Holomorphic Sectional Curvature of Complex Finsler Manifolds

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition.


Introduction
In this paper, we study the holomorphic sectional curvature of complex Finsler manifolds (see Definition 2.3). For a general complex manifold, there is a natural and intrinsic Finsler pseudo-metric, i.e., Kobayashi metric k M (see Definition 3.6), which is the maximum pseudo-metric among the pseudo metrics satisfying the decreasing property. This metric defines a Kobayashi pseudo-distance, and a complex manifold is called Kobayashi hyperbolic if the Kobayashi pseudo-distance is a distance in com-Partially supported by NSFC (Grant Nos. 11221091, 11571184).
B Xueyuan Wan xwan@chalmers.se 1 Department of Mathematical Sciences, Chalmers University of Technology, 41296 Gothenburg, Sweden mon sense. It is known that a complex Finsler manifold is Kobayashi hyperbolic if its holomorphic sectional curvature is bounded from above by a negative constant. As an application, the moduli space of canonically polarized complex manifolds is Kobayashi hyperbolic [23,25].
In hyperbolic geometry a conjecture of Kobayashi asserts that the canonical bundle is ample if the manifold is hyperbolic [19, p. 370]. Wu and Yau [28] proved the ampleness of the canonical bundle for a projective manifold admitting a Kähler metric with negative holomorphic sectional curvature. Later on, Tosatti and Yang [26] proved this result without projective condition and asked if it still true for a compact Hermitian manifold. In [33], Yau conjectured that an algebraic manifold is of general type if and only if it admits a complex Finsler metric with strongly negative holomorphic sectional curvature which may be degenerate along a subvariety. For the case of quasi-negative holomorphic sectional curvature, the ampleness of the canonical bundle also had been proved in [12,29]. In the proofs of above results, they essentially used Yau's Schwarz Lemma [31]. This is also a motivation for us to study the Schwarz Lemma in the case of complex Finsler manifolds.

Proposition 1.1 Let M be a complex manifold of dimension n, G 1 and G 2 be two complex Finsler metrics on M. For any point (z, [v]) ∈ P(T M), one has
where π * : T (T M) → T M is the differential of π : T M → M.
A simple maximum principle argument immediately gives a direct proof for the Schwarz Lemma of [24,Theorem 4.1]. For the case of a complete Riemann surface (M, G 1 ), by using Proposition 1.1, we obtain the following generalization of Yau's Schwarz Lemma [ If (M, G 1 ) is a unit disc with a Poincaré metric (which is complete) and by the definition of Kobayashi metric, one has 4k 2 N ≥ −K 2 G 1 . This implies that N is Kobayashi hyperbolic by definition.
For a compact Kähler manifold with positive holomorphic sectional curvature, Yau [32, Problem 67] asked if the manifold has negative Kodaira dimension. Yang [34] gave a affirmative answer to this problem. More precisely, if (M, ω) is a Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature, then the Kodaira dimension κ(M) = −∞. Naturally, one may ask whether this result holds for a strongly pseudoconvex complex Finsler manifold, namely, whether for a compact strongly pseudoconvex Finsler manifold (M, G) with semi-positive but not identically zero holomorphic sectional curvature one has that κ(M) = −∞.
For any strongly pseudoconvex complex Finsler metric G, there is a canonical (Finslerian) tensorĜ Here (T M) o denotes that the set of all non-zero holomorphic tangent vectors of M, {δv i } n i=1 is a local holomorphic frame of V * (for the definitions of δv i and V * see (2.12)). By taking covariant derivative of the Finslerian tensorĜ along the anti- Hermitian metric satisfies the condition∂Ĝ(P) = 0. Moreover, there are also many non-Hermitian strongly pseudoconvex complex Finsler metrics which satisfy the condition (see Example 2.6).

Holomorphic Sectional Curvature of Complex Finsler Manifolds
In this section, we shall fix notation and recall some basic definitions and facts on complex Finsler manifolds. For more details we refer to [1,4,5,10,13,18,27]. Let M be a complex manifold of dimension n, and let π : T M → M be the holomorphic tangent bundle of M. Let z = (z 1 , . . . , z n ) be a local coordinate system in M, and let { ∂ ∂z i } 1≤i≤n denote the corresponding natural frame of T M. So any element in T M can be written as where we adopt the summation convention of Einstein. In this way, one gets a local coordinate system on the complex manifold T M: (M, G) is called a (strongly pseudoconvex) complex Finsler manifold if G is a (strongly pseudoconvex) complex Finsler metric.
Clearly, any Hermitian metric on M is naturally a strongly pseudoconvex complex Finsler metric on it.
We write to denote the differentiation with respect to v i ,v j , z i ,z j , and we denote (G¯j i ) the inverse matrix of (G ij ). In the following lemma we collect some useful identities related to a Finsler metric G.

Lemma 2.2 ([10,18])
The following identities hold for any (z, v) ∈ E o , λ ∈ C: (2.6) Let = {w ∈ C||w| < 1} be a unit disc. For any holomorphic map ϕ : → M, one can define a conformal metric on the disc by ). The Gaussian curvature K ϕ * G of ϕ * G is given by Then one can define the holomorphic sectional curvature of (M, G) as follows.
If G is a strongly pseudoconvex complex Finsler metric on M, then there is a canonical h-v decomposition of the holomorphic tangent bundle T (T M) o of (T M) o (see [10, §5] or [13, §1]). (2.10) In terms of local coordinates, With respect to the h-v decomposition (2.12), the (1, 1)-form 2π ∂∂ log G has the following decomposition.

13)
where we denote For the holomorphic vector bundle which are well defined. Indeed, for two local coordinate neighborhoods (U α , For a strongly pseudoconvex complex Finsler metric G, one can also define the holomorphic sectional curvature by where the supremum is taken over all the holomorphic maps ϕ : → M satisfying Proof For reader's convenient, we give a direct proof here. For any holomorphic ϕ : → M with ϕ(0) = z, ϕ (0) = λv for some λ ∈ C * , it induces a holomorphic map Similarly, the holomorphic map ϕ * also induces a holomorphic map (ϕ * ) * : Therefore, one has For a given strongly pseudoconvex complex Finsler metric G, there is a canonical (Finslerian) tensorĜ, By taking conjugation, one gets (2.26) In applications, one may consider a class of strongly pseudoconvex complex Finsler metrics G, which satisfies∂Ĝ In terms of local coordinates, Eq.

Proposition 3.1 Let M be a complex manifold of dimension n. Let G 1 and G 2 be two complex Finsler metrics on M. For any point (z, [v]) ∈ P(T M), then
where the supremum is taken over all the holomorphic map ϕ : → M with ϕ(0) = z, ϕ (0) = λv for some λ ∈ C * .
where the last inequality holds since G 1 is strongly pseudoconvex.
As an application of the above proposition, we give a direct proof of the following Schwarz Lemma.

Corollary 3.4 ([24, Theorem 4.1]) Let f be a holomorphic map between two complex Finsler manifolds (M, G) and (N ,
which is a smooth function on P(T M Note that By the assumptions for K G 1 and K G 2 , one obtains Next we consider the case of a holomorphic map from a complete Riemann surface to a Finsler manifold with negative holomorphic sectional curvature. We assume that (M, G) is a complete Riemann surface, the fundamental form is and ds 2 = 2λdz ⊗ dz. The holomorphic sectional curvature of G is Since dim M = 1, one has that P(T M) M and hence u is a smooth function on M. For the case of max z∈M u(z) = 0, the inequality (3.10) is obvious. So one may assume that max z∈M u(z) = u(z 0 ) > 0 for some z 0 ∈ M.
where infinimum is taken all curve of C 1 -class on M.
A complex manifold M is said to be Kobayashi hyperbolic if the pseudo-distance d K M is the distance in the strict sense.
As a simple application, we have the following interesting result.
where k N is the Kobayashi metric of N . In particular, N is Kobayashi hyperbolic.
Proof We equip the r -disc (r ) with the complete Poincaré metric G = 2λdz ⊗ dz, λ = r 2 2(r 2 −|z| 2 ) . It is well known that its Gaussian curvature is K G = −4 < 0. By Theorem 3.5, one has Therefore, By the definition of k N , one has From (3.20) and K 2 < 0, d K N is a distance in the strict sense. By Definition 3.6, N is Kobayashi hyperbolic.

Semi-positive Holomorphic Sectional Curvature
In this section, we assume that the holomorphic sectional curvature of a strongly pseudoconvex complex Finsler manifold (M, G) is semi-positive but not identically zero, i.e., K G ≥ 0 and there exists a point (z 0 , Firstly, we review Berndtsson's curvature formula of direct image bundles (cf. [7][8][9]20,21]).

Curvature of Direct Image Bundles
Let π : X → M be a holomorphic fibration with compact fibers, L a relative ample line bundle over X , i.e., there exists a metric (weight) φ of L such that √ −1∂∂φ| X z > 0 for any z ∈ M, X z := π −1 (z). We denote by (z; w) = (z 1 , . . . , z n ; w 1 , . . . , w m ) a local admissible holomorphic coordinate system of X with π(z; w) = z, where m denotes the dimension of fibers. For any smooth function φ on X , we denote where (φβ α ) is the inverse matrix of (φ αβ ). By a routine computation, one shows easily that { δ δz i } 1≤i≤n spans a well-defined horizontal subbundle of T X . Let {dz i ; δw α } denote the dual frame of { δ δz i ; ∂ ∂w α }. One has which is clearly a horizontal real (1, 1)-form on X . By a direct computation, one has We consider the direct image sheaf Then E is a holomorphic vector bundle. In fact, for any point p ∈ M, taking a local coordinate neighborhood (U ; {z i }) of p, then φ + β n i=1 |z i | 2 is a metric on the line bundle L → X | U , whose curvature is 208 X. Wan By taking β large enough, the curvature of φ + β n i=1 |z i | 2 is positive. By the same argument as in [7, §4, p. 542], there exists a local holomorphic frame for E. So E is a holomorphic vector bundle.
Following Berndtsson (cf. [7][8][9]), we define the following L 2 -metric on the direct image bundle E := π * (K X /M + L). Let {u A } 1≤A≤rankE be a local frame of E. Set (4.5) Note that u A can be written locally as where s L is a local holomorphic frame of L with e −φ = |s L | 2 , and so locally where dw = dw 1 ∧ · · · ∧ dw m is the fiber volume.

Application on Cotangent Bundles
In this subsection, we will apply Theorem 4.1 to the cotangent bundle E = T * M. In this case, X = P(T M), gives a local coordinate system of P(T M) by (1) denote the dual local holomorphic section of s O P(T M) (−1) . By (4.10), there exist local frame s L of L and local frame s π * det T M of π * det T M on U k such that (4.14) Let G be a strongly pseudoconvex complex Finsler metric on M, it induces a metric on O P(T M) (−1) by |v k | 2 on U k . Let g = (g ij ) be a Hermitian metric on M. From (4.14), there exists a metric φ L on L by where det g := det(g ij ).
Combining (4.15) with Lemma 4.2, one has on U k . It is known that H 0 (P n−1 , O P n−1 (k)) can be identified as the space of homogeneous polynomials of degree k in n variables. Therefore, the sections of H 0 (X z , O P(T M) (1)| X z ) are of the form On the other hand, there is a canonical element which is well defined since v i and ∂ ∂z i are covariant to each other. By (4.18), (4.19), and Theorem 4.1, we obtain Now we deal with the second term in the RHS of (4.20). By Theorem (4.1),  where the third equality holds since q * ( δ δz l ) = δ φ δz l .