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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), cf.\ [Ume73].
\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
$$
\det({\,}^T\Theta_{E,h_1}+\omega\otimes\Id_{E^*})^{1/r}=e^f\,{\omega^n\over n!}.
\leqno(2.3)
$$
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''
$$
\det({\,}^T\Theta_{E,h_t}+t\,\omega\otimes\Id_{E^*})^{1/r}=
e^{tf+c(t)}\,{\omega^n\over n!},\quad t\in [0,1],
\leqno(2.4)
$$
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. 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 ask for a condition of
the form $\Theta_{E,h}^0\wedge\omega^{n-1}=0$, but such a condition
requires the polystability of $E$ with respect to $\omega$, and in
any case, it has no reason to be satisfied by $h_1$.
Instead, we can consider a perturbed trace free Hermitian-Yang-Mills
equation of the form
$$
\Theta_{E,h_t}^0\wedge\omega^{n-1}=\Theta_{E,h_1}^0\wedge\omega^{n-1}
+\varepsilon(\log u_t)\,\omega^n,\quad\varepsilon>0,\leqno(2.5)
$$
where $u_t>0$ is the Hermitian endomorphism of $E$ of determinant $1$
such that\break $|v|^2_{h_t}=C_t\langle u_t\cdot v,v\rangle_{h_1}$ for
every $v$ in $E$ (so that $u_1=\Id_E$). It is clear that (2.5) is equi\-valent
to a matrix equation in $\Hom(E,E)^0$ (trace free endomorphisms of $\Hom(E,E)$),
and it can be seen that the combination of (2.4) and (2.5) is a non linear
elliptic system. The addiction of a ``friction term''
$\log u_t$ in the ordinary Hermitian-Yang-Mills equation makes it always
solvable (cf.\ work of Uhlenbeck-Yau [UhY86]). Here, this might be more
involved to check since our system of equations contains one
essentially non linear equation (plus $r^2-1$ essentially linear ones,
in the sense that the order 2 part is linear). If such a universal
solva\-bility property could be proved -- e.g.\ via an extension of Yau's
iteration technique [Yau78] to get 0 order estimates for Monge-Amp\`ere
equations -- then the Griffiths conjecture would find a positive answer.
\bigskip\bigskip

\centerline{\bigbf References}
\vskip8pt

\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 [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

\end

