# Positively curved Finsler metrics on vector bundles

Kuang-Ru Wu

###### Abstract

We construct a convex and strongly pseudoconvex Kobayashi positive Finsler
metric on a vector bundle $E$ under the assumption that the symmetric power of
the dual $S^{k}E^{*}$ has a Griffiths negative $L^{2}$-metric for some $k$.
The proof relies on the negativity of direct image bundles and the Minkowski
inequality for norms. As a corollary, we show that given a strongly
pseudoconvex Kobayashi positive Finsler metric, one can upgrade to a convex
Finsler metric with the same property. We also give an extremal
characterization of Kobayashi curvature for Finsler metrics.

##  1 Introduction

Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex
manifold $X$ of dimension $n$. Denote by $E^{*}$ the dual bundle of $E$. Let
$P(E^{*})$ be the projectivized bundle of $E^{*}$, $O_{P(E^{*})}(-1)$ the
tautological line bundle over $P(E^{*})$, and denote by $O_{P(E^{*})}(1)$ the
dual bundle of $O_{P(E^{*})}(-1)$. A vector bundle $E$ is said to be ample if
the line bundle $O_{P(E^{*})}(1)$ is ample, see [Har66].

Meanwhile, for a Hermitian holomorphic vector bundle $(E,H)$, Griffiths
[Gri69] introduced a notion of positivity using the curvature of the metric
$H$, which is now called Griffiths positivity. When $E$ is a line bundle, the
existence of a Griffiths positive Hermitian metric on $E$ is equivalent to the
ampleness of $E$; essentially by the Kodaira embedding theorem. When $E$ is of
higher rank, the existence of a Griffiths positive metric implies ampleness
(see e.g. [Kob87, p. 89] or [Dem12, p. 283]); however, the converse statement
known as the Griffiths conjecture is still open (see earlier results [Ume73,
CF90] and recent developments [Nau17, Dem20, Pin21, Fin21]).

On the other hand, Kobayashi [Kob75] defined a notion of curvature for
strongly pseudoconvex Finsler metrics on holomorphic vector bundles, and he
showed that the dual bundle $E^{*}$ is ample if and only if $E$ has a strongly
pseudoconvex Kobayashi negative Finsler metric. Moreover, he raised the
question whether the ampleness of $E$ is equivalent to the existence of a
strongly pseudoconvex Kobayashi positive Finsler metric on $E$. One direction
is proved in [FLW18] where the authors show the existence of such a Finsler
metric on $E$ implies the ampleness of $E$.

The motivation of this paper is twofold. One is the remaining part of the
Kobayashi conjecture, and the other is whether Kobayashi’s conjecture implies
Griffiths’ conjecture. Although we do not solve either of the problems, some
progress is made in the results below. Before we state our results, we recall
a natural way to produce Hermitian metrics on direct image bundles.

Let $h$ be a Hermitian metric on $O_{P(E)}(1)$ and $\\{\omega_{z}\\}_{z\in X}$
be a family of Hermitian metrics on the fibers $P(E_{z})$. Given a positive
integer $k$, there is a canonical isomorphism

$\Phi_{k,z}:S^{k}E^{*}_{z}\to H^{0}(P(E_{z}),O_{P(E_{z})}(k)),$

see for example [Dem12, p. 278, Theorem 15.5]. We can define an $L^{2}$-metric
$H_{k}$ on $S^{k}E^{*}$ using $h^{k}$ and $\\{\omega_{z}\\}$

(1.1) $\displaystyle
H_{k}(u,v):=\int_{P(E_{z})}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(v))\omega_{z}^{r-1},\textup{
for $u\textup{ and }v\in S^{k}E^{*}_{z}$.}$

######  Theorem 1.

If the Hermitian metric $H_{k}$ in (1.1) is Griffiths negative for some $k$,
then $E^{*}$ admits a convex Finsler metric that is strongly plurisubharmonic
on $E^{*}\smallsetminus\\{\text{zero section}\\}$; moreover, the bundle $E$
admits a convex and strongly pseudoconvex Finsler metric with positive
Kobayashi curvature.

The idea behind the proof of Theorem 1 is straightforward. The $k$-th root of
$H_{k}$ induces a Finsler metric $F_{k}$ on $E^{*}$. This Finsler metric is
convex by the Minkowski inequality, and it is strongly plurisubharmonic on
$E^{*}\smallsetminus\\{\text{zero section}\\}$ by Griffiths negativity. The
convexity of $F_{k}$ is of great importance (see [Dem99, Remark 2.7] for the
necessity of convexity). Adding a small Hermitian metric to $F_{k}$ and taking
the dual, the resulting Finsler metric on $E$ is what we want (see Section 3
for details).

Whether the assumption in Theorem 1 can be deduced from ampleness of $E$ is
still unknown (of course, an affirmative answer would solve Kobayashi’s
conjecture). Nonetheless, we emphasize that the ampleness of $E$ does imply
$S^{k}E^{*}$ has a Griffiths negative Hermitian metric $\mathcal{H}_{k}^{*}$
for $k$ large; such a Hermitian metric is the dual of $\mathcal{H}_{k}$ on
$S^{k}E$ constructed as in (1.1) except one works on the dual side
$O_{P(E^{*})}(1)$ (this is the well-known positivity of direct image bundles,
see [Ber09a, MT07, MT08, LSY13, LY14]). However the $k$-th root of
$\mathcal{H}^{*}_{k}$ is not convex (see Section 4 for examples). This lack of
convexity is why we work on $O_{P(E)}(1)$ and use negativity of direct image
bundles (see also [Ber09b, Section 3]). Another natural approach is to take
the $k$-th root of $\mathcal{H}_{k}$. The resulting Finsler metric on $E$ is
convex by Minkowski’s inequality, but it is unclear if this metric is
Kobayashi positive. (One can perhaps utilize the intimate connection between
the optimal $L^{2}$ estimates/extensions and the positivity of curvature e.g.
[GZ15, BL16, HPS18, DWZZ18, ZZ18]).

One case in which the assumption in Theorem 1 holds is when $E$ has a strongly
pseudoconvex Finsler metric with positive Kobayashi curvature. Indeed, such a
Finsler metric induces a Hermitian metric $h$ on $O_{P(E)}(1)$ whose curvature
has signature $(r-1,n)$ (see Lemma 5). Then following the observation in
[LSY13, Section 4.2], the curvature of $H_{k}$ in (1.1) is Griffiths negative
for $k$ large. We therefore have

######  Corollary 2.

If $E$ has a strongly pseudoconvex Finsler metric with positive Kobayashi
curvature, then $E$ admits a convex and strongly pseudoconvex Finsler metric
with positive Kobayashi curvature.

Assuming Kobayashi’s conjecture is true, Corollary 2 could be seen as a small
step towards Griffiths’ conjecture in the sense that the Finsler metric can be
made convex, namely a norm, on each fiber (even if not a norm induced by an
inner product).

Using Theorem 1 and Corollary 2, we can give a different proof for the known
direction in Kobayashi’s conjecture ([FLW18]). Indeed, given a strongly
pseudoconvex Kobayashi positive Finsler metric on $E$, we can construct a
Finsler metric on $E^{*}$ which is strongly plurisubharmonic on
$E^{*}\smallsetminus\\{\text{zero section}\\}$, so the induced metric on
$O_{P(E^{*})}(-1)$ has negative curvature, thus $E$ is ample.

We will go over the basics on Kobayashi curvature of Finsler metrics in
Section 2. Theorem 1 and Corollary 2 are proved in Section 3. In Section 4, we
give explicit examples about convexity of inner products. In the last section,
we give an extremal characterization of Kobayashi curvature following [Lem15,
Lem17], where Lempert defined curvature for a much broader class of metrics.

I am indebted to László Lempert for many critical comments and suggestions.
Part of the paper was done at Purdue University, and I would like to thank the
support by NSF grant DMS-1764167. I am grateful to I-Hsun Tsai and Xiaokui
Yang for their interest in this work. Thanks are also due to Academia Sinica
and National Center for Theoretical Sciences for providing a stimulating
environment.

##  2 Kobayashi curvature of Finsler metrics

There is a sizable literature for Finsler metrics on vector bundles, for
example [Kob75, Kob96, Aik98, Aik04, CW03, FLW16, FLW17]. Therefore, we will
be very brief on the review; one can consult the above articles for details.
We will use [Kob96], an updated version of [Kob75].

###  2.1 Finsler metrics

Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex
manifold $X$ of dimension $n$. For a vector $\zeta\in E_{z}$, we will
symbolically write $(z,\zeta)\in E$. A smooth Finsler metric $F$ on the vector
bundle $E\to X$ is a real-valued function on $E$ such that

$\displaystyle(1)\text{ $F$ is continuous on $E$ and smooth away from the zero
section of $E$}.$ $\displaystyle(2)\text{ For $(z,\zeta)\in E$},F(z,\zeta)\geq
0,\text{ and equality holds if and only if $\zeta=0$}.$
$\displaystyle(3)\text{ $F(z,\lambda\zeta)=|\lambda|F(z,\zeta)$, for
$\lambda\in\mathbb{C}$}.$

Since $F^{2}$ will be used more often than $F$ in our computations, we denote
$F^{2}$ by $G$ throughout the paper.

Denote by $P(E)$ the projectivized bundle of $E$, and by $O_{P(E)}(-1)$ the
tautological line bundle over $P(E)$. Let $p$ be the projection from $P(E)$ to
$X$, and denote the pull-back bundle $p^{*}E$ by $\tilde{E}$. In summary,

${O_{P(E)}(-1)\subset\tilde{E}}$${E}$${P(E)}$${X.}$$\scriptstyle{\tilde{p}}$$\scriptstyle{p}$

For a vector $\zeta\in E_{z}$, we denote by $[\zeta]$ the equivalence class of
$\zeta$ in $P(E_{z})$, and we will write $(z,[\zeta])\in P(E)$. The pull-back
bundle $\tilde{E}=\coprod_{(z,[\zeta])\in P(E)}(z,[\zeta])\times E_{z}$, and
we will denote an element in $\tilde{E}$ by $(z,[\zeta],Z)$ with $Z\in E_{z}$.
There is a one-to-one correspondence between smooth Finsler metrics on $E$ and
smooth Hermitian metrics on $O_{P(E)}(-1)$, furnished by $\tilde{p}$. If $F$
is a Finsler metric on $E$ with $G=F^{2}$ and we denote the corresponding
Hermitian metric on $O_{P(E)}(-1)$ by $h_{G}$, then the correspondence is

(2.1) $h_{G}(z,[\zeta],\zeta)=G(z,\zeta).$

We use $(z_{1},...,z_{n})$ for local coordinates on $X$, and
$(\zeta_{1},...,\zeta_{r})$ for local fiber coordinates on $E$ with respect to
a holomorphic frame $\\{s_{1},...,s_{r}\\}$. So
$(z_{1},...,z_{n},\zeta_{1},...,\zeta_{r})$ is a coordinate system on $E$. Let
$F$ be a Finsler metric on $E$ with $G=F^{2}$. We write

$\displaystyle G_{i}=\partial G/\partial\zeta_{i}\,,\text{ }\text{
}G_{\bar{j}}=\partial G/\partial\bar{\zeta}_{j}\,,\text{ }\text{
}G_{i\bar{j}}=\partial^{2}G/\partial\zeta_{i}\partial\bar{\zeta}_{j}\,,$
$\displaystyle G_{i\alpha}=\partial G_{i}/\partial z_{\alpha}\,,\text{ }\text{
}G_{i\bar{j}\bar{\beta}}=\partial
G_{i\bar{j}}/\partial\bar{z}_{\beta}\,,\text{ etc.,}$

with Latin letters $i,j$ for the fiber direction and Greek letters
$\alpha,\beta$ for the base direction. The following formulas will be used
later; their derivation can be found in [Kob96, p. 147]. For
$\lambda\in\mathbb{C}$,

(2.2) $\displaystyle G_{k\bar{j}}(z,\lambda\zeta)=G_{k\bar{j}}(z,\zeta).$
(2.3) $\displaystyle
G(z,\zeta)=\sum_{i}G_{i}(z,\zeta)\zeta_{i}=\sum_{i,j}G_{i\bar{j}}(z,\zeta)\zeta_{i}\bar{\zeta}_{j}.$

###  2.2 Strongly pseudoconvex Finsler metrics

A Finsler metric $F$ is said to be

$\displaystyle(1)$ $\displaystyle\text{ strongly pseudoconvex if
$(F_{i\bar{j}})$ is positive definite on $E\smallsetminus\\{\text{zero
section}\\}$}.$ $\displaystyle(2)$ $\displaystyle\text{ convex if $F$
restricted to each fiber $E_{z}$ is convex},$ $\displaystyle\text{namely the
fiberwise real Hessian of $F$ is positive semidefinite on
$E\smallsetminus\\{\text{zero section}\\}$}.$ $\displaystyle(3)$
$\displaystyle\text{ strongly convex if the fiberwise real Hessian of $G$ is
positive on $E\smallsetminus\\{\text{zero section}\\}$}.$

Note that in (1), the requirement $(F_{i\bar{j}})>0$ is equivalent to
$(G_{i\bar{j}})>0$, which can be shown by using the formulas in [Kob96, (3.3)
and (3.10)]. A Hermitian metric always satisfies (1), (2), and (3). Also
notice that a ’convex’ Finsler metric in [Kob75] is what we call ’strongly
pseudoconvex’ here. Let us recall a lemma of strong pseudoconvexity ([Kob96,
p. 150 Theorem (1)]).

######  Lemma 3.

Let $F$ be a Finsler metric on $E$ and $h_{G}$ be its corresponding Hermitian
metric on $O_{P(E)}(-1)$. Then $F$ is strongly pseudoconvex if and only if the
curvature $\Theta(h_{G})$ of $h_{G}$ restricted to each fiber of $P(E)\to X$
is negative definite.

In the rest of this subsection, we will assume the Finsler metric $F$ is
strongly pseudoconvex. There is a natural Hermitian metric $\tilde{G}$ on the
pull-back bundle $\tilde{E}$ (see [Kob75]). In terms of local coordinates, the
Hermitian metric $\tilde{G}$ is given by

$\tilde{G}_{(z,[\zeta])}(Z,Z)=\sum_{i,j}G_{i\bar{j}}(z,\zeta)Z_{i}\bar{Z}_{j},\text{
for $Z=\sum^{r}_{i=1}Z_{i}s_{i}(z)\in E_{z}$}.$

By (2.3), the metric $\tilde{G}$ restricted to $O_{P(E)}(-1)$ is the same as
the Hermitian metric $h_{G}$ on $O_{P(E)}(-1)$ coming from $G$ on $E$ through
the one-to-one correspondence (2.1).

Now $(\tilde{E},\tilde{G})$ is a Hermitian holomorphic vector bundle, so we
can talk about its Chern curvature $\Theta$, an
$\operatorname{End}\tilde{E}$-valued $(1,1)$-form on $P(E)$. With respect to
the metric $\tilde{G}$, the bundle $\tilde{E}$ has a fiberwise orthogonal
decomposition $O_{P(E)}(-1)\oplus O_{P(E)}(-1)^{\perp}$, and so $\Theta$ can
be written as a block matrix. Let $\Theta|_{O_{P(E)}(-1)}$ denote the block in
the matrix $\Theta$ corresponding to $\operatorname{End}(O_{P(E)}(-1))$. Since
$O_{P(E)}(-1)$ is a line bundle, $\Theta|_{O_{P(E)}(-1)}$ is a $(1,1)$-form on
$P(E)$.

######  Definition 4.

The $(1,1)$-form $\Theta|_{O_{P(E)}(-1)}$ is called the Kobayashi curvature of
the strongly pseudoconvex Finsler metric $F$. Kobayashi positivity (or
negativity) means the positivity (or negativity) of $\Theta|_{O_{P(E)}(-1)}$.

Following the computation in [Kob96, Section 5], we have

######  Lemma 5.

Let $F$ be a strongly pseudoconvex Finsler metric on $E$ and $h_{G}$ be the
corresponding Hermitian metric on $O_{P(E)}(-1)$. Then $F$ is Kobayashi
positive (or negative) if and only if $\Theta(h_{G})$ has signature $(n,r-1)$
(or $(0,n+r-1)$).

We give a local expression of the Kobayashi curvature that will be used in the
final section. We focus on a local chart $\\{(z,[\zeta]):\zeta_{r}\neq 0\\}$
of $P(E)$, and let $\zeta_{i}/\zeta_{r}=w_{i}$ for $1\leq i\leq r-1$. The
curvature $\Theta$ of $\tilde{G}$ can be locally written as

$\Theta=\sum_{\alpha,\beta}R_{\alpha\bar{\beta}}\,dz_{\alpha}\wedge
d\bar{z}_{\beta}+\sum_{\alpha,l}P_{\alpha\bar{l}}\,dz_{\alpha}\wedge
d\bar{w}_{l}+\sum_{k,\beta}\mathcal{P}_{k\bar{\beta}}\,dw_{k}\wedge
d\bar{z}_{\beta}+\sum_{k,l}Q_{k\bar{l}}\,dw_{k}\wedge d\bar{w}_{l},$

where $R_{\alpha\bar{\beta}},P_{\alpha\bar{l}},\mathcal{P}_{k\bar{\beta}}$,
and $Q_{k\bar{l}}$ are endomorphisms of $\tilde{E}$. Since $O_{P(E)}(-1)$ is a
line bundle,

$\displaystyle\Theta|_{O_{P(E)}(-1)}$
$\displaystyle=\sum_{\alpha,\beta}\frac{\tilde{G}(R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dz_{\alpha}\wedge
d\bar{z}_{\beta}+\sum_{\alpha,l}\frac{\tilde{G}(P_{\alpha\bar{l}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dz_{\alpha}\wedge
d\bar{w}_{l}$
$\displaystyle+\sum_{k,\beta}\frac{\tilde{G}(\mathcal{P}_{k\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dw_{k}\wedge
d\bar{z}_{\beta}+\sum_{k,l}\frac{\tilde{G}(Q_{k\bar{l}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dw_{k}\wedge
d\bar{w}_{l},$

and by [Kob96, Formula (5.8)] the last three terms vanish, therefore

(2.4)
$\Theta|_{O_{P(E)}(-1)}=\sum_{\alpha,\beta}\frac{\tilde{G}(R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dz_{\alpha}\wedge
d\bar{z}_{\beta}.$

##  3 Proofs of Theorem 1 and Corollary 2

Recall that the $L^{2}$-metric $H_{k}$ on $S^{k}E^{*}$ is

(3.1) $\displaystyle
H_{k}(u,v):=\int_{P(E_{z})}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(v))\omega_{z}^{r-1},\textup{
for $u\textup{ and }v\in S^{k}E^{*}_{z}$,}$

where $h$ is a Hermitian metric on $O_{P(E)}(1)$ and $\\{\omega_{z}\\}_{z\in
X}$ is a family of Hermitian metrics on the fibers $P(E_{z})$, and
$\Phi_{k,z}$ is the canonical isomorphism from $S^{k}E^{*}_{z}$ to
$H^{0}(P(E_{z}),O_{P(E_{z})}(k))$.

###### Proof of Theorem 1.

According to the assumption, the Hermitian metric $H_{k}$ is Griffiths
negative for some $k$; equivalently, $\log H_{k}(s,s)$ is strongly
plurisubharmonic for any nonvanishing holomorphic section $s$ of $S^{k}E^{*}$.

If for a vector $v\in E^{*}$, we denote its $k$-th tensor power by $v^{k}\in
S^{k}E^{*}$, then one can define a Finsler metric $F_{k}$ on $E^{*}$ by
setting $F_{k}(v)=H_{k}(v^{k},v^{k})^{1/2k}$. By Griffiths negativity of
$H_{k}$, $F_{k}$ is strongly plurisubharmonic on the total space
$E^{*}\smallsetminus\\{\text{zero section}\\}$.

Next, we will use the $L^{2}$-integral feature of $H_{k}$ to prove that the
Finsler metric $F_{k}$ is convex i.e., it is convex on each fiber $E^{*}_{z}$.
Fix $z\in X$ and two vectors $u,v\in E^{*}_{z}$. Let $\\{s_{1},...,s_{r}\\}$
be a holomorphic frame of $E$ around $z$ with fiber coordinates
$(\zeta_{1},...,\zeta_{r})$, and $\\{t_{1},...,t_{r}\\}$ be the dual frame on
$E^{*}$. The frame $\\{t_{i}\\}$ on $E^{*}$ induces a frame $\\{t^{\alpha}\\}$
on $S^{k}E^{*}$ with $t^{\alpha}=t_{1}^{\alpha_{1}}\cdot\cdot\cdot
t_{r}^{\alpha_{r}}$ and $\alpha_{1}+...+\alpha_{r}=k$. On $P(E_{z})$, we
consider the chart $\\{[\zeta_{1},...,\zeta_{r}]:\zeta_{r}\neq 0\\}$, then

$e:=\frac{\sum_{i}\zeta_{i}s_{i}}{\zeta_{r}}$

is a holomorphic frame of $O_{P(E_{z})}(-1)$ with dual frame $e^{*}$ for
$O_{P(E_{z})}(1)$ and frame $(e^{*})^{k}$ for $O_{P(E_{z})}(k)$. The global
section $\Phi_{k,z}(t^{\alpha})\in H^{0}(P(E_{z}),O_{P(E_{z})}(k))$ can be
locally written as

$\frac{\zeta^{\alpha}}{\zeta_{r}^{k}}(e^{*})^{k},\text{where
$\zeta^{\alpha}=\zeta_{1}^{\alpha_{1}}\cdot\cdot\cdot\zeta_{r}^{\alpha_{r}}$.
}$

Therefore, if $E^{*}_{z}\ni u=\sum_{i}u_{i}t_{i}$ and $v=\sum_{i}v_{i}t_{i}$
with $u_{i},v_{i}\in\mathbb{C}$, then

(3.2) $\displaystyle
h^{k}\big{(}\Phi_{k,z}((u+v)^{k}),\Phi_{k,z}((u+v)^{k})\big{)}=\frac{\bigr{|}\sum_{i}(u_{i}+v_{i})\zeta_{i}\bigr{|}^{2k}}{|\zeta_{r}|^{2k}}h(e^{*},e^{*})^{k}.$

If we denote by $h^{*}$ the dual metric on $O_{P(E)}(-1)$, then
$h(e^{*},e^{*})=|\zeta_{r}|^{2}/h^{*}(\sum_{i}\zeta_{i}s_{i},\sum_{i}\zeta_{i}s_{i})$.
So (3.2) becomes

(3.3) $\displaystyle
h^{k}\big{(}\Phi_{k,z}((u+v)^{k}),\Phi_{k,z}((u+v)^{k})\big{)}=\frac{\bigr{|}\sum_{i}(u_{i}+v_{i})\zeta_{i}\bigr{|}^{2k}}{h^{*}(\sum_{i}\zeta_{i}s_{i},\sum_{i}\zeta_{i}s_{i})^{k}};$

the right hand side of formula (3.3) is meaningful on the entire $P(E_{z})$,
and we will abbreviate the denominator by $(h^{*})^{k}$ in the computation
below.

Using formula (3.3), we have

$\displaystyle F_{k}(u+v)$
$\displaystyle=H_{k}\big{(}(u+v)^{k},(u+v)^{k}\big{)}^{1/2k}$
$\displaystyle=\bigr{(}\int_{P(E_{z})}h^{k}\big{(}\Phi_{k,z}((u+v)^{k}),\Phi_{k,z}((u+v)^{k})\big{)}\omega_{z}^{r-1}\bigr{)}^{1/2k}$
$\displaystyle=\bigr{(}\int_{P(E_{z})}\frac{\bigr{|}\sum_{i}(u_{i}+v_{i})\zeta_{i}\bigr{|}^{2k}}{(h^{*})^{k}}\omega_{z}^{r-1}\bigr{)}^{1/2k}$
$\displaystyle\leq\bigr{(}\int_{P(E_{z})}\big{(}\frac{\bigr{|}\sum_{i}u_{i}\zeta_{i}\bigr{|}}{(h^{*})^{1/2}}+\frac{\bigr{|}\sum_{i}v_{i}\zeta_{i}\bigr{|}}{(h^{*})^{1/2}}\big{)}^{2k}\omega_{z}^{r-1}\bigr{)}^{1/2k}$
$\displaystyle\leq\bigr{(}\int_{P(E_{z})}\frac{\bigr{|}\sum_{i}u_{i}\zeta_{i}\bigr{|}^{2k}}{(h^{*})^{k}}\omega_{z}^{r-1}\bigr{)}^{1/2k}+\bigr{(}\int_{P(E_{z})}\frac{\bigr{|}\sum_{i}v_{i}\zeta_{i}\bigr{|}^{2k}}{(h^{*})^{k}}\omega_{z}^{r-1}\bigr{)}^{1/2k}$
$\displaystyle=H_{k}(u^{k},u^{k})^{1/2k}+H_{k}(v^{k},v^{k})^{1/2k}$
$\displaystyle=F_{k}(u)+F_{k}(v),$

where the second inequality is due to the Minkowski inequality. We have just
shown that $F_{k}(u+v)\leq F_{k}(u)+F_{k}(v)$, and this implies convexity
$F_{k}(\theta u+(1-\theta)v)\leq\theta F_{k}(u)+(1-\theta)F_{k}(v)$ for
$0\leq\theta\leq 1$. So $F_{k}$ is a convex Finsler metric which is strongly
plurisubharmonic on $E^{*}\smallsetminus\\{\text{zero section}\\}$.

For the second part of the theorem, since the Finsler metric $F_{k}$ from
above is convex and strongly plurisubharmonic, after adding a small Hermitian
metric we obtain a strongly convex and strongly plurisubharmonic Finsler
metric on $E^{*}$, whose dual metric has transversal Levi signature $(r,n)$
according to [Dem99, Theorem 2.5] and [Som78] (see below for the definition of
transversal Levi signature). By Lemma 6 below we obtain a convex and strongly
pseudoconvex Kobayashi positive Finsler metric on $E$. ∎

######  Lemma 6.

If a Finsler metric $F$ has tranversal Levi signature $(r,n)$, then $F$ is
strongly pseudoconvex and has positive Kobayashi curvature.

###### Proof.

This lemma has appeared in Proposition 1.8 in the first version of [FLW18]. We
give a simplified proof. Recall that a Finsler metric $F$ is said to have
transversal Levi signature $(r,n)$, if at every point $(z,\zeta)\in
E\smallsetminus\\{\text{zero section}\\}$, the Levi form
$i\partial\bar{\partial}F$ on $E$ is positive definite along the fiber $E_{z}$
and negative definite on some $n$-dimensional subspace $W\subset
T^{1,0}_{(z,\zeta)}E$ which is transversal to the fiber $E_{z}$.

Since $F_{i\bar{j}}>0$, the metric $F$ is strongly pseudoconvex. Let
$\pi:E\smallsetminus\\{\text{zero section}\\}\to P(E)$ be the quotient map. It
is not hard to verify $\pi^{*}\Theta(h_{G})=-\partial\bar{\partial}\log G$.
For any nonzero $e\in W$,

$\displaystyle\Theta(h_{G})(\pi_{*}e,\overline{\pi_{*}e})=\pi^{*}\Theta(h_{G})(e,\overline{e})=-2\partial\bar{\partial}\log
F(e,\overline{e})$
$\displaystyle=-2(\frac{\partial\bar{\partial}F}{F}-\frac{\partial
F\wedge\bar{\partial}F}{F^{2}})(e,\overline{e})\geq-2\frac{\partial\bar{\partial}F}{F}(e,\overline{e})>0.$

As a consequence, $\pi_{*}W$ is of dimension $n$, and $\Theta(h_{G})$ is
positive definite on $\pi_{*}W$. By Lemma 3 and a dimension count,
$\Theta(h_{G})$ has signature $(n,r-1)$, thus $F$ is Kobayashi positive by
Lemma 5. ∎

###### Proof of Corollary 2.

By Lemmas 3 and 5, a strongly pseudoconvex Kobayashi positive Finsler metric
$F$ on $E$ induces a Hermitian metric $h$ on $O_{P(E)}(1)$ such that the
curvature $\Theta(h)$ restricted to each fiber $P(E_{z})$ is positive, and
$\Theta(h)$ has signature $(r-1,n)$. With this $h$ on $O_{P(E)}(1)$ and fixing
a family of Hermitian metrics $\\{\omega_{z}\\}_{z\in X}$ on the fibers
$P(E_{z})$, we have the Hermitian metric $H_{k}$ in (1.1) on $S^{k}E^{*}$.

Given a point $(z_{0},\zeta_{0})\in E$, we can find a frame $\\{s_{i}\\}$ of
$E$ such that $G_{i\bar{j}}(z_{0},\zeta_{0})=\delta_{ij}$ and
$G_{i\bar{j}\alpha}(z_{0},\zeta_{0})=G_{i\bar{j}\bar{\beta}}(z_{0},\zeta_{0})=0$
(such a frame can be obtained by (5.11) in [Kob96]). In this frame,

$\Theta(h)|_{(z_{0},[\zeta_{0}])}=-\sum_{\alpha,\beta}\frac{\partial^{2}\log
h}{\partial z_{\alpha}\partial\bar{z}_{\beta}}dz_{\alpha}\wedge
d\bar{z}_{\beta}-\sum_{i,j}\frac{\partial^{2}\log h}{\partial
w_{i}\partial\bar{w}_{j}}dw_{i}\wedge d\bar{w}_{j}.$

Since $\Theta(h)|_{P(E_{z_{0}})}$ is positive and $\Theta(h)$ has signature
$(r-1,n)$, the matrix

$\bigr{(}-\frac{\partial^{2}\log h}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\bigr{)}$

is negative at $(z_{0},[\zeta_{0}])$ hence negative in a neighborhood of
$(z_{0},[\zeta_{0}])$ in $P(E)$.

Therefore, $P(E)$ can be covered by a finite number of open sets
$\\{U_{m}\\}$; each $U_{m}$ corresponds to a frame of $E$ such that the
corresponding matrix $(-\partial^{2}\log h/\partial
z_{\alpha}\partial\bar{z}_{\beta})$ is negative in $U_{m}$. For a fixed
$U_{m}$, we write

$\omega_{z}=\sqrt{-1}\sum_{i,j}g_{i\bar{j}}(z,w)dw_{i}\wedge d\bar{w}_{j},$

so $\omega_{z}^{r-1}=\det(g_{i\bar{j}})\bigwedge_{l}(\sqrt{-1}dw_{l}\wedge
d\bar{w}_{l})$ that we will abbreviate as $\det(g)dw\wedge d\bar{w}$. By
taking $k$ large, the following matrix can be made negative

(3.4) $\big{(}\frac{\partial^{2}-k\log h-\log\det(g)}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\big{)}.$

Since there are only finitely many $U_{m}$, we can take $k$ large such that
the corresponding (3.4) in each $U_{m}$ is negative.

For a fixed point $z_{0}\in X$, we partition $P(E_{z_{0}})=\bigcup_{l}V_{l}$
with $V_{l}$ in $U_{m}$ for some $m$. Let us denote the curvature of $H_{k}$
by $\Theta$ an $\operatorname{End}S^{k}E^{*}$-valued (1,1)-form. Given a
nonzero $u\in S^{k}E^{*}_{z_{0}}$, we would like to show the (1,1)-form
$H_{k}(\Theta u,u)$ is negative, namely, $H_{k}(\Theta
u,u)(\eta,\bar{\eta})<0$ for $0\neq\eta\in T^{1,0}_{z_{0}}X$. First, we extend
$u$ to a local holomorphic section whose covariant derivative (with respect to
$H_{k}$) at $z_{0}$ is zero. We still denote the holomorphic extension by $u$.
A simple computation shows, at $z_{0}$,

(3.5) $\partial\bar{\partial}H_{k}(u,u)(\eta,\bar{\eta})=-H_{k}(\Theta
u,u)(\eta,\bar{\eta}).$

Meanwhile, if we denote $p$ as the projection from $P(E)$ to $X$, then by
definition

(3.6) $\displaystyle\partial\bar{\partial}H_{k}(u,u)$
$\displaystyle=\partial\bar{\partial}p_{*}\big{(}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))\omega_{z}^{r-1}\big{)}=p_{*}\partial\bar{\partial}\big{(}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))\omega_{z}^{r-1}\big{)}$
$\displaystyle=\sum_{l}\int_{V_{l}}\partial\bar{\partial}\big{(}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))\omega_{z}^{r-1}\big{)}.$

Let us compute
$\partial\bar{\partial}\big{(}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))\omega_{z}^{r-1}\big{)}$
in $U_{m}$. In $U_{m}$, we can write $\Phi_{k,z}(u)$ as $fe$ with $f$ a
scalar-valued holomorphic function and $e$ a frame for $O_{P(E)}(k)$. So,
$h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))=|f|^{2}h^{k}(e,e)$, and we will abbreviate
$h^{k}(e,e)$ by $h^{k}$. Moreover, we have $\omega_{z}^{r-1}=\det(g)dw\wedge
d\bar{w}$. Therefore, in $U_{m}$,

$\displaystyle\partial\bar{\partial}\big{(}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))\omega_{z}^{r-1}\big{)}$
$\displaystyle=\partial\bar{\partial}\big{(}|f|^{2}h^{k}\det(g)dw\wedge
d\bar{w}\big{)}$
$\displaystyle=\sum_{\alpha,\beta}\frac{\partial^{2}|f|^{2}h^{k}\det(g)}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}dz_{\alpha}\wedge d\bar{z}_{\beta}\wedge
dw\wedge d\bar{w}.$

Let us denote $h^{k}\det(g)$ by $e^{-\phi}$ temporarily. Let
$\eta=\sum_{\alpha}\eta_{\alpha}\partial/\partial z_{\alpha}\in
T^{1,0}_{z_{0}}X$. A straightforward computation shows

(3.7) $\sum_{\alpha,\beta}\frac{\partial^{2}|f|^{2}e^{-\phi}}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\eta_{\alpha}\bar{\eta}_{\beta}=\big{(}e^{-\phi}\bigr{|}\sum_{\alpha}\frac{\partial
f}{\partial
z_{\alpha}}\eta_{\alpha}-f\sum_{\alpha}\eta_{\alpha}\frac{\partial\phi}{\partial
z_{\alpha}}\bigr{|}^{2}-|f|^{2}e^{-\phi}\sum_{\alpha,\beta}\frac{\partial^{2}\phi}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\eta_{\alpha}\bar{\eta}_{\beta}\big{)}.$

The matrix in the last term of (3.7)

$\frac{\partial^{2}\phi}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}=\frac{\partial^{2}-k\log
h-\log\det(g)}{\partial z_{\alpha}\partial\bar{z}_{\beta}}$

is negative by (3.4), so (3.7) is positive, and

(3.8)
$\int_{V_{l}}\partial\bar{\partial}\big{(}h^{k}(\Phi_{k,z}(u),\Phi_{k,z}(u))\omega_{z}^{r-1}\big{)}(\eta,\bar{\eta})>0.$

Hence by (3.5) and (3.6), $H_{k}(\Theta u,u)(\eta,\bar{\eta})<0$. Therefore,
the Hermitian metric $H_{k}$ on $S^{k}E^{*}$ is Griffith negative for $k$
large. The proof is complete by using Theorem 1.

The above curvature computation is a combination of ideas from [LSY13, Section
4.2], [Ber09b, Section 3]. ∎

##  4 Convexity of the $k$-th root of an inner product

Given a complex vector space $V$ of dimension $r$, let $S^{k}V$ be the $k$-th
symmetric power of $V$. For an inner product $H$ on $S^{k}V$, the map $V\ni
v\mapsto H(v^{k},v^{k})$ is in general not convex, and $V\ni v\mapsto
H(v^{k},v^{k})^{1/2k}$ is even less so.

######  Example 7.

Take $r=2$ and a basis $\\{e_{1},e_{2}\\}$ for $V$, then
$\\{e_{1}^{2},e_{2}^{2},e_{1}e_{2}\\}$ is a basis for $S^{2}V$. Consider an
inner product $H$ on $S^{2}V$ with matrix representation under
$\\{e_{1}^{2},e_{2}^{2},e_{1}e_{2}\\}$ equal to

$\begin{pmatrix}1/2&0&0\\\ 0&1/2&0\\\ 0&0&1\end{pmatrix}.$

We write $v\in V$ as $v=v_{1}e_{1}+v_{2}e_{2}$ with
$v_{1}=x_{1}+\sqrt{-1}y_{1}$, $v_{2}=x_{2}+\sqrt{-1}y_{2}$, and $x_{j},y_{j}$
real. So

$H(v^{2},v^{2})=|v_{1}^{2}|^{2}/2+|v_{2}^{2}|^{2}/2+|2v_{1}v_{2}|^{2}=(x_{1}^{2}+y_{1}^{2})^{2}/2+(x_{2}^{2}+y_{2}^{2})^{2}/2+4(x_{1}^{2}+y_{1}^{2})(x_{2}^{2}+y_{2}^{2}),$
$\begin{pmatrix}\frac{\partial^{2}H}{\partial x_{i}\partial
x_{j}}\end{pmatrix}_{i,j}=\begin{pmatrix}4x_{1}^{2}+2(x_{1}^{2}+y_{1}^{2})+8(x_{2}^{2}+y_{2}^{2})&16x_{1}x_{2}\\\
16x_{1}x_{2}&4x_{2}^{2}+2(x_{2}^{2}+y_{2}^{2})+8(x_{1}^{2}+y_{1}^{2})\end{pmatrix}.$

At $x_{1}=x_{2}=1$ and $y_{1}=y_{2}=0$, this 2-by-2 matrix equals

$\begin{pmatrix}14&16\\\ 16&14\end{pmatrix}$

which is not semipositive. So the map $v\mapsto H(v^{2},v^{2})$ is not convex.

The next example shows that if the $k$-th root of an inner product is convex,
the $k$-th root of the dual inner product is in general not convex.

######  Example 8.

Let $V$ be a complex vector space of dimension two with basis
$\\{e_{1},e_{2}\\}$. Then $\\{e_{1}^{2},e_{2}^{2},e_{1}e_{2}\\}$ is a basis
for $S^{2}V$. Let $\\{e_{1}^{*},e_{2}^{*}\\}$ be the dual basis for $V^{*}$,
and we use $(\zeta_{1},\zeta_{2})$ to denote vectors in $V^{*}$. Let $h$ be an
inner product on $V^{*}$ such that $\\{e_{1}^{*},e_{2}^{*}\\}$ is orthonormal.
This $h$ induces on $O_{P(V^{*})}(1)$ a Hermitian metric which we still denote
by $h$; the curvature $\omega$ of $h$ is a Kähler form on $P(V^{*})$. Through
the isomorphism $\Phi:S^{2}V\to H^{0}(P(V^{*}),O_{P(V^{*})}(2))$, we define an
inner product $H$ on $S^{2}V$ by, for $\xi,\eta\in S^{2}V$,

$H(\xi,\eta)=\int_{P(V^{*})}h^{2}(\Phi(\xi),\Phi(\eta))\omega.$

On the chart $\\{[\zeta_{1},\zeta_{2}]:\zeta_{2}\neq 0\\}$ of $P(V^{*})$, we
have a local frame

$e:=\frac{\zeta_{1}}{\zeta_{2}}e_{1}^{*}+e_{2}^{*}$

for $O_{P(V^{*})}(-1)$, and a dual frame $e^{*}$ for $O_{P(V^{*})}(1)$.
Therefore, the global sections have local expressions

(4.1)
$\displaystyle\Phi(e_{1}^{2})=\frac{\zeta_{1}^{2}}{\zeta_{2}^{2}}(e^{*})^{2},\text{
}\Phi(e_{2}^{2})=(e^{*})^{2},\text{ and
}\Phi(e_{1}e_{2})=\frac{\zeta_{1}}{\zeta_{2}}(e^{*})^{2}.$

Moreover, if we denote $\zeta_{1}/\zeta_{2}$ by $z$, then
$h(e^{*},e^{*})=1/(|z|^{2}+1)$ and $\omega=dz\wedge d\bar{z}/(|z|^{2}+1)^{2}$.
Having these information, one can compute the matrix representation of $H$ in
terms of $\\{e_{1}^{2},e_{2}^{2},e_{1}e_{2}\\}$. For example,

$\displaystyle
H(e_{1}^{2},e_{2}^{2})=\int_{\mathbb{C}}z^{2}\frac{1}{(|z|^{2}+1)^{2}}\frac{dz\wedge
d\bar{z}}{(|z|^{2}+1)^{2}}=0,$ $\displaystyle
H(e_{1}^{2},e_{1}^{2})=\int_{\mathbb{C}}|z|^{4}\frac{1}{(|z|^{2}+1)^{2}}\frac{dz\wedge
d\bar{z}}{(|z|^{2}+1)^{2}}=\frac{\pi}{3},$

and similarly
$H(e_{2}^{2},e_{2}^{2})=\pi/3,H(e_{1}e_{2},e_{1}e_{2})=\pi/6,H(e_{1}^{2},e_{1}e_{2})=H(e_{2}^{2},e_{1}e_{2})=0$.
Therefore, the matrix representation under
$\\{e_{1}^{2},e_{2}^{2},e_{1}e_{2}\\}$ is

$\begin{pmatrix}\frac{\pi}{3}&0&0\\\ 0&\frac{\pi}{3}&0\\\
0&0&\frac{\pi}{6}\end{pmatrix}.$

The convexity of the map $V\ni v\mapsto H(v^{2},v^{2})$ follows by the
Minkowski inequality as in the proof of Theorem 1. But the dual metric $H^{*}$
has matrix representation under
$\\{{e^{*}_{1}}^{2},{e^{*}_{2}}^{2},e^{*}_{1}e^{*}_{2}\\}$

$\frac{6}{\pi}\begin{pmatrix}1/2&0&0\\\ 0&1/2&0\\\ 0&0&1\end{pmatrix},$

and we know from the first example that $V^{*}\ni v\mapsto H^{*}(v^{2},v^{2})$
is not convex.

##  5 Extremal characterization of the Kobayashi curvature

In this section we discuss a different way of obtaining the Kobayashi
curvature by following a curvature notion defined in [Lem15, Lem17] (see also
[Roc84]). Finsler metrics are assumed convex in [Lem15, Lem17], while in our
case we assume strong pseudoconvexity. This characterization has the potential
to define curvature for Finsler metrics with weaker convexity.

Let $F$ be a strongly pseudoconvex Finsler metric on $E$ with $F^{2}=G$. For
$z_{0}\in X$, $v\in T^{1,0}_{z_{0}}X$ and $0\neq\zeta\in E_{z_{0}}$, we define

(5.1) $K_{v}(\zeta)=-\inf\partial\bar{\partial}\log
G(\phi)\bigr{|}_{z_{0}}(v,\bar{v}),$

the inf taken over local holomorphic sections $\phi$ of $E$ such that
$\phi(z_{0})=\zeta$.

Consider a tangent vector $\tilde{v}$ to $P(E)$ at $(z_{0},[\zeta])$ such that
$p_{*}(\tilde{v})=v$. By (2.4),
$\Theta|_{O_{P(E)}(-1)}(\tilde{v},\bar{\tilde{v}})$ is independent of the lift
$\tilde{v}$, so we will simply write $\Theta|_{O_{P(E)}(-1)}(v,\bar{v})$.

######  Lemma 9.

$K_{v}(\zeta)=\Theta|_{O_{P(E)}(-1)}(v,\bar{v})$.

###### Proof.

Fix a frame $\\{s_{1},...,s_{r}\\}$ of $E$ around $z_{0}$ such that
$G_{i\bar{j}}(z_{0},\zeta)=\delta_{ij}$ and
$G_{i\bar{j}\alpha}(z_{0},\zeta)=G_{i\bar{j}\bar{\beta}}(z_{0},\zeta)=0$ (see
(5.11) in [Kob96]). Since the vector $\zeta\in E_{z_{0}}$ is nonzero, without
loss of generality we assume its $r$-th component $\zeta_{r}\neq 0$. Suppose
$v=\sum v_{\alpha}\partial/\partial z_{\alpha}$. Under this local
trivialization, the holomorphic section $\phi=(\phi_{1},...,\phi_{r})$ with
$\phi_{r}\neq 0$ near $z_{0}$. The computation will be done on a chart
$\\{(z,[w_{1},...,w_{r-1},1])\\}$ of $P(E)$. The connection form of the Chern
connection of $\tilde{G}$ is an $\operatorname{End}\tilde{E}$-valued
$(1,0)$-form on $P(E)$ that we write as
$\sum_{\alpha}A_{\alpha}dz_{\alpha}+\sum_{i}B_{i}dw_{i}$ with $A_{\alpha}$ and
$B_{i}\in\operatorname{End}\tilde{E}$. Moreover, $A_{\alpha}(z_{0},[\zeta])=0$
because of the choice of the frame $\\{s_{i}\\}$. By setting
$D_{z_{\alpha}}=\partial/\partial z_{\alpha}+A_{\alpha}$ and
$D_{w_{i}}=\partial/\partial w_{i}+B_{i}$, we have

$\displaystyle\frac{\partial}{\partial
z_{\alpha}}\tilde{G}_{(z,[w,1])}(\phi(z),\phi(z))=\tilde{G}_{(z,[w,1])}(D_{z_{\alpha}}\phi,\phi),$
$\displaystyle\frac{\partial}{\partial
w_{i}}\tilde{G}_{(z,[w,1])}(\phi(z),\phi(z))=\tilde{G}_{(z,[w,1])}(D_{w_{i}}\phi,\phi).$

By (2.3), $G(\phi)=G(z,\phi(z))=\tilde{G}_{(z,[\phi(z)])}(\phi(z),\phi(z))$
and we will suppress the subscript $(z,[\phi(z)])$ of $\tilde{G}$ for
simplicity. Therefore, by chain rule

$\displaystyle\partial\bar{\partial}G(z,\phi(z))(v,\bar{v})$
$\displaystyle=\sum^{n}_{\alpha,\beta=1}\frac{\partial^{2}}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\tilde{G}(\phi(z),\phi(z))v_{\alpha}\bar{v}_{\beta}$
$\displaystyle=\big{\|}\sum_{\alpha}v_{\alpha}D_{z_{\alpha}}\phi+\sum_{i,\alpha}D_{w_{i}}\phi\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}v_{\alpha}\big{\|}^{2}_{\tilde{G}}$
$\displaystyle-\sum_{\alpha,\beta}v_{\alpha}\bar{v}_{\beta}\big{[}\tilde{G}(R_{\alpha\bar{\beta}}\phi,\phi)+\sum^{r-1}_{i=1}\tilde{G}(P_{\alpha\bar{i}}\phi,\phi)\overline{\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\beta}}}$
$\displaystyle+\sum^{r-1}_{i=1}\tilde{G}(\mathcal{P}_{i\bar{\beta}}\phi,\phi)\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}+\sum^{r-1}_{i,j=1}\tilde{G}(Q_{i\bar{j}}\phi,\phi)\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}\overline{\frac{\partial(\frac{\phi_{j}}{\phi_{r}})}{\partial
z_{\beta}}}\big{]}.$

The last three terms involving $P,\mathcal{P}$, and $Q$ are zero by [Kob96,
(5.8)]. Meanwhile, by the Cauchy–Schwarz inequality

$\displaystyle\big{|}\sum_{\alpha}\frac{\partial}{\partial
z_{\alpha}}G(z,\phi(z))v_{\alpha}\big{|}^{2}=$
$\displaystyle\big{|}\tilde{G}\big{(}\sum_{\alpha}v_{\alpha}D_{z_{\alpha}}\phi+\sum_{i,\alpha}D_{w_{i}}\phi\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}v_{\alpha},\phi\big{)}\big{|}^{2}$ $\displaystyle\leq$
$\displaystyle\big{\|}\sum_{\alpha}v_{\alpha}D_{z_{\alpha}}\phi+\sum_{i,\alpha}D_{w_{i}}\phi\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}v_{\alpha}\big{\|}^{2}_{\tilde{G}}\|\phi\|^{2}_{\tilde{G}}.$

All added up, we see

$\displaystyle\partial\bar{\partial}\log G(z,\phi(z))(v,\bar{v})$
$\displaystyle=\frac{1}{\|\phi\|^{4}_{\tilde{G}}}\big{(}\big{\|}\sum_{\alpha}v_{\alpha}D_{z_{\alpha}}\phi+\sum_{i,\alpha}D_{w_{i}}\phi\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}v_{\alpha}\big{\|}^{2}_{\tilde{G}}\|\phi\|^{2}_{\tilde{G}}-\sum_{\alpha,\beta}v_{\alpha}\bar{v}_{\beta}\tilde{G}(R_{\alpha\bar{\beta}}\phi,\phi)\|\phi\|^{2}_{\tilde{G}}$
$\displaystyle-\big{|}\sum_{\alpha}\frac{\partial}{\partial
z_{\alpha}}G(z,\phi(z))v_{\alpha}\big{|}^{2}\big{)}$
$\displaystyle\geq-\frac{\sum_{\alpha,\beta}v_{\alpha}\bar{v}_{\beta}\tilde{G}(R_{\alpha\bar{\beta}}\phi,\phi)}{\|\phi\|^{2}_{\tilde{G}}}.$

Therefore,

$\displaystyle\partial\bar{\partial}\log
G(z,\phi(z))(v,\bar{v})\geq-\sum_{\alpha,\beta}\frac{\tilde{G}(R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}v_{\alpha}\bar{v}_{\beta};$

the equality holds if we choose
$\phi(z)=(\zeta_{1},...,\zeta_{r})+(\zeta_{1},...,\zeta_{r})\sum_{\alpha}(z-z_{0})_{\alpha}$
under the frame $\\{s_{i}\\}$. Indeed,

$\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}\bigr{|}_{z_{0}}=\frac{\frac{\partial\phi_{i}}{\partial
z_{\alpha}}\phi_{r}-\phi_{i}\frac{\partial\phi_{r}}{z_{\alpha}}}{\phi_{r}^{2}}\bigr{|}_{z_{0}}=0,\text{
and }A_{\alpha}(z_{0},[\zeta])=0.$

So, at $(z_{0},[\zeta])$,

$\sum_{\alpha}v_{\alpha}D_{z_{\alpha}}\phi+\sum_{i,\alpha}D_{w_{i}}\phi\frac{\partial(\frac{\phi_{i}}{\phi_{r}})}{\partial
z_{\alpha}}v_{\alpha}=\sum_{\alpha}v_{\alpha}\zeta\text{ is parallel with
}\phi(z_{0})=\zeta.$

As a result,

$K_{v}(\zeta)=\sum_{\alpha,\beta}\frac{\tilde{G}(R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}v_{\alpha}\bar{v}_{\beta}=\Theta|_{O_{P(E)}(-1)}(v,\bar{v}).$

∎

One immediate consequence of Lemma 9 is that the Kobayashi curvature is
seminegative if and only if $\log G(z,\phi(z))$ is plurisubharmonic for any
local holomorphic section $\phi$ (which is also equivalent to $G$ being
plurisubharmonic on the total space $E$).

## References

  * [Aik98] Tadashi Aikou, _A partial connection on complex Finsler bundles and its applications_ , Illinois J. Math. 42 (1998), no. 3, 481–492. MR 1631260
  * [Aik04] , _Finsler geometry on complex vector bundles_ , A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 83–105. MR 2132658
  * [Ber09a] Bo Berndtsson, _Curvature of vector bundles associated to holomorphic fibrations_ , Ann. of Math. (2) 169 (2009), no. 2, 531–560. MR 2480611
  * [Ber09b] , _Positivity of direct image bundles and convexity on the space of Kähler metrics_ , Journal of Differential Geometry 81 (2009), no. 3, 457–482.
  * [BL16] Bo Berndtsson and László Lempert, _A proof of the Ohsawa-Takegoshi theorem with sharp estimates_ , J. Math. Soc. Japan 68 (2016), no. 4, 1461–1472. MR 3564439
  * [CF90] F. Campana and H. Flenner, _A characterization of ample vector bundles on a curve_ , Math. Ann. 287 (1990), no. 4, 571–575. MR 1066815
  * [CW03] J.-G. Cao and Pit-Mann Wong, _Finsler geometry of projectivized vector bundles_ , J. Math. Kyoto Univ. 43 (2003), no. 2, 369–410. MR 2051030
  * [Dem99] Jean-Pierre Demailly, _Pseudoconvex-concave duality and regularization of currents_ , Several complex variables (Berkeley, CA, 1995–1996), Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 233–271. MR 1748605
  * [Dem12] , _Complex analytic and differential geometry_ , https://www-fourier.ujf-grenoble.fr/ demailly/manuscripts/agbook.pdf, 2012.
  * [Dem20] , _Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles_ , arXiv preprint arXiv:2002.02677 (2020).
  * [DWZZ18] Fusheng Deng, Zhiwei Wang, Liyou Zhang, and Xiangyu Zhou, _New characterizations of plurisubharmonic functions and positivity of direct image sheaves_ , arxiv:1809.10371 (2018).
  * [Fin21] Siarhei Finski, _On Monge-Ampère volumes of direct images_ , International Mathematics Research Notices (2021), 1–24.
  * [FLW16] Huitao Feng, Kefeng Liu, and Xueyuan Wan, _Chern forms of holomorphic Finsler vector bundles and some applications_ , Internat. J. Math. 27 (2016), no. 4, 1650030, 22. MR 3491047
  * [FLW17] , _A Donaldson type functional on a holomorphic Finsler vector bundle_ , Math. Ann. 369 (2017), no. 3-4, 997–1019. MR 3713533
  * [FLW18] , _Complex Finsler vector bundles with positive Kobayashi curvature_ , arXiv preprint arXiv:1811.08617 (2018).
  * [Gri69] Phillip A. Griffiths, _Hermitian differential geometry, Chern classes, and positive vector bundles_ , Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 185–251. MR 0258070
  * [GZ15] Qi’an Guan and Xiangyu Zhou, _A solution of an $L^{2}$ extension problem with an optimal estimate and applications_, Ann. of Math. (2) 181 (2015), no. 3, 1139–1208. MR 3296822
  * [Har66] Robin Hartshorne, _Ample vector bundles_ , Inst. Hautes Études Sci. Publ. Math. (1966), no. 29, 63–94. MR 193092
  * [HPS18] Christopher Hacon, Mihnea Popa, and Christian Schnell, _Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and P $\breve{a}$un_, Local and global methods in algebraic geometry, Contemp. Math., vol. 712, Amer. Math. Soc., Providence, RI, 2018, pp. 143–195. MR 3832403
  * [Kob75] Shoshichi Kobayashi, _Negative vector bundles and complex Finsler structures_ , Nagoya Math. J. 57 (1975), 153–166. MR 377126
  * [Kob87] , _Differential geometry of complex vector bundles_ , Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987, Kanô Memorial Lectures, 5. MR 909698
  * [Kob96] , _Complex Finsler vector bundles_ , Finsler geometry (Seattle, WA, 1995), Contemp. Math., vol. 196, Amer. Math. Soc., Providence, RI, 1996, pp. 145–153. MR 1403586
  * [Lem15] László Lempert, _A maximum principle for Hermitian (and other) metrics_ , Proc. Amer. Math. Soc. 143 (2015), no. 5, 2193–2200. MR 3314125
  * [Lem17] , _Extrapolation, a technique to estimate_ , Functional analysis, harmonic analysis, and image processing: a collection of papers in honor of Björn Jawerth, Contemp. Math., vol. 693, Amer. Math. Soc., Providence, RI, 2017, pp. 271–281. MR 3682614
  * [LSY13] Kefeng Liu, Xiaofeng Sun, and Xiaokui Yang, _Positivity and vanishing theorems for ample vector bundles_ , J. Algebraic Geom. 22 (2013), no. 2, 303–331. MR 3019451
  * [LY14] Kefeng Liu and Xiaokui Yang, _Curvatures of direct image sheaves of vector bundles and applications_ , Journal of Differential Geometry 98 (2014), no. 1, 117–145.
  * [MT07] Christophe Mourougane and Shigeharu Takayama, _Hodge metrics and positivity of direct images_ , J. Reine Angew. Math. 606 (2007), 167–178. MR 2337646
  * [MT08] , _Hodge metrics and the curvature of higher direct images_ , Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 6, 905–924. MR 2504108
  * [Nau17] Philipp Naumann, _An approach to Griffiths conjecture_ , arXiv preprint arXiv:1710.10034 (2017).
  * [Pin21] Vamsi Pritham Pingali, _A note on Demailly’s approach towards a conjecture of Griffiths_ , arxiv:2102.02496 (2021).
  * [Roc84] Richard Rochberg, _Interpolation of Banach spaces and negatively curved vector bundles._ , Pacific J. Math. 110 (1984), no. 2, 355–376.
  * [Som78] Andrew John Sommese, _Concavity theorems_ , Math. Ann. 235 (1978), no. 1, 37–53. MR 486637
  * [Ume73] Hiroshi Umemura, _Some results in the theory of vector bundles_ , Nagoya Math. J. 52 (1973), 97–128. MR 337968
  * [ZZ18] Xiangyu Zhou and Langfeng Zhu, _An optimal $L^{2}$ extension theorem on weakly pseudoconvex Kähler manifolds_, J. Differential Geom. 110 (2018), no. 1, 135–186. MR 3851746

Institute of Mathematics, Academia Sinica, Taipei, Taiwan

<EMAIL_ADDRESS>