Positively curved Finsler metrics on vector bundles II

We show that if $E$ is an ample vector bundle of rank at least two with some curvature bound on $O_{P(E^*)}(1)$, then $E^*\otimes \det E$ is Kobayashi positive. The proof relies on comparing the curvature of $(\det E^*)^k$ and $S^kE$ for large $k$ and using duality of convex Finsler metrics. Following the same thread of thought, we show if $E$ is ample with similar curvature bounds on $O_{P(E^*)}(1)$ and $O_{P(E\otimes \det E^*)}(1)$, then $E$ is Kobayashi positive. With additional assumptions, we can furthermore show that $E^*\otimes \det E$ and $E$ are Griffiths positive.


Introduction
Let E be a holomorphic vector bundle of rank r over a compact complex manifold X of dimension n.We denote the dual bundle by E * and its projectivized bundle by P(E * ).The vector bundle E is said to be ample if the line bundle O P(E * ) (1) over P(E * ) is ample.On the other hand, E is called Griffiths positive if E carries a Griffiths positive Hermitian metric.Moreover, E is called Kobayashi positive if E carries a strongly pseudoconvex Finsler metric whose Kobayashi curvature is positive (we will give a quick review on Finsler metrics and Kobayashi curvature in Section 2A; or see [Wu 2022, Section 2]).
There are two conjectures made by Griffiths [1969] and Kobayashi [1975] regarding the equivalence of ampleness and positivity: (1) If E is ample, then E is Griffiths positive.
By Kodaira's embedding theorem, ampleness of a line bundle is equivalent to the existence of a positively curved metric on the line bundle.So, the conjectures of Griffiths and Kobayashi can be rephrased: Given a positively curved metric on O P(E * ) (1), can we construct a positively curved Hermitian/Finsler metric on E? In this paper, we show that it is so, by imposing curvature bounds on tautological line bundles of P(E * ) and P(E).Since Hermitian metrics on O P(E * ) (1) are in one-to-one correspondence with Finsler metrics on E * , these curvature bounds can also be written in terms of Kobayashi curvature.
We first consider a relevant case where the picture is clearer.It is known that, for rank of E at least 2: (1) If E is Griffiths positive, then E * ⊗ det E with the induced metric is Griffiths positive.
(2) If E is ample, then E * ⊗ det E is ample.
The first fact can be found in [Demailly 2012, p. 346, Theorem 9.2] and the second in [Hartshorne 1966, Corollary 5.3] together with the isomorphism (see Appendix) If we follow the guidance of Griffiths and Kobayashi, we would ask whether or not the ampleness of E implies Griffiths/Kobayashi positivity of E * ⊗ det E for r ≥ 2. Our first result is that this can be achieved by imposing curvature bounds on O P(E * ) (1).
Let q : P(E * ) → X be the projection.Let g be a metric on O P(E * ) (1) whose curvature restricted to a fiber (g)| P(E * z ) is positive for all z ∈ X .For a tangent vector η ∈ T 1,0 z X and a point [ζ ] ∈ P(E * z ), we consider tangent vectors η to P(E * ) at (z, [ζ ]) such that q * ( η) = η, namely the lifts of η to T 1,0  (z,[ζ ]) P(E * ).Then we define the function where the infimum taken over all the lifts of η to T 1,0 (z, [ζ ]) P(E * ).This infimum is actually a minimum, see (2-3).On the other hand, since such a metric g corresponds to a strongly pseudoconvex Finsler metric on E * , and if we denote its Kobayashi curvature by θ(g) a (1, 1)-form on P(E * ), then The term on the right is independent of the choice of lifts η (we will prove (1-2) in Section 2A).
Theorem 1. Assume r ≥ 2 and the line bundle O P(E * ) (1) has a positively curved metric h and a metric g with (g)| P(E * z ) > 0 for all z ∈ X .If there exist a Hermitian metric on X and a constant M ∈ [1, r ) such that the following inequalities of (1, 1)-forms hold: We can of course choose g to be h in Theorem 1, but having two different metrics seems more flexible.The proof of Theorem 1 relies on two observations.First, starting with g and h on O P(E * ) (1), we construct two Hermitian metrics on S k E and det E respectively.The curvature of the induced metric on S k E ⊗ (det E * ) k can be shown to be Griffiths negative for k large (see Section 3 for details).The second observation (see [Wu 2022]) is that since the induced metric on S k E ⊗ (det E * ) k is basically an L 2 -metric, its k-th root is a convex Finsler metric on E ⊗ det E * which is also strongly plurisubharmonic on the total space minus the zero section.After perturbing this Finsler metric and taking duality, we get a convex and strongly pseudoconvex Finsler metric on E * ⊗ det E whose Kobayashi curvature is positive.So the bundle E * ⊗ det E is Kobayashi positive.Notice that the Finsler metric we find is actually convex.
The reason why we impose , M and inequalities (1-3) and (1-4) in Theorem 1 is the following.On the bundle With small changes on the proof, one can write down a variant of Theorem 1 where the conclusion is about the Kobayashi positivity of E * ⊗ (det E) l (see the end of Section 3).Now let us go back to the original conjecture of Kobayashi and adapt the proof of Theorem 1 to this case.Let p : P(E) → X be the projection.We recall under the canonical isomorphism P(E ⊗ det E * ) ≃ P(E), the line bundle O P(E⊗det E * ) (1) corresponds to the line bundle O P(E) (1) ⊗ p * det E (see [Kobayashi 1987, p. 86, Proposition 3.6.21]).Let g be a metric on O P(E) (1)⊗ p * det E with (g)| P(E z ) > 0 for all z ∈ X .For a tangent vector η ∈ T 1,0 z X and a point [ξ ] ∈ P(E z ), we similarly have where η ′ are the lifts of η to T 1,0 (z,[ξ ]) P(E).Meanwhile, such a metric g corresponds to a strongly pseudoconvex Finsler metric on E ⊗ det E * and we denote its Kobayashi curvature by θ(g) a (1, 1)-form on P(E).As before, (1-5) inf Theorem 2. Assume r ≥ 2 and O P(E * ) (1) has a positively curved metric h and O P(E) (1) ⊗ p * det E has a metric g with (g)| P(E z ) > 0 for all z ∈ X .If there exist a Hermitian metric on X and a constant M ∈ [1, r ) such that Since the ampleness of E implies ampleness of E * ⊗ det E, one choice for g in Theorem 2 is a positively curved metric on O P(E) (1) ⊗ p * det E, but how much this choice helps is unknown to us.The proof of Theorem 2 follows the same scheme as in Theorem 1.We first use h and g to construct Hermitian metrics on det E and ) k is Griffiths negative for k large (see Section 4).Then by taking k-th root, perturbing, and taking duality, we obtain a convex, strongly pseudoconvex, and Kobayashi positive Finsler metric on E.
1A. Griffiths positivity.The conclusions in Theorems 1 and 2 are about Finsler metrics.For their Hermitian counterpart, we need additional assumptions.The reason is that in Theorems 1 and 2, taking large tensor power of various bundles helps us eliminate the curvature of the relative canonical bundles K P(E * )/ X and K P(E)/ X , and after getting the desired estimates we take k-th root to produce Finsler metrics.However, the step of taking k-th root produces only Finsler, not Hermitian metrics.So the first step of taking large tensor power is not allowed if one wants Hermitian metrics.
Let us be more precise.For a metric g on O P(E * ) (1) with (g)| P(E * z ) > 0 for all z ∈ X , we denote (g)| P(E * z ) by ω z for the moment.The relative canonical bundle K P(E * )/ X has a metric induced from {ω r −1 z } z∈X and we denote the corresponding curvature by γ g , a (1, 1)-form on P(E * ).For η ∈ T 1,0 z X and where the supremum taken over all the lifts of η to T 1,0 (z,[ζ ]) P(E * ).The supremum is a maximum under a suitable assumption, see (2-9).Moreover, for z ∈ X , the restriction γ g | P(E * z ) is actually the negative of Ricci curvature −Ric ω z of the metric ω z on P(E * z ).Any Hermitian metric G on E * will induce a metric g on O P(E * ) (1) with (g)| P(E * z ) > 0 and γ g | P(E * z ) < 0 for all z ∈ X .Indeed, in this case, (g is the Fubini-Study metric and its Ricci curvature is positive, so γ g | P(E * z ) < 0. Furthermore, for any η ∈ T 1,0 z X and any [ζ ] ∈ P(E * z ), (1-8) sup (we will prove (1-8) in Section 2B).
Theorem 3. Assume r ≥ 2 and the line bundle O P(E * ) (1) has a positively curved metric h and a metric g induced from a Hermitian metric G on E * .If there exist a Hermitian metric on X and a constant M ∈ [1, r ) such that ⊗ det E is Griffiths positive.
Theorem 3 could be seen as a Hermitian analogue of Theorem 1.To state a Hermitian analogue of Theorem 2, we use again the isomorphism between Theorem 4. Suppose that r ≥ 2 and O P(E * ) (1) has a positively curved metric h and O P(E) (1) ⊗ p * det E has a metric g induced from a Hermitian metric G on E ⊗ det E * .If there exist a Hermitian metric on X and a constant M ∈ [1, r ) such that M p * ≥ −(r + 1) θ (g) + p * (det G), (1-11) In all the theorems above, the existence of the metric h comes from ampleness of E. So the real assumptions lie in (g, , M) and the inequalities they have to satisfy.To weaken or remove these inequalities, one possible direction is to use geometric flows as in [Naumann 2021;Wan 2022;Ustinovskiy 2019;Li et al. 2021].Another possible direction is to use the interplay between the optimal L 2 -estimates and the positivity of curvature (see [Guan and Zhou 2015;Berndtsson and Lempert 2016;Lempert 2017;Hacon et al. 2018;Zhou and Zhu 2018]).
One example where the assumptions of all the theorems above are satisfied is given by E = L 9 ⊕ L 8 ⊕ L 7 with L a positive line bundle.The triple (9, 8, 7) or the rank r = 3 is not that important; the point is to make sure the eigenvalues of the curvature with respect to some positive (1, 1)-form do not spread out too far.This example also indicates that a reasonable choice for is probably related to c 1 (det E).
A more sophisticated example, related to approximate Hermitian-Yang-Mills metrics [Jacob 2014;Misra and Ray 2021;Li et al. 2021], is semistable ample vector bundles over Riemann surfaces (see Section 7 for details of the examples).
The proof of Theorem 1 is given in Section 3 and almost as a corollary we prove Theorem 2 in Section 4. The proof of Theorem 3 in Section 5 is a modification of Theorem 1 but we still write out the details.In Section 6, we prove Theorem 4 based on Section 5.

Preliminaries
2A. Finsler metrics.We will use some facts about Finsler metrics on vector bundles which can be found in [Kobayashi 1975;1996;Cao and Wong 2003;Aikou 2004;Wu 2022].First, we recall the definition of Finsler metrics.Let E * be a holomorphic vector bundle of rank r over a compact complex manifold X .For a vector ζ ∈ E * z , we symbolically write (z, ζ ) ∈ E * .A Finsler metric G on the vector bundle E * → X is a real-valued function on E * such that: (1) G is smooth away from the zero section of E * . (

and equality holds if and only if
A Finsler metric G on E * is said to be: (1) Strongly pseudoconvex if the fiberwise complex Hessian of G is positive definite on E * \ {zero section}, namely ( we can define a Hermitian metric G on the pull-back bundle q * E * , where q : P(E * ) → X is the projection, as follows.For a vector Z in the fiber , where the Z on the right-hand side is viewed as a tangent vector to E * z at ζ by the identification of vector spaces q ) is a Hermitian holomorphic vector bundle, so we can talk about its Chern curvature , an End q * E * -valued (1, 1)-form on P(E * ).With respect to the metric G, the bundle q * E * has a fiberwise orthogonal decomposition and so can be written as a block matrix.Let | O P(E * ) (−1) denote the block in the matrix corresponding to End(O P(E * ) (−1)).Since O P(E * ) (−1) is a line bundle, ), and it is called the Kobayashi curvature of the Finsler metric G.We will use θ (g) to denote the Kobayashi curvature In order to relate the Kobayashi curvature θ (g) to the curvature (g) of g, we consider coordinates normal at one point.Given a point (z 0 , [ζ 0 ]) ∈ P(E * ), there exists a holomorphic frame where we use {ζ i } for the fiber coordinates on E * with respect to the frame {s i } and {z α } for the local coordinates on X (such a frame can be obtained by (5.11) in [Kobayashi 1996]).Moreover if is a Hermitian metric on X , then by a linear transformation in the z-coordinates, we can make without affecting (2-2).We will call this coordinate system normal at the point Around the point (z 0 , [ζ 0 ]) ∈ P(E * ), we assume the local coordinates (z 1 , . . ., z n , w 1 , . . ., w r −1 ) are given by ) and g(e * , e * ) = e −φ .Then, the curvature (g) can be written locally as Note that the terms ∂ 2 φ ∂z α ∂w j := φ α j vanish at (z 0 , [ζ 0 ]) by (2-2) and the fact For a tangent vector η ∈ T 1,0 z 0 X , we can write η = α η α ∂ ∂z α .For the lifts η of η to T 1,0 (z 0 ,[ζ 0 ]) P(E * ), we have ) and the matrix (φ i j ) is positive.On the other hand, using the same coordinate system, the curvature of G can be written as where R α β , P α l , ᏼ k β , and Q k l are endomorphisms of q * E * .By [Wu 2022, (2.4)], for any lift η of η to T 1,0 where the last equality is by [Kobayashi 1996, (5.16)].
2B. Hermitian metrics.This subsection is a special case of Section 2A and it will be used in the proofs of Theorems 3 and 4. Let G be a Hermitian metric on the bundle E * .The pull-back bundle q * E * → P(E * ) with the pull-back metric q * G induces a metric g * on the subbundle O P(E * ) (−1).We denote the dual metric on O P(E * ) (1) by g.
Let be a Hermitian metric on X and z 0 a point in X with local coordinates There exists a holomorphic frame {s i } for E * around z 0 such that where z 0 corresponds to the origin in the local coordinates.We use {ζ i } for the fiber coordinates with respect to the frame {s i }.For a point (z 0 , [ζ 0 ]) ∈ P(E * ), we assume the local coordinates (z 1 , . . ., z n , w 1 , . . ., w r −1 ) around (z 0 , [ζ 0 ]) are given by is a holomorphic frame for O P(E * ) (−1) and The z α -derivative of g * (e, e) is g * (e, e) z α = O((1 + |w| + |w| 2 )|z|), and hence the w i -derivatives of g * (e, e) z α of any order are zero when evaluated at z 0 .Therefore, if we denote g * (e, e) by e φ , then at z 0 (2-5) φ α j = φ αi j = φ αi j k = 0 and (log det(φ i j )) α k = 0.
In this coordinate system, the curvature (g) is For a tangent vector η ∈ T 1,0 z 0 X , we can write η = α η α ∂ ∂z α .For the lifts η of η to T 1,0 because φ α j = 0 at z 0 and the matrix (φ i j ) is positive.Since G is a Hermitian metric, the corresponding Kobayashi curvature is which is equal to the negative of (2-6) by Section 2A.
Using the same coordinate system, the restriction (g The matrix (log det(φ i j )) i j is negative because it represents the negative of the Ricci curvature of the Fubini-Study metric on P(E * z ).Moreover, the terms (log det(φ i j )) α j = 0 at z 0 by (2-5).As a result, for a tangent vector η ∈ T 1,0 z 0 X with η = η α ∂ ∂z α in this coordinate system, we have where η are the lifts of η to T 1,0 Finally, the metric on K P(E * )/ X induced from {( (g)| P(E * z ) ) r −1 } z∈X can be identified with the metric (g * ) r ⊗ q * (det G * ) under the isomorphism (see [Kobayashi 1987, p. 85, Proposition 3.6.20]).This fact can be verified at one point using the normal coordinates above.Therefore, (2-10) So, for any η ∈ T 1,0 z X and any This is formula (1-8) that we promise to prove in the introduction.
2C. Convexity.Let E be a holomorphic vector bundle of rank r over a compact complex manifold X .Given a Hermitian metric H k on the symmetric power S k E, we can define a Finsler metric on E by assigning to u ∈ E length H k (u k , u k ) 1/2k .We will denote this Finsler metric by H Lemma 5. Let F 1 be a vector bundle and F 2 a line bundle over X .Assume F 2 carries a Hermitian metric H .We also assume, for some k, S k F 1 carries a Hermitian metric H k such that the induced Finsler metric H Then the Finsler metric Since F 2 is a line bundle, there is a canonical isomorphism between the bundles S k (F 1 ⊗ F 2 ) and S k F 1 ⊗ F k 2 which we use implicitly in the statement of Lemma 5. Roughly speaking, Lemma 5 indicates that convexity is not affected by tensoring with a line bundle.
Proof.Fix p ∈ X .The fiber F 2 | p is a one dimensional vector space and we let e be a basis.For x and y ∈ F 1 ⊗ F 2 | p , we can write x = x ⊗ e and y = ỹ ⊗ e where Therefore the Finsler metric Direct image bundles.We recall how to construct Hermitian metrics on direct image bundles and compute their curvature.Let g be a Hermitian metric on O P(E * ) (1) with curvature (g).Denote the restriction of the curvature to a fiber, (g)| P(E * z ) by ω z for z ∈ X , and assume ω z > 0 for all z ∈ X .With the canonical isomorphism (see [Demailly 2012, p. 278, Theorem 15.5]), we define a Hermitian metric Let us denote by k the curvature of H k .Fixing z ∈ X and u ∈ S k E z , in order to estimate the (1, 1)-form H k ( k u, u), we first extend the vector u to a local holomorphic section ũ whose covariant derivative at z with respect to H k equals zero.A straightforward computation shows But H k ( ũ, ũ)(z) for z near z can also be written as the push-forward where q : P(E * ) → X is the projection, so Similarly, we can use a metric on O P(E) (1) ⊗ p * det E to construct Hermitian metrics on S k E * ⊗ (det E) k .The formula is similar to (2-11), and we use bold symbols to highlight the change.Let g be a metric on O P(E) (1) ⊗ p * det E with curvature (g).Denote the restriction of the curvature to a fiber (g)| P(E z ) by ω z for z ∈ X .Assume ω z > 0 for all z ∈ X .With the canonical isomorphism We also have a curvature formula similar to (2-12).
2E. Berndtsson's positivity theorem.Let h be a metric on O P(E * ) (1) with curvature (h) > 0. Denote (h)| P(E * z ) by ω z for z ∈ X .We are going to define a Hermitian metric on det E using the metric h.The relative canonical bundle K P(E * )/ X has a metric induced from {ω r −1 z } z∈X .With h r on O P(E * ) (r ) and the isomorphism K P(E * )/ X ⊗ O P(E * ) (r ) ≃ q * det E, there is an induced metric ρ on q * det E. Using the canonical isomorphism we define a Hermitian metric H on det E by By Berndtsson's theorem [Berndtsson 2009a], this metric H is Griffiths positive, but it is the inequality that leads to this fact we will use.We follow the presentation in [Liu et al. 2013, Section 4.1] (see also [Berndtsson 2009b, Section 2]).Denote the curvature of H by .Fix z ∈ X , v ∈ det E z and η ∈ T 1,0 z X .For a local holomorphic frame of E * around z, we denote by {ζ i } the fiber coordinates with respect to this frame, and by {z α } the local coordinates on X .Around P(E * z ) in P(E * ), we have homogeneous coordinates [ζ 1 , . . ., ζ r ] which induce local coordinates (w 1 , . . ., w r −1 ).For a local frame e * of O P(E * ) (1), we denote h(e * , e * ) by e −φ and write the tangent vector η = η α ∂ ∂z α .The inequality that leads to Berndtsson's theorem is where and (φ i j ) is the inverse matrix of (φ i j ).Since det E is a line bundle, the curvature is a (1, 1)-form, and so If we further assume H (v, v) = 1, then the left-hand side of (2-15) becomes − (η, η).

Proof of Theorem 1
Recall that h and g are metrics on O P(E * ) (1) that satisfy the assumptions in Theorem 1 and the inequalities (1-3) and (1-4).We use the metric h to construct a Hermitian metric H on det E as in (2-14), and the metric g to construct Hermitian metrics H k on S k E as in (2-11).The number k is yet to be determined.
We start with the metric g.Given a point (z 0 , [ζ 0 ]) ∈ P(E * ), we have the normal coordinate system from Section 2A.In this coordinate system, let us introduce the following n-by-n matrix-valued function: where g(e * , e * ) = e −φ .By continuity, there is a neighborhood U of (z 0 , For this U , there is a positive integer k 0 such that for k ≥ k 0 and in U Let us summarize what we have done so far: Lemma 6.Given a point (z 0 , [ζ 0 ]) ∈ P(E * ), there exist a coordinate neighborhood U of (z 0 , [ζ 0 ]) in P(E * ) and a positive integer k 0 such that in U and for k ≥ k 0 .
By Lemma 6, since P(E * z 0 ) is compact, we can find on P(E * z 0 ) finitely many points {(z 0 , [ζ l ])} l each of which corresponds to a coordinate neighborhood U l in P(E * ) and a positive integer k l such that the corresponding (3-3) holds, and P(E * z 0 ) ⊂ l U l .Denote max l k l by k max .The point z 0 has a neighborhood W in X such that for z ∈ W , the fiber P(E * z ) can be partitioned as m V m with each V m in U l for some l.By shrinking W , we can assume that for each U l the corresponding where ε := r −M 5(r +M) .Recall the Hermitian metrics H k on S k E in (2-11) constructed using the metric g.Denote by k the curvature of H k .We claim the following lemma (one can also use the asymptotic expansion in [Ma and Zhang 2023] to deduce the lemma).
Proof.As in Section 2D, we extend the vector u ∈ S k E z to a local holomorphic section ũ whose covariant derivative at z equals zero, and we have In the last equality, we partition the fiber P(E * z ) as m V m with each V m in U l for some l.In a fixed V m ⊂ U l , using the coordinate system of U l , we can write k,z ( ũ) as f (e * ) k with f a scalar-valued holomorphic function and e * a local frame for Meanwhile, recall the curvature (g) = ∂ ∂φ.By Stokes' theorem and a count on degrees, we have Note that the integrands in (3-6) are written in the local coordinates of corresponding U l .A direct computation shows Using the coordinate system of U l , the tangent vector η = α η α ∂ ∂z α at z induces a tangent vector by ηl .According to (1-2), (1-3), and (2-3), we see Therefore, (3-7) becomes , where we use (3-4) in the second inequality.So, (3-6) becomes We turn now to the metric h.The argument about h is similar to that about g, and it will be used in Theorems 2, 3, and 4. Given a point (z 0 , [ζ 0 ]) ∈ P(E * ), we have the normal coordinate system from Section 2A with respect to the metric h.In this coordinate system, let us introduce the n-by-n matrix-valued function where h(e * , e * ) = e −φ and (φ i j ) is the inverse matrix of (φ i j ).By continuity, there is a neighborhood U of (z 0 , In summary: By Lemma 8, since P(E * z 0 ) is compact, we can find on P(E * z 0 ) finitely many points {(z 0 , [ζ l ])} l each of which corresponds to a coordinate neighborhood U l in P(E * ) such that the corresponding (3-12) holds, and P(E * z 0 ) ⊂ l U l .The point z 0 has a neighborhood W ′ in X such that for z ∈ W ′ , the fiber P(E * z ) can be partitioned as m V m with each V m in U l for some l.By shrinking W ′ , we can assume that for each U l the corresponding where ε := r −M 5(r +M) .Recall the Hermitian metric H on det E in (2-14) constructed using the metric h.Denote by the curvature of H .We claim: Lemma 9.For z ∈ W ′ and η ∈ T 1,0 z X , we have Proof.Using (2-15) and assuming where we again partition P(E * z ) as m V m with each V m in U l for some l.Note that the integrands in (3-15) are written in the local coordinates of corresponding U l .In a fixed V m ⊂ U l , we have η = α η α ∂ ∂z α , and by (3-12) we see In U l , the tangent vector η = α η α ∂ ∂z α at z induces a tangent vector Denote the lifts of η l to T 1,0 (z 0 ,[ζ l ]) P(E * ) by ηl .By (1-2), (1-4), and (2-3), we see Therefore, (3-16) becomes where we use (3-13) in the second inequality.So, (3-15) becomes -11), and H on det E in (2-14).Since (det E * ) k is a line bundle, we can identify End(S k E ⊗ (det E * ) k ) with End(S k E), and the curvature of the metric where k and are the curvature of H k and H respectively.We claim that for k ≥ k max and in W ∩ W ′ a neighborhood of z 0 , the metric H k ⊗ (H * ) k is Griffiths negative.Indeed, as a result of Lemmas 7 and 9, for The term on the right is negative after some computation using ε = r −M 5(r +M) .So, we have proved the claim that for k ≥ k max and in W ∩ W ′ ⊂ X , the metric H k ⊗ (H * ) k is Griffiths negative.Since X is compact, H k ⊗ (H * ) k is Griffiths negative on the entire X for k large enough.Now we fix k such that the Hermitian metric H k ⊗ (H * ) k on the bundle is Griffiths negative on X .The Hermitian metric H k by construction is an L 2integral, so its k-th root is a convex Finsler metric on E (see [Wu 2022, proof of Theorem 1] for details).By Lemma 5, the k-th root of H k ⊗(H * ) k is a convex Finsler metric on E ⊗ det E * which we denote by F.Moreover, this Finsler metric F is strongly plurisubharmonic on E ⊗ det E * \ {zero section} due to Griffiths negativity of H k ⊗ (H * ) k .By adding a small Hermitian metric, we can assume F is strongly convex and strongly plurisubharmonic.
In general, the Kobayashi curvature of Finsler metrics do not behave well under duality [Demailly 1999, Remark 2.7].But since our Finsler metric F is strongly convex, the dual Finsler metric of F is in fact strongly pseudoconvex and Kobayashi positive (this duality result is originally due to Sommese [1978] and Demailly [1999, Theorem 2.5].See also [Wu 2022, proof of Theorem 1 and Lemma 6]).In summary, the dual Finsler metric of F is a convex, strongly pseudoconvex, and Kobayashi positive Finsler metric on E * ⊗ det E. Hence the proof of Theorem 1 is complete.With slight modification on the proof, one has the following variant of Theorem 1.
Theorem 10.Assume r ≥ 2 and the line bundle O P(E * ) (1) has a positively curved metric h and a metric g with (g)| P(E * z ) > 0 for all z ∈ X .If there exist a Hermitian metric on X and a constant M ≥ 1 such that the following inequalities of (1, 1)forms hold Mq * ≥ −θ (g), (3-18) then for any positive integer l > M/r , the bundle E * ⊗(det E) l is Kobayashi positive.

Proof of Theorem 2
The proof is similar to what we do in Section 3 except that we are dealing with not only P(E * ) but P(E) here.The metric h is used to define a Hermitian metric H on det E as in (2-14).The metric g is used to define Hermitian metrics H k on S k E * ⊗ (det E) k as in (2-13).Fix z 0 in X .For the metric h on O P(E * ) (1), we follow the path that leads to Lemma 9 in Section 3 to deduce a neighborhood W ′ of z 0 in X such that for z ∈ W ′ and η ∈ T 1,0 z X , the curvature of H satisfies (4-1) with ε = r −M 5(r +M) .For the metric g on O P(E) (1) ⊗ p * det E, we replace O P(E * ) (1) → P(E * ) in Section 3 with O P(E⊗det E * ) (1) → P(E ⊗det E * ) and use the canonical isomorphism between O P(E⊗det E * ) (1) → P(E ⊗det E * ) and O P(E) (1)⊗ p * det E → P(E).Then following the argument leading to Lemma 7, we obtain a positive integer k max and a neighborhood W of z 0 in X such that for k As a result of (4-1) and (4-2), we deduce that, for k In a fixed V m ⊂ U l , we can write 1,z ( ũ) as f e * with f a scalar-valued holomorphic function and e * a local frame for O P(E * ) (1).So, Meanwhile, recall the curvature (g) = ∂ ∂φ.By Stokes' theorem and a count on degrees, we have By (1-8), (1-9), (2-6), and (2-9), we see as in Lemma 12, we get for η ∈ T 1,0 z 0 X , and On the bundle (E * ⊗ det E) ⊗ det E * , there is a Hermitian metric H 1 ⊗ H * with curvature 1 − ⊗ Id E * ⊗det E .As a result of (6-1) and (6-2), we deduce that for η ∈ T 1,0 z 0 X , and the term on the right is negative.So the Hermitian metric H 1 ⊗ H * is Griffiths negative at z 0 an arbitrary point.Hence H 1 ⊗ H * is Griffiths negative on X , and the bundle E is Griffiths positive.
For all four theorems, we will use this metric h on O P(E * ) (1) and take to be 7 .So q * ≤ −θ(h) always holds.The choice of g will be different from case to case.For Theorem 1, we choose g to be h, and hence by (7-1) and = 7 we get (7-2) q * ≤ −θ (h) = −θ (g) ≤ 9 7 q * .To fulfill the assumption of Theorem 1, we can choose M = 9 7 which is in the interval [1, 3).
We choose M = 12 7 which is in [1, 3).We choose M = 20 7 which is in [1, 3).Example 14.Let X be a compact Riemann surface with a Hermitian metric ω.Let E be an ω-semistable ample vector bundle of rank r over X .The assumptions in Theorems 1, 2, 3, and 4 are all satisfied in this case.We will explain for only Theorems 2 and 4. Theorems 1 and 3 can be verified similarly.By [Li et al. 2021, Theorem 1.7, Remark 1.8, and Theorem 1.11], there exists a constant c > 0 such that for any δ > 0, there exists a Hermitian metric H δ on E satisfying (7-8) where ω is the contraction with respect to ω.Since X is a Riemann surface, ω locally is multiplication by a positive function.
For Theorem 4, we choose δ = c 9r .We still have (7-10).The Hermitian metric G on E ⊗ det E * is taken to be H δ ⊗ det H * δ , so we get So the assumption of Theorem 4 is satisfied.
In light of [Li et al. 2021, Theorem 1.7], it is possible to modify our theorems so that semistability is not needed in this example.
the curvature of the induced metric is roughly bounded above by k m a m c m − r k m b m c m where a m and b m are some positive integrals with m a m = m b m = 1, and c m are positive numbers related to the curvature of h.It does not seem possible to us that the upper bound k m a m c m − r k m b m c m can be made negative without any assumption.So we introduce and M to control the upper bound.