Good Geometry on the Curve Moduli

42(2008), 699–724

Good Geometry on the Curve Moduli

Dedicated to Heisuke Hironaka

By

Kefeng Liu^∗, XiaofengSun^∗∗and Shing-TungYau^∗∗∗

Contents

§1. Introduction

§2. Background and Notation

§3. Goodness of the Weil-Petersson Metric

§4. Dual Nakano Negativity of the Weil-Petersson Metric

§5. L²-Cohomology and Rigidity

§6. Goodness of the Ricci and Perturbed Ricci Metrics References

§1. Introduction

This paper is written to dedicate to Professor Hironaka for his outstanding contributions to mathematics, especially in algebraic geometry. His leadership in Asia has inspired many generations of asian mathematicians. We wish many young mathematicians will continue to follow his footsteps in this grand subject.

In this paper we describe some of our recent results in the asymptotic analysis of various K¨ahler metrics and their curvatures on the moduli spaces of Riemann surfaces. These works will enable us to use differential geometric techniques to study various algebraic geometric and topological problems about

Communicated by K. Saito. Received January 30, 2007.

2000 Mathematics Subject Classification(s): 14D20, 14H20, 32G13, 53C55.

The authors are supported by NSF grants.

∗Center of Mathematical Sciences, Zhejiang University, Hangzhou, China; Department of Mathematics, University of California at Los Angeles.

∗∗Department of Mathematics, Lehigh University.

∗∗∗Center of Mathematical Sciences, Zhejiang University, Hangzhou, China; Department of Mathematics, Harvard University.

c 2008 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

the moduli spaces. We believe that the new results here may give a new way to study the intersection theory on various moduli spaces.

In his work [11], Mumford defined the goodness condition to study the cur- rents of Chern forms defined by a singular Hermitian metric on a holomorphic bundle over a quasi-projective manifold. The goodness condition is a growth condition of the Hermitian metric near the compactification divisor of the base manifold. The major property of a good metric is that the currents of its Chern forms define the Chern classes of this bundle. For details, please see Section 2.

Except for the symmetric spaces discussed by Mumford in [11], several nat- ural bundles over moduli spaces of Riemann surfaces give beautiful and useful examples. In [20], Wolpert showed that the metric induced by the hyperbolic metric on the twisted relative tangent bundle over the total space of moduli space of hyperbolic Riemann surfaces is good. Later it was shown by Trapani [16] that the metric induced by the Weil-Petersson metric on the determinant line bundle of the logarithmic cotangent bundle of the Deligne-Mumford mod- uli space is good. In both cases, the bundles involved are line bundles in which cases it is easier to estimate the connection and curvature. Other than these, very few examples of natural good metrics are known.

The goodness of the Weil-Petersson metric has been a long standing open problem. In this paper, we describe our solution of this problem. In fact we will present proofs of the goodness of the metrics induced by the Weil-Petersson metric, as well as the Ricci and perturbed Ricci metrics on the logarithmic cotangent bundle over the compactified moduli space of Riemann surfaces (the DM moduli spaces). These works depend on our very accurate estimates of the asymptotic of the curvature and connection forms of these metrics in [6] and [7]

together with the estimates of derivatives of the hyperbolic metric on Riemann surfaces. The computations and proofs are quite involved and very subtle. We will also present our proof of the dual Nakano negativity of the Weil-Petersson metric. In [13] Schumacher proved the strong negativity of the Weil-Petersson metric in the sense of Siu. Dual Nakano negativity is stronger than the strong negativity in the sense of Siu and several interesting consequences will follow.

For example the goodness, combining with the dual Nakano negativity of the Weil-Petersson metric, gives rise to interesting geometric consequences such as infinitesimal rigidity of the complex structure on the moduli spaces. We feel that our results open a new way to study the intersection theory on the moduli spaces by using differential geometric techniques, and also to apply index theory to the study of the geometry and topology of the moduli spaces and Teichm¨uller spaces.

Now we briefly describe the organization of this note. In Section 2 we recall some known works on the geometry of moduli spaces which include the degeneration of Riemann surfaces and hyperbolic metrics, the Ricci, perturbed Ricci and K¨ahler-Einstein metrics as well as their curvature properties, the asymptotic of Weil-Petersson metric, the Ricci and perturbed Ricci metrics as established in [6] and [7]. We also review Mumford’s definition of good metrics.

In Section 3 we describe the main ideas of proving the goodness of the Weil- Petersson metric and in Section 4 we describe the proof of the dual Nakano negativity of the Weil-Petersson metric. In Section 5 we apply these results to derive the vanishing theorem about certain cohomology groups and the in- finitesimal rigidity of the moduli spaces. Our definition of theL² cohomology generalizes the usual one with trivial bundle coefficients. Here we have used both the Weil-Petersson metric and the Ricci metric and their goodness.

Finally in Section 6 we present the goodness of the Ricci and the perturbed Ricci metrics. We remark that our previous results about the asymptotic be- havior of the K¨ahler-Einstein metrics on the moduli spaces already imply an orbifold Chern number inequality for the logarithmic cotangent bundle which should give new information about positive divisors on the DM moduli spaces.

In this paper, we will only give the main ideas and sketch the proofs of the results presented. For details and precise estimates, we refer the reader to [8]

and [9].

We would like to thank professors H.-D. Cao, R. Schoen and E. Viehweg for their help and encouragement.

§2. Background and Notation

In this section we review the necessary backgrounds and setup our nota- tions. Most of the results can be found in [6], [7], [20] and [11].

Let Mg,k be the moduli space of Riemann surfaces of genus g with k punctures such that 2g−2 +k > 0. We know there is a unique hyperbolic metric on such a Riemann surface. To simplify the computation, throughout this paper, we will assume k = 0 and g ≥ 2 and work on Mg. Most of the results can be trivially generalized toMg,k.

By the Riemann-Roch theorem, we know that the complex dimension of the moduli space isn = dim_CMg = 3g−3. Given a Riemann surface X of genus g ≥ 2, we denote byλ the unique hyperbolic (K¨ahler-Einstein) metric onX. Letzbe local holomorphic coordinate on X. We normalizeλ:

∂z∂zlogλ=λ.

(2.1)

Let (s1,· · ·, sn) be local holomorphic coordinates on Mg near a point p and let Xs be the corresponding Riemann surfaces. Let ρ : TsMg → H¹(X_s, T X_s)∼= H^0,1(X_s, T X_s) be the Kodaira-Spencer map. Then the har- monic representative ofρ

∂

∂si

is given by ρ

∂

∂si

=∂_z

−λ⁻¹∂_si∂_zlogλ ∂

∂z⊗dz=B_i. (2.2)

If we letai =−λ⁻¹∂s_i∂zlogλand let Ai =∂zai, then the harmonic lift vi of

∂s∂_i is given by

vi = ∂

∂s_i +ai ∂

∂z. (2.3)

The well-known Weil-Petersson metricω_{W P} =

√−1

2 h_ijdsi∧dsj onMg is defined to be

h_ij(s) =

Xs

A_iA_j dv (2.4)

wheredv= ^√₂⁻¹λdz∧dzis the volume form onX_s. It was proved by Ahlfors that the Ricci curvature of the Weil-Petersson metric is negative. The upper bound of the Ricci curvature of the Weil-Petersson metric was conjectured by Royden and was proved by Wolpert [18].

In our work [6] we defined the Ricci metricω_τ: ωτ =−Ric(ω_{W P}) (2.5)

and the perturbed Ricci metricω_eτ:

ω_eτ =ωτ+Cω_{W P} (2.6)

whereC is a positive constant. These new K¨ahler metrics have good curvature and asymptotic properties and play important roles in our study.

Now we describe the curvature formulae of these metrics. Please see [6]

and [7] for details. We denote byf_ij =AiAj where each Ai is the harmonic Beltrami differential corresponding to the local holomorphic vector field _∂s^∂

i. It is clear thatf_ijis a function onX. We let=−∂_z∂_zbe the Laplace operator, letT = (+ 1)⁻¹ be the Green operator and let e_ij =T(f_ij). The functions e_ij andf_ij are building blocks of these curvature formulae.

The curvature formula of the Weil-Petersson metric was given by R_ijkl=−

X_s

(e_ijf_kl+e_ilf_kj)dv.

(2.7)

This formula was first established by Wolpert [18] and was generalized by Siu [15] and Schumacher [14] to higher dimensions. A short proof can be found in [6].

To describe the curvature formulae of the Ricci and perturbed Ricci met- rics, we need to introduce several operators. We first define the operator ξ_k :C^∞(X_s)→C^∞(X_s) by

ξ_k(f) =∂^∗(i(B_k)∂f) =−λ⁻¹∂_z(A_k∂_zf) =−A_kK₁K₀(f) (2.8)

whereK₀, K₁are the Maass operators [18], [6].

It was proved in[6] thatξk is the commutator of the Laplace operator and the Lie derivative in the directionvk:

(+ 1)vk−vk(+ 1) =vk−vk=ξk. (2.9)

We also need the commutator of the operatorvk andvl. In [6] we defined the operatorQ_kl:C^∞(X_s)→C^∞(X_s) by

Q_kl(f) = [v_l, ξ_k](f) =P(e_kl)P(f)−2f_klf+λ⁻¹∂_zf_kl∂_zf (2.10)

where P : C^∞(X_s) → Γ(Λ^1,0(T^0,1X_s)) is the operator defined by P(f) =

∂_z(λ⁻¹∂_zf).

The terms appeared in the curvature formulae of the Ricci and perturbed Ricci metrics are formally symmetric with respect to indices. For convenience, we recall the symmetrization operator defined in [6].

Definition 2.1. LetU be any quantity which depends on indicesi, k, α, j, l, β. The symmetrization operatorσ₁ is defined by taking the summation of all orders of the triple (i, k, α). Similarly,σ₂is the symmetrization operator of j andβ andσ₁ is the symmetrization operator ofj,l andβ.

Let R_ijkl and P_ijkl be the curvature tensors of the Ricci and perturbed Ricci metrics respectively. In [6] we established the following curvature formulae of these metrics:

R_ijkl =−h^αβ

σ₁σ₂

X_s

(T(ξ_k(e_ij))ξ_l(e_αβ) +T(ξ_k(e_ij))ξ_β(e_αl)

dv

−h^αβ

σ₁

Xs

Q_kl(e_ij)e_αβdv

+τ^pqh^αβh^γδ

σ1

Xs

ξk(eiq)e_αβ dv σ1

Xs

ξ_l(e_pj)e_γδ)dv +τ_pjh^pqR_iqkl

(2.11)

and

P_ijkl=−h^αβ

σ₁σ₂

X_s

T(ξ_k(e_ij))ξ_l(e_αβ) +T(ξ_k(e_ij))ξ_β(e_αl)

dv

−h^αβ

σ₁

Xs

Q_kl(e_ij)e_αβdv

+τ^pqh^αβh^γδ

σ1

Xs

ξk(eiq)e_αβ dv σ1

Xs

ξ_l(e_pj)e_γδ)dv +τ_pjh^pqR_iqkl+CR_ijkl.

(2.12)

It is easy to derive information of the sign of the curvature of the Weil- Petersson metric from its curvature formula (2.7). However, the curvature formulae of the Ricci and perturbed Ricci metrics are too complicated to use directly. Thus we need to look at the asymptotic behavior of these metrics.

We now recall geometric construction of the DM moduli space which is due to Earle-Marden and the degeneration of hyperbolic metrics. Please see [6] and [18] for details.

Let Mg be the Deligne-Mumford compactification of Mg and let D = Mg\ Mg. It was shown in [2] thatDis a divisor with only normal crossings.

A pointy ∈D corresponds to a stable nodal surface Xy. A pointp∈Xy is a node if there is a neighborhood ofpwhich is isometric to the germ{(u, v)|uv= 0, |u|,|v|<1} ⊂C². Let p₁,· · ·, p_m∈X_y be the nodes. X_y is stable if each connected component ofX_y\ {p₁,· · ·, p_m}has negative Euler characteristic.

Fix a point y ∈ D, we assume the corresponding Riemann surface Xy

hasm nodes. Now for any point s ∈ Mg lying in a neighborhood ofy, the corresponding Riemann surfaceX_scan be decomposed into the thin part which is a disjoint union ofmcollars and the thick part where the injectivity radius with respect to the K¨ahler-Einstein metric is uniformly bounded from below.

There are two kinds of local holomorphic coordinate on a collar or near a node. We first recall the rs-coordinate defined by Wolpert in [20]. In the node case, given a nodal surfaceX with a nodep∈X, we leta, bbe two punctures which are glued together to formp.

Definition 2.2. A local coordinate chart (U, u) near a is called rs- coordinate if u(a) = 0 where u maps U to the punctured disc 0 < |u| < c with c >0, and the restriction toU of the K¨ahler-Einstein metric on X can be written as _2|u|2(log¹ _|u|)2|du|². The rs-coordinate (V, v) near b is defined in a similar way.

In the collar case, given a closed surface X, we assume there is a closed geodesicγ⊂Xsuch that its lengthl=l(γ)< c_∗wherec_∗is the collar constant.

Definition 2.3. A local coordinate chart (U, z) is called rs-coordinate at γ if γ ⊂U where z maps U to the annulusc⁻¹|t|¹² <|z| < c|t|¹², and the K¨ahler-Einstein metric onX can be written as ¹₂(_log^π_|t||¹_z|csc^π_log^log_|^|_t^z_|^|)²|dz|².

The existence of collar was due to Keen [5]. We formulate this theorem in the following:

Lemma 2.1. Let X be a closed surface and let γ be a closed geodesic on X such that the length l of γ satisfies l < c_∗. Then there is a collar Ω on X with holomorphic coordinatez defined onΩsuch that

(1) zmapsΩ to the annulus{¹_ce^−2πl2 <|z|< c} forc >0;

(2) the K¨ahler-Einstein metric onX restricted to Ωis given by 1

2u²r⁻²csc²τ

|dz|² (2.13)

whereu= _2π^l ,r=|z|andτ =ulogr;

(3) the geodesicγ is given by the equation|z|=e^−πl2;

(4) the constant c has a lower bound such that the area of Ω is bounded from below by a universal constant.

We call such a collarΩa genuine collar.

Now we describe the pinching coordinate chart of Mg near the divisorD [20]. LetX0 be a nodal surface corresponding to a codimension mboundary point and letp₁,· · · , p_m be the nodes of X₀. Then X₀ =X₀\ {p₁,· · · , p_m} is a union of punctured Riemann surfaces. Fix rs-coordinate charts (U_i, η_i) and (V_i, ζ_i) at p_i for i = 1,· · ·, m such that all the U_i and V_i are mutually disjoint. Now pick an open set U0 ⊂ X0 such that the intersection of each connected component ofX0 andU0 is a nonempty relatively compact set and the intersectionU₀∩(U_i∪V_i) is empty for alli. Now pick Beltrami differentials ν_m+1,· · · , ν_n which are supported inU₀ and span the tangent space at X₀ of the deformation space of X₀. Let ∆ⁿ_ε^−m ⊂ C^n−m be the polydisc of radius ε. For t = (t_m+1,· · ·, t_n) ∈ ∆ⁿ_ε^−m, let ν(t) = n

i=m+1t_iν_i. We assume

|t|= (n

i=m+1|ti|²)¹² small enough such that |ν(t)|<1. The nodal surface

X0,tis obtained by solving the Beltrami equation∂w=ν(t)∂w. Sinceν(t) is supported inU0, (Ui, ηi) and (Vi, ζi) are still holomorphic coordinates onX0,t. By the theory of Ahlfors and Bers [1] and Wolpert [20] we can assume that there are constantsδ, c >0 such that when|t|< δ,η_i andζ_i are holomorphic coordinates on X_0,t with 0 < |η_i| < c and 0 < |ζ_i| < c. Now we assume t = (t₁,· · ·, t_m) has small norm. We do the plumbing construction onX_0,t

to obtain X_t = X_t,t. For each i = 1,· · ·, m, we remove the discs {0 <

|ηi| ≤ ^|t_c^i|} and{0<|ζi| ≤ ^|t_c^i|} fromX0,t and identify{^|t_c^i| <|ηi|< c} with {^|t_c^i| <|ζ_i|< c} by the rule η_iζ_i =t_i. This defines the surfaceX_t. The tuple t = (t, t) = (t1,· · ·, tm, tm+1,· · ·, tn) are the local pinching coordinates for the manifold cover of Mg. We call the coordinates ηi (or ζi) the plumbing coordinates onX_t,s and the collar{^|t_c^i| <|η_i|< c}the plumbing collar.

Remark2.1. From the estimate of Wolpert [19], [20] on the length of short geodesic, we haveu_i=_2π^li ∼ −_log^π_|ti|.

In [6] we first proved the equivalence of canonical metrics onMg: Theorem 2.1. All the canonical metrics on the moduli spaceMg: the Teichm¨uller-Kobayashi metric, the Carath´eodory metric, the induced Bergman metric, the asymptotic Poincar´e metric, the McMullen metric, the Ricci metric, the perturbed Ricci metric and the K¨ahler-Einstein metric are equivalent.

The new metrics we defined have nice curvature properties which can be used to control the K¨ahler-Einstein metric. In [6] and [7] we proved

Theorem 2.2. The Ricci and perturbed Ricci metrics are complete K¨ahler metrics with Poincar´e growth. These metrics and the K¨ahler-Einstein metric have bounded geometry on the Teichm¨uller space Tg. Furthermore, all the covariant derivatives of the curvature of the K¨ahler-Einstein metric are bounded. The Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants.

We also derived in [6] and [7] the precise asymptotic of the Weil-Petersson, Ricci and perturbed Ricci metrics and their curvature. This is one of the key components in the proof of the goodness of these metrics. For the Weil- Petersson and Ricci metrics we have

Theorem 2.3. Let (t, s) = (t₁,· · ·, t_m, s_m+1,· · ·, s_n) be the pinching coordinates near a codimension mboundary point in Mg. Let hand τ be the Weil-Petersson and Ricci metrics respectively. Then the Weil-Petersson metric has the asymptotic:

(1) hⁱⁱ= 2u⁻_i ³|ti|²(1 +O(u0))andh_ii= ¹_2|t^u_i³ⁱ_|2(1 +O(u0))for1≤i≤m;

(2) h^ij =O(|t_it_j|)andh_ij =O _u3

iu³_j

|t_it_j|

, if1≤i, j≤mandi=j;

(3) h^ij =O(1) andh_ij =O(1), ifm+ 1≤i, j≤n;

(4) h^ij =O(|ti|)andh_ij=O _u3

|t_ii|

ifi≤m < j;

(5) h^ij =O(|t_j|)andh_ij =O _u3

j

|tj|

ifj≤m < i whereu0=m

j=1uj+n

j=m+1|sj|. The Ricci metric has the asymptotic:

(1) τ_ii= _4π³_2|t^u2i

i|²(1 +O(u₀))andτⁱⁱ =^4π₃^2|t_uⁱ₂^|2

i (1 +O(u₀)), ifi≤m;

(2) τ_ij =O _u2

iu²_j

|t_it_j|(ui+uj)

andτ^ij=O(|titj|), ifi, j≤mandi=j;

(3) τ_ij =O u²_i

|t_i|

andτ^ij =O(|t_i|), ifi≤m andj≥m+ 1;

(4) τ_ij =O(1), ifi, j≥m+ 1.

The holomorphic sectional curvature of the Ricci metric has the asymptotic:

(1) R_iiii=−_8π^3u4|t⁴ⁱi|⁴(1 +O(u₀))if i≤m;

(2) R_iiii=O(1) ifi > m.

We also have a weak curvature estimate of the Ricci metric. Let

Λ_i= _u

|tii| if i≤m 1 if i > m.

Then

(1) R_ijkl=O(1)if i, j, k, l > m;

(2) R_ijkl =O(Λ_iΛ_jΛ_kΛ_l)O(u₀) if at least one of these indices i, j, k, l is less than or equal tomand they are not all equal to each other.

The asymptotic of the perturbed Ricci metric and its curvature can be found in [6] and [7]. Also, precise estimates of the full curvature tensor of the Weil-Petersson, Ricci and perturbed Ricci metrics, which will be used in the proof of their goodness, can be found in [8] and [9].

Stronger estimates of the asymptotic of these metrics lead to the Mum- ford’s goodness condition of singular Hermitian metrics on vector bundles over quasi-projective manifolds. We recall the definition and basic properties of good metrics from [11].

LetX be a quasi-projective variety of dim_CX =kobtained by removing a divisorD of normal crossings from a closed smooth projective varietyX. Let E be a holomorphic vector bundle of ranknoverX and E=E |X. Lethbe a Hermitian metric onE which may be singular near D.

We cover a neighborhood ofD⊂X by finitely many polydiscs U_α=

∆^k,(z₁,· · · , z_k)

α∈A

such thatV_α=U_α\D= (∆^∗)^m×∆^k−m. Namely,U_α∩D={z₁· · ·z_m= 0}. We letU =

α∈AU_αandV =

α∈AV_α. On eachV_αwe have the local Poincar´e metric

ωP,α=

√−1 2

_m

i=1

1

2|z_i|²(log|z_i|)²dzi∧dzi+ k i=m+1

dzi∧dzi

.

Definition 2.4. Letη be a smooth localp-form defined onV_α.

• We sayη has Poincar´e growth if there is a constantCα>0 depending on η such that

|η(t1,· · · , tp)|²≤Cα

p i=1

ti²ω_P,α

for any pointz∈V_α andt₁,· · ·, t_p∈T_zX.

• We sayη is good if bothηand dηhave Poincar´e growth.

Definition 2.5. An Hermitian metric honE is good if for all z ∈V, assumingz ∈V_α, and for all basis (e₁,· · ·, e_n) of E over U_α, if we let h_ij = h(e_i, e_j), then

• h_ij,(deth)⁻¹≤C(m

i=1log|z_i|)²ⁿ for someC >0 andn≥1.

• The local 1-forms

∂h·h⁻¹

αγ are good onV_α. Namely the local connec- tion and curvature forms ofhhave Poincar´e growth.

It is easy to see the following basic properties of good metrics:

• The definition of Poincar´e growth is independent of the choice of U_α or local coordinates on it.

• A formη∈A^p(X) with Poincar´e growth defines ap-current [η] on X. In

fact we have

X|η∧ξ|<∞ for anyξ∈A^k−p(X).

• If bothη ∈A^p(X) andξ ∈A^q(X) have Poincar´e growth, thenη∧ξ has Poincar´e growth.

• For a good formη∈A^p(X), we have d[η] = [dη].

The importance of a good metric on E is that we can compute the Chern classes ofE via the Chern forms of h as currents. Namely, with the growth assumptions on the metric and its derivatives, we can integrate by part, so Chern-Weil theory still holds. In [11] Mumford has proved:

Theorem 2.4. Given an Hermitian metric h on E, there is at most one extensionE of E toX such thathis good.

Theorem 2.5. Ifhis a good metric onE, the Chern formsc_i(E, h)are good forms. Furthermore, as currents, they represent the corresponding Chern classesci(E)∈H²ⁱ(X,C).

In the following sections, we will discuss the goodness of the above metrics and their applications.

§3. Goodness of the Weil-Petersson Metric

From Theorem 2.3, it is very natural to consider the metrics induced by the Weil-Petersson, Ricci, perturbed Ricci and K¨ahler-Einstein metrics on the logarithmic extension E = T^∗

Mg(logD) of the cotangent bundle T_M^∗ _g to the DM moduli spaceMg.

We first give a general discussion of the goodness condition of the metric onEinduced by a K¨ahler metricgonMg. LetD=Mg\Mg be the compact- ification divisor and letp∈Dbe a codimensionmboundary point inMg with the corresponding stable nodal surfaceX0,0. Letn= 3g−3 be the dimension ofMg. Let (t1,· · ·, tn) be the pinching coordinates nearpwhere (t1,· · ·, tm) corresponding to the degeneration directions.

For any K¨ahler metricg onMg, let g^∗ be the induced metric on E. We know that

dt₁

t1 ,· · ·,dt_m

tm , dt_m+1,· · · , dt_n (3.1)

is a local holomorphic frame of E. Under this frame, the metric g^∗ and its inverse are given by

g_ij^∗ =















1

t_it_jg^ij i, j≤m

t1_ig^ij i≤m < j

1

t_jg^ij j≤m < i g^ij i, j > m (3.2)

and

(g^∗)^ij=













titjg_ij i, j≤m t_ig_ij i≤m < j t_jg_ij j≤m < i g_ij i, j > m.

(3.3)

Now we define two quantities. Let

D^k_i =













t_k

ti i, k≤m t_k k≤m < i

t1_i i≤m < k 1 i, k > m.

(3.4)

Letvi=−_log^π_|ti| fori≤mand letui=_2π^li whereliis the length of the geodesic loop on thei-th collar ofX =Xt. We have that

ui=vi(1 +O(vi)).

Now we let

Λ_i = _u

|t_ii| i≤m 1 i > m.

(3.5)

LetA= (A^k_i) be the connection form ofg^∗ whereiis the row index andk is the column index. We have

A^k_i =

p





j

∂_pg_ij^∗

(g^∗)^kj



dt_p. (3.6)

By (3.2), (3.3) and (3.4), we have A^k_i =−

p



D_i^k

j

∂_pg_kj

g^ij



dt_p (3.7)

ifi=k ori=k > m. We also have Aⁱ_i=−

p=i





j

∂pg_ij

g^ij



dtp−



1 ti +

j

∂ig_ij

g^ij



dti

(3.8)

ifi=k≤m.

To prove the goodness of g^∗, the first order estimates are reduced to

D_i^k

j

∂_pg_kj

g^ij

=O(Λ_p) (3.9)

ifi=k ori=k > mor p=i=k≤mand

1 t_i +

j

∂_ig_ij

g^ij

=O(Λ_i) (3.10)

ifi=k=p≤m.

For the estimates on theg^∗ itself, we need to show that g_ij^∗, (detg^∗)⁻¹≤C

_m

i=1

log|t_i| 2n

. (3.11)

By (3.2) we have (detg^∗)⁻¹=|t₁· · ·t_m|²(detg), inequality (3.11) is equiv- alent to

g_ij^∗, |t1· · ·tm|²(detg)≤C _m

i=1

log|ti| 2n

. (3.12)

The second order estimates are reduced to show that dA^k_i has Poincar´e growth for any choice ofi, k. Since

dA=∂A+∂A=∂A−A∧A,

if each entry ofAhas Poincar´e growth, then each entry ofA∧Ahas Poincar´e growth. Thus we need to show that each entry of∂Ahas Poincar´e growth.

By (3.7) and (3.8), sinceDⁱ_i= 1, we have

∂A^k_i =D^k_i∂_q





j

∂_pg_kj

g^ij



dt_p∧dt_q =−D_i^kg^ijR_kjpqdt_p∧dt_q (3.13)

whereR_kjpq is the curvature ofg. Thus we need to show that D_i^kg^ijR_kjpq=O(Λ_pΛ_q).

(3.14)

By collecting the above argument, we have

Lemma 3.1. The metricg^∗onT^∗

Mg(logD)induced by a K¨ahler metric g on Mg is good if and only if the estimates (3.9), (3.10), (3.12) and (3.14) hold.

In this section, we will focus on the goodness of the metric induced by the Weil-Petersson metrich. The main theorem is

Theorem 3.1. The metrich^∗on the logarithm cotangent bundleEover the DM moduli space induced by the WP metric is good in the sense of Mumford.

Thus the Chern forms ofh^∗, as currents, are equal to the Chern classes ofE.

We now sketch the proof of this theorem in three steps: the zero-th order, first order and second order estimates. The details are in [8]. In the following, we take the metricg_ij to be the Weil-Petersson metrich_ij. We use the same notation as in our paper [6].

We first consider the zero-th order estimate. This follows directly from Theorem 2.3.

Lemma 3.2. The inequality(3.12)hold for the Weil-Petersson metrich.

Proof. By Theorem 2.3 and (3.2), (3.3), we have h^∗_ij =

2u⁻_i³(1 +O(u₀)) i=j≤m

O(1) otherwise

and

|t₁· · ·t_m|²(deth)≤C _m

i=1

u_i 3

.

It is easy to see that (3.12) hold.

Now we prove the first order estimates. In order to compute the connection of the induced metric h^∗, it is easier to prove the formula (3.6) directly with the metric h. We use the estimate of Masur [10] and refine the estimates of Schumacher [13] and Trapani [16].

By the work of [6] we know that h^ij=

X

ϕ_iϕ_j λ² dv (3.15)

whereϕ_i is the holomorphic quadratic differential corresponding to dt_i andλ is the KE metric onX. In order to compute the connection forms of the WP metric, we need to estimate the derivatives of eachϕi andλ.

Unlike the approach in [6], here we take plumbing coordinate and plumbing collar rather than rs-coordinate and genuine collar because we need the trivi- alization of the collars. It is easier to compute the derivative of the hyperbolic metric by using the plumbing coordinate on the degeneration collars.

We first change coordinate on the collars. Let (t₁,· · ·, t_n) be the pinching coordinates near a codimensionmboundary pointpin the DM moduli. Letz_i andw_i be the plumbing coordinates on the i-th collar ofX_twith i≤m. Let ri =|zi|, θi = argzi, ri =|wi|and θi = argwi. We know that ziwi =ti. Let Ωⁱ_c be the i-th plumbing collar of sizecwith a fixed 0< c <1. Namely,

Ωⁱ_c={z_i|c⁻¹|t_i| ≤r_i≤c}={w_i|c⁻¹|t_i| ≤r_i≤c}. We denote by Ω_c the union of all collars: Ω_c = m

i=1Ωⁱ_c. We also define the half collars Ωⁱ⁺_c and Ωⁱ_c⁻ by

Ωⁱ⁺_c ={zi| |ti|¹² ≤ri ≤c} and

Ωⁱ_c⁻={z_i |c⁻¹|t_i| ≤r_i≤ |t_i|¹²}={w_i| |t_i|¹² ≤r_i ≤c}.

To compute the derivative ofϕ, by our works in [6] and the work of Masur [10], we have the expansion of ϕ_i on the plumbing collars. Let ∆ⁿ_δ be the closed polydisc in Cⁿ such that the radius of each disk isδ > 0. We assume the pinching coordinatest= (t1,· · ·, tn) is defined fort∈∆ⁿ_δ. By shrinking δ we have

Lemma 3.3. Letk≤mand letzk andwk be the plumbing coordinates on thek-th collar Ω^k_c0 withc < c0<1 fixed. Then onΩ^k_c0 we have

(1) ϕ_i=−^t_πⁱ_z¹2

k(p^k_i(z_k) +q_i^k(z_k))if i≤mandi=k;

(2) ϕ_i= _z¹₂

k(p^k_i(z_k) +q^k_i(z_k))ifi > m;

(3) ϕk =−^t_π^k_z¹2

k(1 +p^k_k(zk) +q^k_k(zk)).

There is a constantM >0such that in the above formulae, the functionsp^k_i, q^k_i satisfy

(1) p^k_i =_∞

s=1a^k_is(t)z_k^s such that each a^k_is(t) is a holomorphic function of the multi-variablet and_∞

s=1|a^k_is(t)|c^s₀≤M fort∈∆ⁿ_δ; (2) q_i^k =

s≤−1a^k_is(t)t⁻_k^sz_k^s such that a^k_is(t) is holomorphic in t and

s≤−1|a^k_is(t)|c⁻₀^s≤M fort∈∆ⁿ_δ.

There are similar expansions by using the wk coordinates. Furthermore, on X\Ωc we have

ϕi=

O(|ti|) i≤m O(1) i > m.

For the proof of this lemma, please see [10]. We also have the estimates of the derivatives ofp^k_i andq_i^k:

Lemma 3.4. Let 0 < c < c0 be a fixed constant. On the collar Ω^k_c we have

(1) ^∂p_∂t^ki

j =_∞

s=1∂a^k_is(t)

∂tj z_k^s such that_∞

s=1^∂a_∂t^kis_j^(t)c^s≤M₁ fort∈∆ⁿ_δ

2; (2) ^∂q_∂t^ki

j =

s≤−1

∂a^k_is(t)

∂t_j t⁻_k^sz^s_k such that

s≤−1^∂a∂t^kis_j^(t)c^−s≤M1 fort∈∆ⁿδ

andj=k; 2

(3) ^∂q_∂t^ki

k = _t¹

k

s≤−1b^k_is(t)t⁻_k^sz^s_k where b^k_is(t) = t_k^∂a_∂t^kis(t)

k − sa^k_is(t) and

s≤−1|b^k_is(t)|c^−s≤M₁.

HereM₁ is a constant depending onM, c, c₀, δ, n.

By combining the above two lemmas, we can get desired estimates of the derivatives of each quadratic differentialϕ_i.

We then estimate the KE metricλ and its derivatives on each Riemann surface. The following estimate ofλis due to Masur [10]. The following lemma, although is not sharp, will be enough for our purpose.

Lemma 3.5. For each 1≤i≤m, there is a constantα >0 such that, onΩⁱ⁺_c , we have

1 α

1

r²_i(logr_i)² ≤λ≤α 1 r_i²(logr_i)² and onΩⁱ_c⁻, we have

1 α

1

r²_i(logr_i)² ≤λ≤α 1 r_i²(logr_i)². The estimate the derivative ofλis more subtle. We have

Lemma 3.6. Letλbe the KE metric on the Riemann surfaceX =Xt. On each collar Ω^k_c, λ has a unique representation in term of the plumbing coordinatez_k. Then

∂

∂ti (logλ|Ωc)

=O(Λ_i).