The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $\pi_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for large $k$. As an application we prove that the analytic torsion $\tau_k(\bar{\partial})$ satisfies $\partial\bar{\partial}\log(\tau_k(\bar{\partial}))^2=o(k^{n-1})$, where $n$ is the dimension of fibers.


Introduction
Let π : X → M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over X , i.e. there is a smooth metric (weight) φ on L such that the first Chern form √ −1 2π ∂∂φ is positive (1, 1)-form along each fiber. One may consider the following direct image bundle E k = π * (L k ⊗ K X /M ).
Here K X /M = K X ⊗ π * K −1 M denotes the relative canonical line bundle. The bundle E k is equipped with the canonical L 2 -metric (1.1) see [6][7][8]. Here |u| 2 e −φ is defined as follows: u can be written locally as u = u dv ∧ e, where e is a local holomorphic frame for L| X y , X y = π −1 (y), and |u| 2 e −φ := ( √ −1) n 2 |u | 2 |e| 2 dv ∧ dv = ( √ −1) n 2 |u | 2 e −φ dv ∧ dv, where dv = dv 1 ∧ · · · ∧ dv n . In [8,Theorem 1.2] Berndtsson computed the curvature E k of the L 2 -metric and consequently proved Nagano positivity. More precisely, where the definitions of c(φ), μ α and are given in Theorem 2.2. In particular, if π : X → M is a trivial fibration, Berndtsson [7,Theorem 4.1,4.2] obtained an asymptotic of tr E k /d k up to o(1), d k = rankE k . In view of the relation of the L 2 -curvature with analytic torsion and Quillen metric it is a natural and interesting problem to find the lower order terms in the asymptotic of tr E k for a general fibration. We solve the problem up to reminder term of order o(k n−1 ).

Theorem 1.1 For any vector ζ ∈ T y M, we have
We refer to the Sect. 3.1 for the definitions and notation. We note that the first two terms in the expansion above were proved by Ma-Zhang [22]. The leading terms of the first summand in (1.2) is studied by Sun [24] and the second by Berndtsson [7] (in the different setup of trivial fibration with variation of Kähler metrics).
We shall then compare our expansion for the L 2 -curvature with the Quillen curvature. So let D y =∂ y +∂ * y be the Dirac operator acting on A 0, * (X y , L k ⊗ K X /M ) of (0, * )-forms, where X y is endowed with the restricted Hermitian metric ( √ −1∂∂φ)| X y . For any 0 < b < c, The Quillen metric • Q on the determinant line λ (see Definition 2.5) is patched by the L 2 -metric | • | b on λ b (see (2.13)) and the analytic torsion τ k (∂), i.e. (1.4) where b > 0 is a sufficiently small constant. In their papers [11][12][13], J.-M. Bismut, H. Gillet and C. Soulé computed the curvature of Quillen metric for a locally Kähler family and obtained the differential form version of Grothendieck-Riemann-Roch Theorem. More precisely, they proved that as holomorphic bundles, and the curvature is (1,1) .
Since L is a relatively ample line bundle over X , by Kodaira vanishing theorem, We expand also the Quillen curvature c 1 (λ, • Q ) and compare it with (1.3). We prove Theorem 1.2 Up to terms of order o(k n−1 ) the Quillen curvature −c 1 (λ, • Q ) has the same expansion (1.3) as for the L 2 -curvature.
As application we shall find the asymptotics of the variation of the analytic torsion. From (3.56) we have for b > 0 a sufficiently small constant, where det • k denotes the natural induced L 2 -metric on line bundle det E k and ((| • | b ) 2 ) * denotes the dual metric of (| • | b ) 2 . Using (1.4) and (1.7) we have furthermore As an immediate consequence of Theorem 1.1 and Theorem 1.2 we have Here the asymptotic (1.9) is understood as (∂∂ log(τ k (∂)) 2 )(ζ,ζ ) = o(k n−1 ) for any vector ζ ∈ T M.

Remark 1.4
The asymptotic of analytic torsion has been studied by [5,10]. It is proved in [10,Theorem 8] the coefficients of k n , k n log k are topological invariants. After a preliminary version of this paper was finished, we were informed by Xiaonan Ma of the paper [19, Theorem 1.1, 1.2] by Finski where the coefficients of k n−1 , k n−1 log k in the analytic torsion τ k have also been computed, which implies then (1.9). However, our method here is completely different from the methods in [10,19].

Remark 1.5
For the case of M is the Teichmüller space of compact Riemann surfaces of genus g ≥ 2 and L is the relative canonical line bundle, Corollary 1.3 was proved in [18]; in this case the analytic torsion τ k actually decays exponentially in k.
We proceed to explain briefly our method for the expansion in Theorem 1.1. By the formula (1.2), the first Chern form c 1 (E k , • k ) is the trace of an integral operator, and in the paper [22] X. Ma and W. Zhang found the expansion of the diagonal of the kernel of the operators, proving a local index formula. To find the third order term, i.e. the coefficient of k n−1 , seems a difficult task and requires much more effort. The trace of first summand in (1.2) is relatively easy to handle, however the second summand involves Toeplitz operators with symbols being differential operators. A major ingredient of our method is the following expansion (see Lemma 3.1), . The corresponding contribution of each term above to the L 2curvature c 1 (E k , • k ) will be effectively treated by using further the asymptotic expansion of Bergman Kernel for bundles [26,Theorem 4.2].
We note that generally it is always interesting to study variations of complex or Kähler structures and connections on bundles of cohomology spaces over moduli spaces. The most well-known case might be the Siegel moduli space parametrising polarized Abelian varieties. This aspect has been very much studied in mathematical physics; see e.g. [1][2][3] and references therein.
This article is organized as follows. In Sect. 2, we fix notation and recall some basic facts on Berndtsson's curvature formula of L 2 -metric, the asymptotic expansion of Bergman kernel for bundles, analytic torsion and Quillen metric. In Sect. 3, we find the expansion of c 1 (E k , • k ) and prove Theorem 1.1. We also give the expansion of −c 1 (λ, • Q ) and prove Theorem 1.2. By comparing with their expansions, we prove Corollary 1.3.

Preliminaries
We shall fix notation and recall some necessary background material.
Let π : X → M be a holomorphic fibration with compact fibres and L a relatively ample line bundle over X . We denote by (z; v) = (z 1 , . . . , z m ; v 1 , . . . , v n ) a local admissible For any smooth function φ on X , we denote By a routine computation, one can show that { δ δz α } 1≤α≤m spans a well-defined horizontal subbundle of T X .
Let {dz α ; δv k } denote the dual frame of δ δz α ; ∂ ∂v i . Then Moreover, the differential operators are well-defined. For any φ ∈ F + (L), the geodesic curvature c(φ) is defined by which is a horizontal real (1, 1)-form on X . The following lemma confirms that the geodesic curvature c(φ) of φ is indeed well-defined.
Lemma 2.1 [17] The following decomposition holds, Proof This is a direct computation, Following Berndtsson (cf. [6][7][8]) we consider the direct image bundle E := π * (K X /M ⊗ L), and define the following L 2 -metric on E: for y ∈ M, X y = π −1 (y), and u ∈ E y ≡ H 0 (X y , (L ⊗ K X /M ) y ), Note that u can be written locally as u = u dv ∧ e, where e is a local holomorphic frame for L| X , and so locally where dv = dv 1 ∧ · · · ∧ dv n is the fiber volume. By the definition of∂ V , we have which is in the Kodaira-Spencer class ρ( ∂ ∂z α | y ) ∈ H 1 (X y , T X y ). The following theorem was proved by Berndtsson in [8, Theorem 1.2], its proof also can be found in [17, [8] For any y ∈ M the curvature E (u, u) , u ∈ E y , of the Chern connection on E with the L 2 metric is given by Here = ∇ ∇ * + ∇ * ∇ is the Laplacian on L| X y -valued forms on X y defined by the (1, 0)-part of the Chern connection on L| X y .

The Bergman kernels
Let (X , ω) be a compact Kähler manifold of n-dimension, (L, e −φ ) be a Hermitian line bundle over X satisfying We shall eventually replace (X , ω) by the fibers of the fibration in the previous section.
Let (E, H ) be a Hermitian vector bundle over X . There is a natural L 2 -metric on the space Recall the following Tian-Yau-Zelditch expansion of Bergman kernel for bundles. We use the version in [14] and refer [4,16,20,[25][26][27][28] for different variations and proofs.

Theorem 2.3 For a fixed metric H , there is an asymptotic expansion as k → ∞,
where A i ∈ End(E) are determined by the geometry of ω and H . The expansion is in the sense that for any integer l, R ≥ 0, where the norm is computed in the space C l (X , End(E)) of End(E)-valued sections and C l,R,H depends on l, R, ω and H .
The first three coefficients A 0 , A 1 and A 2 have been computed in [26,Theorem 4.2].
Theorem 2.4 [26] (0) A 0 = I d, Here R, Ric and ρ represent the curvature tensor, the Ricci curvature and the scalar curvature The Bergman kernel is From (2.10), (2.11) and Theorem 2.3, the asymptotic expansion of Bergman kernel for the bundle (2.12)

Analytic torsion and Quillen metric
The definitions and results in this subsection can be found in [9,[11][12][13]21,23]. Let π : X → M be a proper holomorphic mapping between complex manifolds X and M, (F, h F ) a holomorphic Hermitian vector bundle on X , ∇ F the corresponding Chern connection, and R F = (∇ F ) 2 its curvature. For any y ∈ M, let X y = π −1 (y) be the fiber over y with Kähler metric g X y depending smoothly on y. The fibers are assumed to be compact.
For any 0 ≤ p ≤ n := dim C X y we put The operator D y =∂ y +∂ * y acts on the fiber E y . For every y ∈ M, the spectrum of D 2 be the sum of the eigenspaces of the operator D 2 y acting on E p y for eigenvalues < b. Let U b be the open set: On each open set U b , K b, p is a smooth finite dimensional vector bundle. Set Define the following line bundle λ b on U b , Define λ (b,c) accordingly as before. Let∂ (b,c) and D (b,c) be the restriction of∂ and D to Since the chain complex The analytic torsion of the chain complex (2.14) was introduced by Ray and Singer [23].

Definition 2.6
The analytic torsion τ (∂ (b,c) ) associated to the acyclic chain complex (2.14) is defined as the positive real number If b is a small constant less than all positive eigenvalues of D 2 y , then we denote The functions θ b y and θ (b,c) y extend into a meromorphic function which is holomorphic at and by [13,Equation 1.32], Then the definition of Quillen metric • Q and Chern form c 1 (λ, • Q ) of the Quillen metric are given by the following theorem. Theorem 2.7 [11][12][13] The metrics

into a smooth Hermitian metric
• Q on the holomorphic line bundle λ. The Chern form of Hermitian line bundle (λ, • Q ) is (1,1) .
The Knudsen-Mumford determinant is defined by The fiber λ K M y is by definition given by We assume that π is locally Kähler, i.e. there is an open covering U of M such that if U ∈ U , π −1 (U ) admits a Kähler metric.
Theorem 2.8 [11][12][13] Assume that π is locally Kähler. Then the identification of the fibers λ y ∼ = λ K M y defines a holomorphic isomorphism of line bundles λ ∼ = λ K M . The Chern form of the Quillen metric on λ ∼ = λ K M is given by (2.17).

The asymptotic of the curvature of direct image bundle
We shall give the expansion of c 1 (E k , • k ) and −c 1 (λ, • Q ) up to o(k n−1 ).
Let π : X → M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over X as in the Sect. 2.1. Denote

The asymptotic of the curvature of L 2 -metric
The curvature of direct image bundle E k = π * (L k ⊗ K X /M ) is, by (2.7), where 3) The following technical expansion of ( + k) −1 will be critical to find the asymptotics of the L 2 -curvature; the main point of this expansion is that the leading term of the contribution to the L 2 -curvature of each term below is effectively found. (Composed with the operators i μ and i * μ it gives an expansion of the Toeplitz operator with symbol i * μ ( + k) −1 i μ on the cohomology space H 0 (X y , L k ⊗ K X y ).)

Lemma 3.1
The resolvent operator ( + k) −1 has the following 7-term-expansion, ( + k) −1 = I + I I + · · · + V I + V I I , (3.4) where Proof The RHS of (3.4), by elementary computations, is which completes the proof. Here the second and last equalities follow from while the third and fourth equalities follow from We shall treat each term in the expansion using the following lemmas. We refer [15,Chapter VII] for the calculus on Kähler manifolds. (X , ω) be a compact Kähler manifold and (L k , e −kφ ) be a Hermitian line bundle over X with

Lemma 3.2 Let
where ∇ * is the adjoint operator of the (1, 0)-part ∇ of Chern connection, We expand the second term and find Furthermore the action on α of the second term in the RHS of (3.7), by the definition of ∇ ∂ ∂v t , is the operator (3.8)

Lemma 3.3
The following identity holds, where we have introduced |μ| 2 R * := μ lt Rt j il μ ij , which need not to be nonnegative, Proof By a direct computation, we find It follows that Thus, using Lemma 3.2 and (3.9), we obtain (3.10) In terms of local coordinates the first term is (3.11) where the fourth equality follows from the definition of μ (3.3) and The second term in the RHS of (3.10) is where in the second equality, (•) * denotes the adjoint operator of •, the fifth equality follows from The sixth equality holds by (3.12) and ∇¯j (φ st ) = ∇¯j (φq i ) = 0. Substituting (3.11) and (3.13) into (3.10) we have In the subsequent text we shall write O(k j ) for any term that is of the order k j and is independent of u.

Lemma 3.4 We have the following expansion
Proof Writing k − = (k − − R * ) + R * and using Lemma 3.2 we have (3.14) For the first term in the RHS of (3.14), we have For the second term in RHS of (3.15), using Lemma 3.2, we obtain (3.16) We substitute (3.15) and (3.16) into (3.14), This completes the proof.

Lemma 3.5 The quadratic form (k − ) 3 i μ u, i μ u has the following expansion
Proof Denote ∇t = ∇ ∂ ∂v t . We have, using Lemma 3.2, that (3.17) where the second equality holds because all the terms in , and so its adjoint operator is in O(1), the fourth equality follows from [∇t , ∇ ] = k∂φt .

Lemma 3.6 The following expansion holds
Proof Similar to the proof in Lemma 3.5 for estimating reminder terms we have (3.18) We prove now Theorem 1.1.

Proof
The curvature formula in (2.7) contains two quadratic forms in u, the first one invovles only |u| 2 and will be treated using Bergman kernel expansion later, the second one involves ( + k) −1 and will be treated first. We have, by the expansion (3.4) ( + k) −1 i μ u, i μ u = I (u) + I I (u) + · · · + V I I (u) We shall treat each term using the lemmas above. First we have (3.19) where |μ| 2 = μ ij μ s t φ is φ¯j t . By Lemma 3.3 and (3.11), the second term is (3.20) Likewise, by (3.11) The fourth term is where the third equality follows from Lemma 3.4. By Lemmas 3.2, 3.3 and 3.4, the fifth term (3.24) Here we have used Lemmas 3.3 and 3.5 to conclude Finally the last term is where the last equality follows from Lemmas 3.4, 3.5 and 3.6.
We estimate the second term in the RHS of (3.26) as (3.27) indeed the third equality follows from Lemma 3.4 and the following equality (3.30) Finally substituting (3.30) into (3.2) we get (3.31) Denote d k = dim H 0 (X y , L k + K X /M ), and let {u j } d k j=1 be an orthogonal basis of H 0 (X y , L k + K X /M ). From (2.12) and (3.56), we have where c(L, φ) = 1 2π ω is the Chern form of the Hermitian line bundle (L, e −φ ). We take the trace to both sides of (3.31) and use the Bergman kernel expansion (3.32), where we have denoted ∇μ 2 = X y |∇μ| 2 ω n n! , μ 2 R * = X y |μ| 2 R * ω n n! . We rewrite the integrals μ 2 R * and ∇μ 2 (of the anti-holomorphic connection ∇μ) in terms of ∂ * μ and the holomorphic connection ∇ μ. By Akizuki-Nakano identity [ where we have denoted μ 2 Ric := X y (μ ij μ s t R il φl k φ¯j t φ ks ) ω n n! , which again needs not to be nonnegative. Therefore, By its definition we find also (3.35) Substituting (3.34) and (3.35) into (3.33) we obtain finally This completes the proof of Theorem 1.1.

Remark 3.7
It is a general fact [22] that the L 2 -curvature c 1 (E k ) above has an expansion in the integer powers of k, so that the lower order term o(k n−1 ) in our statement can be written as O(k n−2 ). Indeed observe that the fractional order O(k n−1− 1 2 )-terms in the proof of Theorem 1.1 are all due to the estimate ( + k) −1 (k − )i μ u ≤ Ck − 1 2 u . However we can use again Lemma 3.1 and prove that the traces of the quadratic forms involving ( + k) −1 (k − ) are actually of integer order instead of fractional order, e.g. the trace of the quadratic form I V (u) in the estimate (3.22) has an expansion of order k −3+n . It might be interesting to find a recursive formula for the coefficients of the expansion c 1 (E k ) following our proof and using the Bergman kernel expansion.

The asymptotic of the curvature of Quillen metric
We compute now the asymptotic of the curvature of Qullien metric and prove Theorem 1.2.
By Theorems 2.7 and 2.8, we have where td is the Todd character forms, which has an expansion, for any Hermitian vector bundle (F, h), the second equality follows from 2π R K X /M . Now we consider the last term in the RHS of (3.37), where the curvature operator R is defined by (3.39) Here the second and third curvature term is In fact, one can prove them in terms of normal coordinates, i.e. φ ij = δ i j , φ ijk = 0 at a fix point, so while second identity in (3.40) holds similarly. We compute the second term in the RHS of (3.38), where the second equality follows from the fact for two real (1, 1)-forms α and β.

An application
In this subsection, we will prove Corollary 1.3. For any positive integer k write temporarily where K X /M is the relative canonical line bundle endowed with the following metric, As in Sect. 2.3, the operator D y =∂ y +∂ * y acts on ⊕ p≥0 A 0, p (X y , F). Take a small constant b > 0 that is smaller than the all positive eigenvalues of D y . Then K b, p y ∼ = H p (X y , F) = H p (X y , K X y ) ⊗ L k ).