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\centerline{\hugebf Hermitian-Yang-Mills approach to the conjecture of}\vskip6pt
\centerline{\hugebf Griffiths on the positivity of ample vector bundles}
\bigskip
\centerline{\bf Jean-Pierre Demailly}
\centerline{Universit\'e Grenoble Alpes, Institut Fourier}
\centerline{December  27, 2019}
\bigskip

\noindent
{\bigbf 1. Introduction}\medskip

Let $X$ be a projective $n$-dimensional manifold. A conjecture due to
Griffiths [Gri69] stipulates that a holomorphic vector bundle $E\to X$
is ample (in the sense of Hartshorne, meaning that that associated
line bundle $\cO_{\bP(E)}(1)$ is ample) if and only if it possess a
Hermitian metric $h$ such that the Chern curvature tensor
$\Theta_{E,h}=i\nabla_{E,h}^2$ is Griffiths positive. In other words,
if $\rank E=r$ and
$$
\Theta_{E,h}=\sum_{1\le j,k\le n,1\le}c_{jk\lambda\mu}dz_j\wedge
d\ol z_k\otimes e_\lambda^*\otimes e_\mu\leqno(1.1)
$$
in terms of holomorphic coordinates $(z_1,\ldots,z_n)$ on $X$ and of
an orthonormal frame $(e_\lambda)_{1\le\lambda\le r}$ of $E$,
the associated quadratic form
$$
\widetilde\Theta_{E,h}(\xi\otimes v):=
\langle\Theta_{E,h}(\xi,\ol\xi)\cdot v,v\rangle=
\sum_{1\le j,k\le n,1\le}c_{jk\lambda\mu}\xi_j\ol\xi_k v_\lambda\ol v_\mu
\leqno(1.2)
$$
takes positive values on non zero tensors $\xi\otimes v\in T_X\otimes E$.
A stronger concept is Nakano positivity, asserting that
$$
\widetilde\Theta_{E,h}(\tau):=
\sum_{1\le j,k\le n,1\le}c_{jk\lambda\mu}\tau_{j\lambda}\ol\tau_{k\mu}>0
\leqno(1.3)
$$
for all non zero tensors $\tau=\sum_{j,\lambda}
\tau_{j\lambda}{\partial\over\partial z_j}\otimes e_\lambda\in T_X\otimes E$.
It is also interesting to consider the curvature tensor of the dual
bundle $E^*$, which happens to be given by the opposite of the transpose
of $\Theta_{E,h}$, i.e.
$$
\Theta_{E^*,h}=-{}^T\Theta_{E,h}=
\sum_{1\le j,k\le n,1\le}c_{jk\mu\lambda}dz_j\wedge
d\ol z_k\otimes (e^*_\lambda)^*\otimes e^*_\mu.
\leqno(1.4)
$$
This leads to the concept of dual Nakano positivity, stipulating that
$$
-\widetilde\Theta_{E^*,h}(\tau)=
\sum_{1\le j,k\le n,1\le}c_{jk\mu\lambda}\tau_{j\lambda}\ol\tau_{k\mu}>0
\leqno(1.5)
$$
for all non zero tensors $\tau=\sum_{j,\lambda}
\tau_{j\lambda}{\partial\over\partial z_j}\otimes e^*_\lambda\in T_X\otimes E^*$.
Well known results due to Kodaira (the Kodaira embedding theorem)
show that
$$
\widetilde\Theta_{E,h}~\hbox{dual Nakano positive}~~\Rightarrow~~
\widetilde\Theta_{E,h}~\hbox{Griffiths positive}~~\Rightarrow~~
E~\hbox{ample}.\leqno(1.6)
$$
Conversely, we may ask for the following:\vskip8pt

\vbox{\noindent
{\bf Basic question.} {\it Does it hold that}\vskip7pt
\line{(1.7)\hfill $E~\hbox{ample}~\Rightarrow~
\widetilde\Theta_{E,h}~\hbox{dual Nakano positive}~?$\hfill}}
\medskip

\noindent A positive answer would clearly settle the Griffiths
conjecture, in an even stronger form. One should observe
that Nakano positivity also implies Griffiths positivity, but is in
general a much stronger condition. As a consequence, one cannot
expect ampleness to imply Nakano positivity. For instance, $T_{\bP^n}$
is known to be ample and Nakano semi-positive, but it is not Nakano
positive, as the Nakano vanishing theorem [Nak55] would then yield
$$H^{n-1,n-1}(\bP^n,\bC)=
H^{n-1}(\bP^n,\Omega^{n-1}_{\bP^n})=
H^{n-1}(\bP^n,K_{\bP^n}\otimes T_{\bP^n})=0.
\leqno(1.8)
$$
On the other hand, it does not seem that there are any known examples of
ample vector bundles that are not dual Nakano positive, thus the above basic
question seems to be open.

So far, the Griffiths conjecture is known to hold when $n=\dim X=1$ or
$r=\rank E=1$ (in which cases, Nakano and dual Nakano positivity
coincide with Griffiths positivity). Proofs can be found in [Ume73,
Theorem 2.6] and in [CaF90]. In both cases, the proof is based on the
existence of Harder-Narasimhan filtrations and the theorem of
Narasimhan-Seshadri for stable vector bundles -- the 1-dimensional
case of the Donaldson-Uhlenbeck-Yau theorem. It is tempting to
investigate whether techniques of gauge theory could be used to
approach the Griffiths conjecture. In this direction, P.~Naumann
[Nau17] recently proposed a K\"ahler-Ricci flow method that starts
with a given Finsler metric of positive curvature, and converges to
a Hermitian metric. It is however quite unclear whether the flow used
by P.~Naumann preserves positivity, so it might very well in the end
produce a Hermitian metric that does not have positive curvature.

Here, we describe a different continuity method based on a natural
determinantal equation, that is designed to enforce positivity of
the curvature, actually in the dual Nakano sense. It would however
remain to prove long time existence of the flow for the said equation
or one of its variants.\bigskip

\noindent
{\bigbf 2. Approach via a combination of Monge-Amp\`ere and\vskip0pt
Hermitian-Yang-Mills equations}\medskip

If $\Theta_{E,h}$ is dual Nakano positive, then the ${1\over r}$-power of
the $(nr)$-dimensional determinant of the corres\-ponding Hermitian quadratic
form on $T_X\otimes E^*$ can be seen as a positive $(n,n)$-form
$$
\det({\,}^T\Theta_{E,h})^{1/r}=\det(c_{jk\mu\lambda})_{(j,\lambda),(k,\mu)}^{1/r}\,
i dz_1\wedge d\ol z_1\wedge\ldots\wedge i dz_n\wedge d\ol z_n.
\leqno(2.1)
$$
Moreover, this $(n,n)$-form does not depend on the choice of coordinates $(z_j)$
on~$X$, nor on the choice of the orthonormal frame $(e_\lambda)$ on $E$
(but the orthonormality of $(e_\lambda)$ is required). Conversely, the idea
is that assigning $\det({\,}^T\Theta_{E,h})^{1/r}$ to be a positive volume form
may enforce the dual Nakano positivity of $\Theta_{E,h}$ if that
assignment is combined with a continuity technique. In fact, let us fix an
arbitrary smooth Hermitian metric $h_1$ on $E$, and a k\"ahler metric
$\omega$ taken in a sufficiently large K\"ahler class, so that
$$
{}^T\Theta_{E,h_1}+\omega\otimes\Id_{E^*}
=\omega\otimes\Id_{E^*}-\Theta_{E^*,h_1^*}>0~~
\hbox{in the sense of Nakano}.\leqno(2.2)
$$
Assuming $E$ ample, one could take e.g.\ $\omega$ in a sufficiently large
multiple $\alpha\,c_1(E)$.
Let $f$ be the (uniquely determined) smooth function on $X$ such that
$$
\det_{T_X\otimes E^*}({\,}^T\Theta_{E,h_1}+\omega\otimes\Id_{E^*})^{1/r}
=e^f\,{\omega^n\over n!}.
\leqno(2.3)
$$
We try to find a time dependent family of metrics $h_t(z)=h(z,t)$ on
the fibers $E_z$ of $E$, $t\in [0,1]$, satisfying the
``matrix-type Monge-Amp\`ere equation''
$$
\det_{T_X\otimes E^*}({\,}^T\Theta_{E,h_t}+t\,\omega\otimes\Id_{E^*})^{1/r}=
e^{tf+c(t)}\,{\omega^n\over n!},\quad t\in [0,1],
\leqno(2.4)
$$
where $t\mapsto c(t)$ is a smooth real valued function. Formula (2.3) shows
that the equation is satisfied for $t=1$ (with $c(1)=0$), and we want to
solve (2.4) backwards, down to $t=0$. If this is possible (with a solution
depending smoothly on $t\in[0,1]$ and no ``explosion''), then the tensor
${}^T\Theta_{E,h_t}+t\,\omega\otimes\Id_{E^*}$ must remain Nakano positive
for all $t\in[0,1]$, and in particular, $h_0$ will define a smooth metric
on $E$ such that $\Theta_{E,h_0}$ is dual Nakano positive. For $r=1$, equation
(2.4) is a standard Monge-Amp\`ere equation. Yau's theorem guarantees the
existence of a time dependent solution
$h_t(z)=h_1(z)\,e^{-\varphi(z,t)}$, with constants $c(t)$ chosen
such that
$$
\int_X e^{tf+c(t)}\,\omega^n=(c_1(E)+t\{\omega\})^n,\quad
\hbox{i.e}~~
c(t)=\log(c_1(E)+t\{\omega\})^n\Big/\int_X e^{tf}\,\omega^n.
\leqno(**)
$$
The metric $h_0$ is then the unique one such that
$\Theta_{E,h_0}>0$ and $(\Theta_{E,h_0})^n=e^{c(0)}\omega^n$.
For $r>1$, however, the equation (2.4) is underdetermined, since the
metric $h(z,t)$ on $E$ depends on $r^2$ real functions ($r^2$ is the
real dimension of the space of Hermitian $r\times r$ matrices), while
(2.4) provides only a rank $1$ real equation on $X$. In order to
enforce a priori estimates on solutions (and hopefully, existence and
uniqueness), we need a matrix equation in real dimension $r^2$, thus
we have to add $r^2-1$ equations that are geometrically
significant.

One possibility is to consider a ``trace free'' Hermitian-Yang-Mills
equation. Define $\Hom(E,E)^0\subset \Hom(E,E)$ to be the subspace of
endomorphisms with trace equal to~$0$, i.e.\ the Lie algebra
of ${\rm SL}(E)$. For any Hermitian metric $h$, let $\Theta_{E,h}^0=\Theta_{E,h}-
{1\over r}\Theta_{\det E,\det h}\otimes\Id_E$, which is the curvature tensor
of $E\otimes(\det E)^{-1/r}$ and satisfies $\Tr_E(\Theta_{E,h}^0)=0=
\Tr_E({\,}^T\Theta_{E,h}^0)$. We might take a family
$(\omega_t)_{t\in[0,1]}$ of K\"ahler metrics and ask for a condition of
the form $\Theta_{E,h_t}^0\wedge\omega_t^{n-1}=A(z,t)$ where $A(z,t)$
is a suitably designed right hand side with values in $\Hom(E,E)^0$.
Taking $A(z,t)=0$ could be achieved only when $E$ is polystable with
respect to~$\omega_t$, a condition which we certainly cannot expect
in general (and in any any case, we have to start at $t=1$ with a metric
that does not satisfy this property). Instead, we can consider a
perturbed trace free Hermitian-Yang-Mills equation of the form
$$
\Theta_{E,h_t}^0\wedge\omega_t^{n-1}=\beta(t)\,\Theta_{E,h_1}^0\wedge\omega_t^{n-1}
+\varepsilon(t)(\log u_t)\,\omega_t^n,\quad\varepsilon>0,\leqno(2.5)
$$
where $\beta,\varepsilon:[0,1]\to\bR$ are smooth functions,
and $u_t>0$ is the Hermitian endomorphism of $E$ of
determinant $1$ such that $|v|^2_{h_t}=C_t\langle u_t\cdot v,v\rangle_{h_1}$
for every $v$ in $E$ (so that $u_1=\Id_E$, $\log u_1=0$). The addition of
a ``friction term'' $\log u_t$ in the ordinary Hermitian-Yang-Mills
equation makes it always solvable, even when $E$ does not enjoy any
stability properties (cf.\ work of Uhlenbeck-Yau [UhY86]). Notice that
(2.5) is trivially fulfilled for $t=1$ if we take $\beta(1)=1$; one might
want to have $\beta$ increasing with $\beta(0)=0$, and 
. We are still free of picking a
family of K\"ahler metrics~$(\omega_t)$. A~rather natural choice is to take
$\omega_t$ closely related to the curvature of $E$, e.g.
$$
\omega_t=
{1\over r}\Tr_{E^*}\Big({\,}^T\Theta_{E,h_t}+t\,\omega\otimes\Id_{E^*}\Big)
={1\over r}\Tr_E\Theta_{E,h_t}+t\,\omega=
{1\over r}\Theta_{\det E,\det h_t}+t\,\omega,
\leqno(2.6)
$$
since (2.4) implies $\omega_t>0$ whenever the solution exists. Also note
that $\omega_0={1\over r}\Theta_{\det E,\det h_0}$. We now prove the
following:\medskip

\noindent
{\bf Theorem.} {\it Let $(E,h_1)$ be a smooth Hermitian holomorphic vector
bundle and $\omega$ a K\"ahler metric so large that
${}^T\Theta_{E,h}+\omega\otimes\Id_{E^*}$ is positive in the
sense of Nakano. We consider a smoothing varying family of
Hermitian metrics $(h_t)_{t\in[0,1]}$ subject to the conditions
$$
\leqalignno{
&\det_{T_X\otimes E^*}({\,}^T\Theta_{E,h_t}+t\,\omega\otimes\Id_{E^*})^{1/r}=
e^{tf+c(t)}\,{\omega^n\over n!},\quad c(t)\in\bR,&(2.4)\cr
&\Theta_{E,h_t}^0\wedge\omega_t^{n-1}=t\,\Theta_{E,h_1}^0\wedge\omega_t^{n-1}
+\varepsilon(\log u_t)\,\omega_t^n,\quad\varepsilon>0,&(2.5)
\cr}$$
where
$\omega_t={1\over r}\Tr_E\Theta_{E,h_t}+t\,\omega$. For any $\varepsilon>0$,
this is a well determined
elliptic system of equations. In~particular, the set of values
$t\in[0,1]$ for which the solution exists is open, and if it exists
for some $t_0$, we have also uniqueness of the solution in a neighborhood
of~$t_0$. If a smooth solution exists on the whole interval $[0,1]$,
then $E$ is dual Nakano positive.}

What would remain to be done, assuming $E$ ample, is that our elliptic
system does not lead to an ``explosion'' of the solution when
$t$ decreases to~$0$. One possibility would be to show that any
explosion at time $t_0>0$ produces a ``destabilizing subsheaf''
$\cS$ contradicting the ampleness of $E/\cS$. One migh try to
combine the ideas of [UhY86] with the iteration technique of [Yau78]
to get 0 order estimates for Monge-Amp\`ere equations.
\medskip

\noindent
{\it Proof of the theorem.}
\bigskip\bigskip

\centerline{\bigbf References}
\vskip8pt

\Bibitem [CaF90]&F. Campana F., Flenner, H.:& A characterization of ample
vector bundles on a curve,& Math.\ Ann.\ {\bf 287} (1990),  571--575&

\Bibitem [Gri69]&Griffiths, P.A:& Hermitian differential geometry, Chern classes and positive vector bundles,& Global Analysis, papers in honor of K. Kodaira, Princeton Univ. Press, Princeton (1969), 181--251&

\Bibitem [Nak55]&Nakano, S.:& On complex analytic vector bundles,& J.~Math.\ Soc.\ Japan {\bf 7} (1955) 1--12&

\Bibitem [NaS65]&Narasimhan, M. S., Seshadri, C. S.:& Stable and unitary
vector bundles on a compact Riemann surface,& Ann.\ of Math.\ {\bf 82} (1965),
540--567&

\Bibitem [Nau18]&Naumann, P.:& An approach to Griffiths conjecture,&
arXiv:1710.10034, math.AG&

\Bibitem [UhY86]&Uhlenbeck, K., Yau, S.T.:& On the existence of 
Hermitian-Yang-Mills connections in stable vector bundles,& Comm.\
Pure and Appl.\ Math.\ {\bf 39} (1986) 258--293&

\Bibitem [Ume73]&Umemura, H.:& Some results in the theory of vector bundles,&
Nagoya Math.\ J.\ {\bf 52} (1973), 97--128&

\Bibitem [Yau78]&Yau, S.T.:& On the Ricci curvature of a complex K\"ahler
manifold and the complex Monge-Amp\`ere equation I,& Comm.\ Pure and Appl.\
Math.\ {\bf 31} (1978) 339--411&
\vskip15pt

\noindent
Jean-Pierre Demailly\hfil\break
Universit\'e Grenoble Alpes, Institut Fourier (Math\'ematiques)\hfil\break
UMR 5582 du C.N.R.S., 100 rue des Maths, 38610 Gi\`eres, France\hfil\break
{\it e-mail:}\/ jean-pierre.demailly@univ-grenoble-alpes.fr

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