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 [Reference HartshorneHar].
Meanwhile, for a Hermitian holomorphic vector bundle
$(E,H)$
, Griffiths [Reference GriffithsGri] 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., [Reference KobayashiKob2, p. 89] or [Reference DemaillyDem2, p. 283]); however, the converse statement known as the Griffiths conjecture is still open (see earlier results [Reference Campana and FlennerCF], [Reference UmemuraUme] and recent developments [Reference DemaillyDem3], [Reference FinskiFin], [Reference NaumannNau], [Reference PingaliPin]).
On the other hand, Kobayashi [Reference KobayashiKob1] 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 [Reference Feng, Liu and WanFLW3] 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 let
$\{\omega _z\}_{z\in X}$
be a family of Hermitian metrics on the fibers
$P(E_z)$
. Given a positive integer k, there is a canonical isomorphism
$$ \begin{align*}\Phi_{k,z}:S^kE^*_z\to H^0(P(E_z),O_{P(E_z)}(k)),\end{align*} $$
see, for example, [Reference DemaillyDem2, p. 278, Th. 15.5]. We can define an
$L^2$
-metric
$H_k$
on
$S^kE^*$
using
$h^k$
and
$\{\omega _z\}$
$$ \begin{align} H_k(u,v):=\int_{P(E_z)} h^k(\Phi_{k,z}(u),\Phi_{k,z}(v) ) \omega_z^{r-1}, \text{ for }u \text{ and } v\in S^kE^*_z. \end{align} $$
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
$E^*\setminus \{\text {zero section}\}$
; 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 kth 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
$E^*\setminus \{\text {zero section}\}$
by Griffiths negativity. The convexity of
$F_k$
is of great importance (see [Reference DemaillyDem1, Rem. 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 §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
$S^kE^*$
has a Griffiths negative Hermitian metric
$\mathcal {H}_k^*$
for k large; such a Hermitian metric is the dual of
$\mathcal {H}_k$
on
$S^kE$
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 [Reference BerndtssonBer1], [Reference Liu, Sun and YangLSY], [Reference Liu and YangLY], [Reference Mourougane and TakayamaMT1], [Reference Mourougane and TakayamaMT2]). However, the kth root of
$\mathcal {H}^*_k$
is not convex (see §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 [Reference BerndtssonBer2, Sec. 3]). Another natural approach is to take the kth 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., [Reference Berndtsson and LempertBL], [Reference Deng, Wang, Zhang and ZhouDWZZ], [Reference Guan and ZhouGZ], [Reference Hacon, Popa and SchnellHPS], [Reference Zhou and ZhuZZ]).
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 [Reference Liu, Sun and YangLSY, Sec. 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 toward 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 [Reference Feng, Liu and WanFLW3]. Indeed, given a strongly pseudoconvex Kobayashi positive Finsler metric on E, we can construct a Finsler metric on
$E^*$
which is strongly plurisubharmonic on
$E^*\setminus \{\text {zero section}\}$
, 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 §2. Theorem 1 and Corollary 2 are proved in §3. In §4, we give explicit examples about convexity of inner products. In the last section, we give an extremal characterization of Kobayashi curvature following [Reference LempertLem1], [Reference LempertLem2], where Lempert defined curvature for a much broader class of metrics.
2 Kobayashi curvature of Finsler metrics
There is a sizable literature for Finsler metrics on vector bundles, for example [Reference AikouAik1], [Reference AikouAik2], [Reference Cao and WongCW], [Reference Feng, Liu and WanFLW1], [Reference Feng, Liu and WanFLW2], [Reference KobayashiKob1], [Reference KobayashiKob3]. Therefore, we are very brief on the review; one can consult the above articles for details. We use [Reference KobayashiKob3], an updated version of [Reference KobayashiKob1].
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 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
-
(1) F is continuous on E and smooth away from the zero section of E.
-
(2) For
$(z,\zeta )\in E, F(z,\zeta )\geq 0$
, and equality holds if and only if
$\zeta =0$
. -
(3)
$F(z,\lambda \zeta )=|\lambda |F(z,\zeta )$
, for
$\lambda \in \mathbb {C}$
.
Since
$F^2$
is 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,
For a vector
$\zeta \in E_z$
, we denote by
$[\zeta ]$
the equivalence class of
$\zeta $
in
$P(E_z)$
, and we 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 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
$$ \begin{align} h_G(z,[\zeta],\zeta)=G(z,\zeta). \end{align} $$
We use
$(z_1,\ldots ,z_n)$
for local coordinates on X, and
$(\zeta _1,\ldots ,\zeta _r)$
for local fiber coordinates on E with respect to a holomorphic frame
$\{s_1,\ldots ,s_r\}$
. So
$(z_1,\ldots ,z_n,\zeta _1,\ldots ,\zeta _r)$
is a coordinate system on E. Let F be a Finsler metric on E with
$G=F^2$
. We write
$$ \begin{align*} G_i=\partial G/\partial \zeta_i\,,\text{ }\text{ }G_{\bar{j}}=\partial G/\partial \bar{\zeta}_j\,,\text{ }\text{ } G_{i\bar{j}}=\partial^2 G/\partial \zeta_i \partial\bar{\zeta}_{j}\,, \\[3pt] G_{i\alpha}=\partial G_i/\partial z_{\alpha}\,,\text{ }\text{ } G_{i\bar{j}\bar{\beta}}=\partial G_{i\bar{j}}/\partial \bar{z}_{\beta}\,,\text{ etc.,} \end{align*} $$
with Latin letters
$i,j$
for the fiber direction and Greek letters
$\alpha ,\beta $
for the base direction. The following formulas are used later; their derivation can be found in [Reference KobayashiKob3, p. 147]. For
$\lambda \in \mathbb {C}$
,
$$ \begin{align} &G_{k\bar{j}}(z,\lambda\zeta)=G_{k\bar{j}}(z,\zeta). \end{align} $$
$$ \begin{align} &G(z,\zeta)=\sum_i G_i(z,\zeta)\zeta_i=\sum_{i,j}G_{i\bar{j}}(z,\zeta)\zeta_i\bar{\zeta}_j. \end{align} $$
2.2 Strongly pseudoconvex Finsler metrics
A Finsler metric F is said to be
-
(1) Strongly pseudoconvex if
$(F_{i\bar {j}})$
is positive definite on
$E\setminus \{\text {zero section}\}$
. -
(2) Convex if F restricted to each fiber
$E_z$
is convex, namely the fiberwise real Hessian of F is positive semidefinite on
$E\setminus \{\text {zero section}\}$
. -
(3) Strongly convex if the fiberwise real Hessian of G is positive on
$E\setminus \{\text {zero section}\}$
.
Note that in (1), the requirement
$(F_{i\bar {j}})>0$
is equivalent to
$(G_{i\bar {j}})>0$
, which can be shown using the formulas in [Reference KobayashiKob3, (3.3) and (3.10)]. A Hermitian metric always satisfies (1)–(3). Also notice that a convex Finsler metric in [Reference KobayashiKob1] is what we call strongly pseudoconvex here. Let us recall a lemma of strong pseudoconvexity [Reference KobayashiKob3, p. 150, Th. (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 assume the Finsler metric F is strongly pseudoconvex. There is a natural Hermitian metric
$\tilde {G}$
on the pull-back bundle
$\tilde {E}$
(see [Reference KobayashiKob1]). In terms of local coordinates, the Hermitian metric
$\tilde {G}$
is given by
$$ \begin{align*}\tilde{G}_{(z,[\zeta])}(Z,Z)=\sum_{i,j}G_{i\bar{j}}(z,\zeta)Z_i\bar{Z}_j, \text{ for }Z=\sum^r_{i=1}Z_i s_i(z)\in E_z.\end{align*} $$
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 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
$\textrm {End}(O_{P(E)}(-1))$
. 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 [Reference KobayashiKob3, Sec. 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 is 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
$$ \begin{align*} \Theta=\sum_{\alpha,\beta}R_{\alpha\bar{\beta}}\,dz_{\alpha} \wedge d\bar{z}_{\beta}+\sum_{\alpha,l}P_{\alpha\bar{l}}\,dz_{\alpha}\wedge d\bar{w}_l+\sum_{k,\beta}\mathcal{P}_{k\bar{\beta}}\,dw_k\wedge d\bar{z}_{\beta}+\sum_{k,l}Q_{k\bar{l}}\,dw_k\wedge d\bar{w}_l, \end{align*} $$
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,
$$ \begin{align*} \Theta|_{O_{P(E)}(-1)}&=\sum_{\alpha,\beta} \frac{\tilde{G}(R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)} \,dz_{\alpha}\wedge d\bar{z}_{\beta}+\sum_{\alpha,l}\frac{\tilde{G}(P_{\alpha\bar{l}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dz_{\alpha}\wedge d\bar{w}_l\\[3pt] &\quad +\sum_{k,\beta}\frac{\tilde{G}(\mathcal{P}_{k\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dw_k\wedge d\bar{z}_{\beta}+\sum_{k,l}\frac{\tilde{G}(Q_{k\bar{l}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dw_k\wedge d\bar{w}_l, \end{align*} $$
and by [Reference KobayashiKob3, Formula (5.8)] the last three terms vanish, therefore
$$ \begin{align} \Theta|_{O_{P(E)}(-1)}=\sum_{\alpha,\beta} \frac{\tilde{G}(R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}\,dz_{\alpha} \wedge d\bar{z}_{\beta}. \end{align} $$
3 Proofs of Theorem 1 and Corollary 2
Recall that the
$L^2$
-metric
$H_k$
on
$S^kE^*$
is
$$ \begin{align} H_k(u,v):=\int_{P(E_z)} h^k(\Phi_{k,z}(u),\Phi_{k,z}(v) ) \omega_z^{r-1}, \text{ for }u \text{ and } v\in S^kE^*_z, \end{align} $$
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
$S^kE^*_z$
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
$S^kE^*$
.
If for a vector
$v\in E^*$
, we denote its kth 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
$E^*\setminus \{\text {zero section}\}$
.
Next, we use the
$L^2$
-integral feature of
$H_k$
to prove that the Finsler metric
$F_k$
is convex that is, it is convex on each fiber
$E^*_z$
. Fix
$z\in X$
and two vectors
$u,v\in E^*_z$
. Let
$\{s_1,\ldots ,s_r\}$
be a holomorphic frame of E around z with fiber coordinates
$(\zeta _1,\ldots ,\zeta _r)$
, and
$\{t_1,\ldots ,t_r\}$
be the dual frame on
$E^*$
. The frame
$\{t_i\}$
on
$E^*$
induces a frame
$\{t^\alpha \}$
on
$S^kE^*$
with
$t^\alpha =t_1^{\alpha _1}\cdot \cdot \cdot t_r^{\alpha _r}$
and
$\alpha _1+\cdots +\alpha _r=k$
. On
$P(E_z)$
, we consider the chart
$\{[\zeta _1,\ldots ,\zeta _r]: \zeta _r\neq 0\}$
, then
$$ \begin{align*}e:=\frac{\sum_i \zeta_i s_i}{\zeta_r}\end{align*} $$
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
$$ \begin{align*}\frac{\zeta^\alpha}{\zeta_r^k}(e^*)^k, \text{where }\zeta^\alpha=\zeta_1^{\alpha_1}\cdot\cdot\cdot \zeta_r^{\alpha_r}. \end{align*} $$
Therefore, if
$E^*_z\ni u=\sum _i u_it_i$
and
$v=\sum _i v_i t_i$
with
$u_i,v_i\in \mathbb {C}$
, then
$$ \begin{align} \begin{aligned} h^k\big( \Phi_{k,z}((u+v)^k),\Phi_{k,z}((u+v)^k) \big)= \frac{\big|\sum_i (u_i+v_i) \zeta_i\big|^{2k}}{|\zeta_r|^{2k}}h(e^*,e^*)^k. \end{aligned} \end{align} $$
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
$$ \begin{align} h^k\big( \Phi_{k,z}((u+v)^k),\Phi_{k,z}((u+v)^k) \big)= \frac{\big|\sum_i (u_i+v_i) \zeta_i\big|^{2k}}{h^*(\sum_i\zeta_i s_i,\sum_i\zeta_i s_i)^k}; \end{align} $$
the right hand side of formula (3.3) is meaningful on the entire
$P(E_z)$
, and we abbreviate the denominator by
$(h^*)^k$
in the computation below.
Using formula (3.3), we have
$$ \begin{align*} F_k(u+v)&=H_k\big((u+v)^k,(u+v)^k\big)^{1/2k}\\[3pt] &=\Big(\int_{P(E_z)}h^k\big( \Phi_{k,z}((u+v)^k),\Phi_{k,z}((u+v)^k) \big) \omega_z^{r-1} \Big)^{1/2k}\\[3pt] &=\Big(\int_{P(E_z)}\frac{ \big|\sum_i (u_i+v_i) \zeta_i\big|^{2k}}{(h^*)^k} \omega_z^{r-1} \Big)^{1/2k}\\[3pt] &\leq l(\int_{P(E_z)} l( \frac{|\sum_i u_i\zeta_ir|}{(h^*)^{1/2}} + \frac{|\sum_i v_i \zeta_ir|}{(h^*)^{1/2}} r) ^{2k} \omega_z^{r-1} r)^{1/2k}\\[3pt] &\leq \Big(\int_{P(E_z)} \frac{\big|\sum_iu_i\zeta_i\big|^{2k}}{(h^*)^k}\omega_z^{r-1} \Big)^{1/2k}+ \Big(\int_{P(E_z)}\frac{ \big|\sum_iv_i\zeta_i\big|^{2k}}{(h^*)^k}\omega_z^{r-1} \Big)^{1/2k}\\[3pt] &=H_k(u^k,u^k)^{1/2k}+H_k(v^k,v^k)^{1/2k} \\[3pt] &=F_k(u)+F_k(v), \end{align*} $$
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
$E^*\setminus \{\text {zero section}\}$
.
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 [Reference DemaillyDem1, Th. 2.5] and [Reference SommeseSom] (see below for the definition of transversal Levi signature). By Lemma 6, 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 [Reference Feng, Liu and WanFLW3]. We give a simplified proof. Recall that a Finsler metric F is said to have transversal Levi signature
$(r,n)$
, if at every point
$(z,\zeta )\in E\setminus \{\text {zero section}\}$
, 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
$\pi :E\setminus \{\text {zero section}\}\to P(E)$
be the quotient map. It is not hard to verify
$\pi ^*\Theta (h_G)=-\partial \bar {\partial }\log G$
. For any nonzero
$e\in W$
,
$$ \begin{align*} \Theta(h_G)(\pi_*e,\overline{\pi_*e})=\pi^*\Theta(h_G)(e,\overline{e}) =-2\partial \bar{\partial }\log F(e,\overline{e})\\[3pt] =-2\Big(\frac{\partial \bar{\partial }F}{F}-\frac{\partial F\wedge \bar{\partial}F }{F^2}\Big)(e,\overline{e})\geq -2\frac{\partial \bar{\partial }F}{F}(e,\overline{e})>0. \end{align*} $$
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
$S^kE^*$
. The number k will be decided later.
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 [Reference KobayashiKob3]). In this frame,
$$ \begin{align*}\Theta(h)|_{(z_0,[\zeta_0])}=-\sum_{\alpha,\beta}\frac{\partial ^2\log h}{\partial z_\alpha \partial \bar{z}_\beta} dz_\alpha \wedge d\bar{z}_\beta-\sum_{i,j}\frac{\partial ^2\log h}{\partial w_i \partial \bar{w}_j} dw_i \wedge d\bar{w}_j.\end{align*} $$
Since
$\Theta (h)|_{P(E_{z_0})}$
is positive and
$\Theta (h)$
has signature
$(r-1,n)$
, the matrix
$$ \begin{align*}\Big(-\frac{\partial ^2\log h}{\partial z_\alpha \partial \bar{z}_\beta}\Big)\end{align*} $$
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
$$ \begin{align*}\omega_z=\sqrt{-1}\sum_{i,j}g_{i\bar{j}}(z,w)dw_i\wedge d\bar{w}_j,\end{align*} $$
so
$\omega _z^{r-1}=\det (g_{i\bar {j}})\bigwedge _l(\sqrt {-1}dw_l\wedge d\bar {w}_l) $
that we abbreviate as
$\det (g) dw\wedge d\bar {w}$
. By taking k large, the following matrix can be made negative
$$ \begin{align} \Big(\frac{\partial^2 -k\log h-\log \det(g)}{\partial z_\alpha\partial \bar{z}_\beta}\Big). \end{align} $$
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. The number k is fixed from now.
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
$\mathrm {End} S^kE^*$
-valued (1,1)-form. Given a nonzero
$u\in S^kE^*_{z_0}$
, 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
$z_0$
,
$$ \begin{align}\partial \bar{\partial}H_k(u,u) (\eta,\bar{\eta})=-H_k(\Theta u,u) (\eta,\bar{\eta}).\end{align} $$
Meanwhile, if we denote p as the projection from
$P(E)$
to X, then by definition
$$ \begin{align} \begin{aligned} \partial \bar{\partial}H_k(u,u)&=\partial \bar{\partial}p_*\big(h^k(\Phi_{k,z}(u),\Phi_{k,z}(u) ) \omega_z^{r-1}\big)=p_*\partial \bar{\partial}\big(h^k(\Phi_{k,z}(u),\Phi_{k,z}(u) ) \omega_z^{r-1}\big)\\[3pt] &=\sum_l \int_{V_l}\partial \bar{\partial}\big(h^k(\Phi_{k,z}(u),\Phi_{k,z}(u) ) \omega_z^{r-1}\big). \end{aligned} \end{align} $$
Let us compute
$\partial \bar {\partial }(h^k(\Phi _{k,z}(u),\Phi _{k,z}(u) ) \omega _z^{r-1})$
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,
$h^k(\Phi _{k,z}(u),\Phi _{k,z}(u) )=|f|^2h^k(e,e)$
, and we will abbreviate abbreviate
$h^k(e,e)$
by
$h^k$
. Moreover, we have
$\omega _z^{r-1}= \det (g) dw\wedge d\bar {w}$
. Therefore, in
$U_m$
,
$$ \begin{align*} \partial \bar{\partial}\big(h^k(\Phi_{k,z}(u),\Phi_{k,z}(u) ) \omega_z^{r-1}\big)&= \partial \bar{\partial}\big(|f|^2h^k \det(g) dw\wedge d\bar{w}\big)\\[3pt] &=\sum_{\alpha,\beta} \frac{\partial^2 |f|^2h^k\det(g)}{\partial z_\alpha \partial\bar{z}_\beta}dz_\alpha\wedge d\bar{z}_\beta\wedge dw\wedge d\bar{w}. \end{align*} $$
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
$$ \begin{align} \sum_{\alpha,\beta}\frac{\partial^2 |f|^2 e^{-\phi}}{\partial z_\alpha \partial\bar{z}_\beta}\eta_\alpha \bar{\eta}_\beta=\Big( e^{-\phi}\big|\sum_\alpha \frac{\partial f}{\partial z_\alpha}\eta_\alpha-f\sum_\alpha \eta_\alpha \frac{\partial \phi}{\partial z_\alpha}\big|^2-|f|^2e^{-\phi}\sum_{\alpha,\beta}\frac{\partial^2 \phi}{\partial z_\alpha\partial \bar{z}_\beta}\eta_\alpha \bar{\eta}_\beta \Big). \end{align} $$
The matrix in the last term of (3.7)
$$ \begin{align*}\frac{\partial^2 \phi}{\partial z_\alpha\partial \bar{z}_\beta}=\frac{\partial^2 -k\log h-\log \det(g)}{\partial z_\alpha\partial \bar{z}_\beta}\end{align*} $$
is negative by (3.4), so (3.7) is positive, and
$$ \begin{align} \int_{V_l}\partial \bar{\partial}\big(h^k(\Phi_{k,z}(u),\Phi_{k,z}(u) ) \omega_z^{r-1}\big)(\eta,\bar{\eta})>0. \end{align} $$
Hence by (3.5) and (3.6),
$H_k(\Theta u,u)(\eta ,\bar {\eta })<0$
. Therefore, the Hermitian metric
$H_k$
on
$S^kE^*$
is Griffith negative for k large. The proof is complete by using Theorem 1.
The above curvature computation is a combination of ideas from [Reference Liu, Sun and YangLSY, Sec. 4.2], [Reference BerndtssonBer2, Sec. 3].
4 Convexity of the kth root of an inner product
Given a complex vector space V of dimension r, let
$S^kV$
be the kth symmetric power of V. For an inner product H on
$S^kV$
, 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
$\{e_1^2,e_2^2,e_1e_2\}$
is a basis for
$S^2V$
. Consider an inner product H on
$S^2V$
with matrix representation under
$\{e_1^2,e_2^2,e_1e_2\}$
equal to
$$ \begin{align*} \begin{pmatrix} 1/2 & 0 & 0\\ 0 & 1/2 & 0\\ 0&0&1 \end{pmatrix}. \end{align*} $$
We write
$v\in V$
as
$v=v_1e_1+v_2e_2$
with
$v_1=x_1+\sqrt {-1}y_1$
,
$v_2=x_2+\sqrt {-1}y_2$
, and
$x_j,y_j$
real. So
$$ \begin{align*}H(v^2,v^2)= |v_1^2|^2/2+ |v_2^2|^2/2+|2v_1v_2|^2= (x_1^2+y_1^2)^2/2+ (x_2^2+y_2^2)^2/2+4(x_1^2+y_1^2)(x_2^2+y_2^2),\end{align*} $$
$$ \begin{align*} \begin{pmatrix}\frac{\partial^2 H}{\partial x_i \partial x_j}\end{pmatrix}_{i,j}= \begin{pmatrix} 4x_1^2+2(x_1^2+y_1^2) +8(x_2^2+y_2^2) & 16x_1x_2\\ 16x_1x_2 & 4x_2^2+2(x_2^2+y_2^2) +8(x_1^2+y_1^2) \end{pmatrix}. \end{align*} $$
At
$x_1=x_2=1$
and
$y_1=y_2=0$
, this 2-by-2 matrix equals
$$ \begin{align*} \begin{pmatrix} 14 & 16\\ 16 & 14 \end{pmatrix}, \end{align*} $$
which is not semipositive. So the map
$v\mapsto H(v^2,v^2)$
is not convex.
The next example shows that if the kth root of an inner product is convex, the kth 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
$\{e_1^2,e_2^2,e_1e_2\}$
is a basis for
$S^2V$
. 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
$\Phi :S^2V\to H^0(P(V^*),O_{P(V^*)}(2))$
, we define an inner product H on
$S^2 V$
by, for
$\xi ,\eta \in S^2V$
,
$$ \begin{align*}H(\xi,\eta)=\int_{P(V^*)}h^2(\Phi(\xi),\Phi(\eta))\omega. \end{align*} $$
On the chart
$\{[\zeta _1,\zeta _2]:\zeta _2\neq 0\}$
of
$P(V^*)$
, we have a local frame
$$ \begin{align*}e:=\frac{\zeta_1}{\zeta_2}e_1^*+e_2^*\end{align*} $$
for
$O_{P(V^*)}(-1)$
, and a dual frame
$e^*$
for
$O_{P(V^*)}(1)$
. Therefore, the global sections have local expressions
$$ \begin{align} \Phi(e_1^2)=\frac{\zeta_1^2}{\zeta_2^2}(e^*)^2,\text{ } \Phi(e_2^2)=(e^*)^2,\text{ and } \Phi(e_1e_2)=\frac{\zeta_1}{\zeta_2}(e^*)^2. \end{align} $$
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
$\{e_1^2,e_2^2,e_1e_2\}$
. For example,
$$ \begin{align*} H(e_1^2,e_2^2)=\int_{\mathbb{C}}z^2\frac{1}{(|z|^2+1)^2}\frac{dz\wedge d\bar{z}}{(|z|^2+1)^2}=0,\\[3pt] H(e_1^2,e_1^2)=\int_{\mathbb{C}}|z|^4\frac{1}{(|z|^2+1)^2}\frac{dz\wedge d\bar{z}}{(|z|^2+1)^2}=\frac{\pi}{3}, \end{align*} $$
and similarly
$H(e_2^2,e_2^2)=\pi /3, H(e_1e_2,e_1e_2)=\pi /6, H(e_1^2,e_1e_2)=H(e_2^2,e_1e_2)=0$
. Therefore, the matrix representation under
$\{e_1^2,e_2^2,e_1e_2\}$
is
$$ \begin{align*} \begin{pmatrix} \frac{\pi}{3}&0&0\\ 0&\frac{\pi}{3}&0\\ 0&0&\frac{\pi}{6} \end{pmatrix}. \end{align*} $$
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
$\{{e^*_1}^2,{e^*_2}^2,e^*_1e^*_2\}$
$$ \begin{align*} \frac{6}{\pi}\begin{pmatrix} 1/2&0&0\\ 0&1/2&0\\ 0&0&1 \end{pmatrix}, \end{align*} $$
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 [Reference LempertLem1], [Reference LempertLem2]] (see also [Reference RochbergRoc]). Finsler metrics are assumed convex in [Reference LempertLem1], [Reference LempertLem2], 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
$$ \begin{align} K_v(\zeta)=-\inf \partial\bar{\partial}\log G(\phi)\big|_{z_0}(v,\bar{v}), \end{align} $$
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 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
$\{s_1,\ldots ,s_r\}$
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 [Reference KobayashiKob3]). Since the vector
$\zeta \in E_{z_0}$
is nonzero, without loss of generality, we assume its rth component
$\zeta _r\neq 0$
. Suppose
$v=\sum v_\alpha \partial /\partial z_\alpha $
. Under this local trivialization, the holomorphic section
$\phi =(\phi _1,\ldots ,\phi _r)$
with
$\phi _r\neq 0$
near
$z_0$
. The computation is done on a chart
$\{(z,[w_1,\ldots ,w_{r-1},1])\}$
of
$P(E)$
. The connection form of the Chern connection of
$\tilde {G}$
is an
$\textrm {End} \tilde {E}$
-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
$B_i\in \mathrm {End} \tilde {E}$
. 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
$$ \begin{align*} &\frac{\partial }{\partial z_\alpha}\tilde{G}_{(z,[w,1])}(\phi(z),\phi(z))=\tilde{G}_{(z,[w,1])}(D_{z_\alpha}\phi,\phi),\\[3pt] &\frac{\partial }{\partial w_i}\tilde{G}_{(z,[w,1])}(\phi(z),\phi(z))=\tilde{G}_{(z,[w,1])}(D_{w_i}\phi,\phi). \end{align*} $$
By (2.3),
$G(\phi )=G(z,\phi (z))=\tilde {G}_{(z,[\phi (z)])}(\phi (z),\phi (z))$
and we suppress the subscript
$(z,[\phi (z)])$
of
$\tilde {G}$
for simplicity. Therefore, by chain rule
$$ \begin{align*} \partial\bar{\partial} G(z,\phi(z))(v,\bar{v})&=\sum^n_{\alpha,\beta=1}\frac{\partial^2 }{\partial z_\alpha \partial \bar{z}_\beta}\tilde{G}(\phi(z),\phi(z))v_\alpha \bar{v}_\beta\\[3pt] &=\big\|\sum_\alpha v_\alpha D_{z_\alpha}\phi+\sum_{i,\alpha}D_{w_i}\phi\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}v_\alpha\big\|^2_{\tilde{G}} \nonumber \\ &\quad -\sum_{\alpha,\beta}v_\alpha \bar{v}_\beta\Big[\tilde{G}(R_{\alpha\bar{\beta}}\phi,\phi)+\sum^{r-1}_{i=1}\tilde{G}(P_{\alpha\bar{i}}\phi,\phi)\overline{\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\beta}}\\[3pt] &\quad +\sum^{r-1}_{i=1}\tilde{G}(\mathcal{P}_{i\bar{\beta}}\phi,\phi)\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}+\sum^{r-1}_{i,j=1}\tilde{G}(Q_{i\bar{j}}\phi,\phi)\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}\overline{\frac{\partial (\frac{\phi_j}{\phi_r})}{\partial z_\beta}}\Big]. \end{align*} $$
The last three terms involving
$P,\mathcal {P}$
, and Q are zero by [Reference KobayashiKob3, (5.8)]. Meanwhile, by the Cauchy–Schwarz inequality
$$ \begin{align*} \big| \sum_\alpha \frac{\partial}{\partial z_\alpha} G(z,\phi(z)) v_\alpha \big|^2=&\big|\tilde{G}\Big(\sum_\alpha v_\alpha D_{z_\alpha}\phi+\sum_{i,\alpha}D_{w_i}\phi\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}v_\alpha,\phi\Big)\big|^2\\[3pt] \leq& \big\|\sum_\alpha v_\alpha D_{z_\alpha}\phi+\sum_{i,\alpha}D_{w_i}\phi\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}v_\alpha\big\|^2_{\tilde{G}} \|\phi\|^2_{\tilde{G}}. \end{align*} $$
All added up, we see
$$ \begin{align*} &\partial \bar{\partial}\log G(z,\phi(z))(v,\bar{v})\\[3pt] &= \frac{1}{\|\phi\|^4_{\tilde{G}}} \Big(\big\|\sum_\alpha v_\alpha D_{z_\alpha}\phi+\sum_{i,\alpha}D_{w_i}\phi\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}v_\alpha\big\|^2_{\tilde{G}} \| \phi \|^2_{\tilde{G}} -\sum_{\alpha,\beta}v_\alpha \bar{v}_\beta \tilde{G}(R_{\alpha\bar{\beta}}\phi,\phi)\| \phi \|^2_{\tilde{G}}\\[3pt] &\quad - \big|\sum_\alpha \frac{\partial}{\partial z_\alpha} G(z,\phi(z)) v_\alpha \big|^2\Big)\\[3pt] &\geq -\frac{\sum_{\alpha,\beta}v_\alpha \bar{v}_\beta\tilde{G}( R_{\alpha\bar{\beta}}\phi,\phi)}{\|\phi\|^2_{\tilde{G}}}. \end{align*} $$
Therefore,
$$ \begin{align*} \partial \bar{\partial}\log G(z,\phi(z))(v,\bar{v})\geq -\sum_{\alpha,\beta}\frac{\tilde{G} (R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}v_\alpha\bar{v}_\beta; \end{align*} $$
the equality holds if we choose
$\phi (z)=(\zeta _1,\ldots ,\zeta _r)+ (\zeta _1,\ldots ,\zeta _r)\sum _\alpha (z-z_0)_\alpha $
under the frame
$\{s_i\}$
. Indeed,
$$ \begin{align*}\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}\big|_{z_0}=\frac{\frac{\partial \phi_i}{\partial z_\alpha}\phi_r-\phi_i \frac{\partial \phi_r}{z_\alpha} }{\phi_r^2}\big|_{z_0}=0, \text{ and } A_\alpha(z_0,[\zeta])=0.\end{align*} $$
So, at
$(z_0,[\zeta ])$
,
$$ \begin{align*}\sum_\alpha v_\alpha D_{z_\alpha}\phi+\sum_{i,\alpha}D_{w_i}\phi\frac{\partial (\frac{\phi_i}{\phi_r})}{\partial z_\alpha}v_\alpha = \sum_\alpha v_\alpha \zeta \text{ is parallel with } \phi(z_0)=\zeta.\end{align*} $$
As a result,
$$ \begin{align*}K_v(\zeta)=\sum_{\alpha,\beta}\frac{\tilde{G} (R_{\alpha\bar{\beta}}\zeta,\zeta)}{\tilde{G}(\zeta,\zeta)}v_\alpha\bar{v}_\beta=\Theta|_{O_{P(E)}(-1)}(v,\bar{v}).\\[-40pt] \end{align*} $$
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).
Acknowledgments
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, Xueyuan Wan, and Xiaokui Yang for their interest in this work. I thank the referees for their reading of this paper. Thanks are also due to Academia Sinica and National Center for Theoretical Sciences for providing a stimulating environment.
