a varying 1-pointed elliptic curve attached to a fixed g − 1 curve with 1 marked point E l of [22](2.2). More precisely, let us denote by H := {z ∈ C | Im(z) > 0}, by Γ := SL(2, Z), and by Since the g − 1 curve in E l is constant we identify E l with the curve H/Γ l = A Set E z := C/(Z⊕zZ), where z ∈ H/Γ, let ξ z be a holomorphic coordinate on E z , so that H 1,0 (E z ) = dξ z . Then a cotangent direction to the curve E l can be identified with dξ z ⊙dξ z and we have: dξ z ⊙dξ z , dξ z ⊙dξ z = 2 dξ z , dξ z 2 , dξ z , dξ z = i Ez dξ z ∧ dξ z = 2Im(z).