\section{An example by J.-P. Demailly}

In this section we would like to explain, following \cite[\S8]{Dem97}, how one can construct examples of compact hyperbolic projective manifolds which nevertheless do not admit any hermitian metric of negative holomorphic sectional curvature. Such examples can be generalized to higher order analogues ---namely \lq\lq$k$-jet curvature\rq\rq{}--- of holomorphic sectional curvature: this will be mentioned at the end of the section, and related conjectures that come out from this picture will be discussed at the end of the chapter.

The first observation is the following algebraic criterium for the nonexistence of a metric with negative holomorphic sectional curvature. Let $X$ be a complex manifold, $C$ be a compact Riemann surface, and $F\colon C\to X$ be a non constant holomorphic map. Let $m_p\in\mathbb N$ be the multiplicity at $p\in C$ of $F$. Clearly, $m_p=1$ except possibly at finitely many points of $C$, and $m_p\ge 2$ if and only if $F$ is not an immersion at $p$. 

\begin{theorem}[Demailly {\cite[Special case of Theorem 8.1]{Dem97}}]\label{algcrit}
Consider \newline$(X,\omega)$ a compact hermitian manifold and let $F\colon C\to X$ be a non constant holomorphic map from a compact Riemann surface $C$ of genus $g=g(C)$ to $X$. Suppose that $\operatorname{HSC}_\omega\le-\kappa$ for some $\kappa\ge 0$. Then,
$$
2g-2\ge\frac{\kappa}{2\pi}\deg_\omega C+\sum_{p\in C}(m_p-1),
$$ 
where $\deg_\omega C=\int_C F^*\omega>0$ is the degree of $C$ with respect to $\omega$.
\end{theorem}

We shall use this theorem especially in the case where $C$ is the normalization of a singular curve in $X$ and $F$ the normalization map. Observe that, in particular, we recover the well-known fact that on a compact hermitian manifold with negative holomorphic sectional curvature there are no rational nor elliptic curves (even singular), and that there are no rational (possibly singular) curves on a compact hermitian manifold with non positive holomorphic sectional curvature.

\begin{proof}
The differential $F'$ of $F$ gives us a map $F'\colon T_C\to F^* T_X$. This map is injective at the level of sheaves, but not necessarily at the level of vector bundle, since $F'$ may vanish at some point. Taking into account these vanishing points counted with multiplicities, we obtain the following injection of vector bundles
$$
F'\colon T_C\otimes\mathcal O_C(D)\to F^*T_X,
$$
where we defined the effective divisor $D$ to be $\sum_{p\in C}(m_p-1)\, p$. 
Thus, via $F'$, we realized  $T_C\otimes\mathcal O_C(D)$ as a subbundle of the hermitian vector bundle $(F^*T_X, F^*\omega)$, and ---as such--- we can endow it with the induced metric $h=F^*\omega|_{T_C\otimes\mathcal O_C(D)}$ (observe the we consider $F^*\omega$ not as a pull-back of differential forms, but as a pull-back of hermitian metrics). 

Now, a local holomorphic frame for $T_C\otimes\mathcal O_C(D)$ around a point $p\in C$ is given by $\eta(t)=1/t^{m_p-1}\,\frac{\partial}{\partial t}$, where $t$ is a holomorphic coordinate centered at $p$. Call $\xi(t)=F'(\eta(t))\in (F^*T_X)_t=T_{X,F(t)}$, so that $\xi$ is a local holomorphic frame for $T_C\otimes\mathcal O_C(D)$ when seen as a subbundle of $F^*T_X$. We have, for the Griffiths curvature of $(T_C\otimes\mathcal O_C(D),h)$,
\begin{multline*}
\langle\Theta(T_C\otimes\mathcal O_C(D),h)(\partial/\partial t,\partial/\partial \bar t\,)\cdot \xi,\xi\rangle_h \\
=\Theta(T_C\otimes\mathcal O_C(D),h)(\partial/\partial t,\partial/\partial \bar t\,)\,\underbrace{||\xi||^2_h}_{=||\xi||^2_\omega}.
\end{multline*}
By the classical Griffiths' formulae, we have the following decreasing property for the Griffiths curvatures:
\begin{multline*}
\langle\Theta(T_C\otimes\mathcal O_C(D),h)(\partial/\partial t,\partial/\partial \bar t\,)\cdot \xi,\xi\rangle_h \\
\le \langle\Theta(F^*T_X,F^*\omega)(\partial/\partial t,\partial/\partial \bar t\,)\cdot \xi,\xi\rangle_{F^*\omega} \\
=  \langle F^*\Theta(T_X,\omega)(\partial/\partial t,\partial/\partial \bar t\,)\cdot \xi,\xi\rangle_{F^*\omega}\\
= \langle\Theta(T_X,\omega)(F'(\partial/\partial t),\overline{F'(\partial/\partial t)}\,)\cdot \xi,\xi\rangle_{\omega}\\
=|t^{m_p-1}|^2\,\langle\Theta(T_X,\omega)(\xi,\bar\xi\,)\cdot \xi,\xi\rangle_{\omega}
\le -\kappa |t^{m_p-1}|^2 ||\xi||^4_\omega,
\end{multline*}
where the last inequality holds since $\langle\Theta(T_X,\omega)(\xi,\bar\xi\,)\cdot \xi,\xi\rangle_{\omega}=||\xi||^4_\omega\,\operatorname{HSC}_\omega(\xi)$. Therefore, we obtain
$$
\Theta(T_C\otimes\mathcal O_C(D),h)(\partial/\partial t,\partial/\partial \bar t\,)\le-\kappa |t^{m_p-1}|^2 ||\xi||^2_\omega=i\kappa\,(F^*\omega)(\partial/\partial t,\partial/\partial \bar t\,),
$$
where by $F^*\omega$ here we mean the pull-back at the level of differential forms. Summing up, we have obtained that
$$
i\,\Theta(T_C\otimes\mathcal O_C(D),h)\le -\kappa\,F^*\omega,
$$
as real $(1,1)$-forms.
But then, 
$$
\int_C\frac{i}{2\pi}\,\Theta(T_C\otimes\mathcal O_C(D),h)\le-\frac{\kappa}{2\pi}\int_C F^*\omega=-\frac{\kappa}{2\pi}\deg_\omega C,
$$
and 
$$
\int_C\frac{i}{2\pi}\,\Theta(T_C\otimes\mathcal O_C(D),h)=\deg(T_C\otimes\mathcal O_C(D))=2-2g+\sum_{p\in C}(m_p-1),
$$
since $\deg(T_C)=2-2g$ by Hurwitz's formula. The statement follows.
\end{proof}

Following Demailly, we shall now exhibit a smooth projective surface which is Kobayashi hyperbolic, with ample canonical bundle, but which cannot admit any hermitian metric with negative holomorphic sectional curvature. It will be constructed as a fibration of Kobayashi hyperbolic curves onto a Kobayashi hyperbolic curve, with at least one \lq\lq very\rq\rq{} singular fiber, which will violate the above criterium.

\begin{proposition}[Cf. {\cite[8.2. Theorem]{Dem97}}]
There is a smooth projective surface $S$ which is hyperbolic (and hence with ample canonical bundle $K_S$) but does not carry any hermitian metric with negative holomorphic sectional curvature. Moreover, given any two smooth compact hyperbolic Riemann surfaces $\Gamma, \Gamma'$, such a surface can be obtained as a fibration $S\to\Gamma$, with hyperbolic fibers, in which (at least) one of the fibers is singular and has $\Gamma'$ as its normalization. 
\end{proposition}

\begin{proof}
Take any compact hyperbolic Riemann surface $\Gamma'$, and let $g=g(\Gamma')\ge 2$ be its genus. Now, we modify it into a singular compact Riemann surface $\Gamma''$ of the same genus, whose normalization is $\Gamma'$. 

In order to do so, consider a pair of positive relatively prime integers $(a,b)$, with $a<b$, an the associated affine plane curve $C$ in $\mathbb C^2$ given by the equation $y^a-x^b=0$, which has a monomial singularity of type $(a,b)$ at $0\in\mathbb C^2$. Its normalization is given by $\mathbb C\ni t\mapsto (t^a,t^b)\in\mathbb C^2$. Choose integers $n,m$ such that $na+mb=1$. Then, the restriction of the rational function on $\mathbb C^2$ defined by $(x,y)\mapsto x^ny^m$ to $C$ gives a holomorphic coordinate on it minus the singular point (this is actually the inverse map of the normalization map outside the singularity). In particular, the set of points $(x,y)\in C$ such that $0<|x|<1$ is biholomorphic to the punctured unit disc.

Now, take a point $x_0\in\Gamma'$ and choose a holomorphic coordinate centered at $x_0$ such that we can select a neighborhood of $x_0$ whose image is the unit disc \textsl{via} this coordinate. Finally, remove the point $x_0$ in order to obtain a holomorphic coordinate chart whose image is the punctured unit disc. By identifying with the punctured unit disc constructed above, we replace this neighborhood of $x_0$ with the set of point $(x,y)\in C$ such that $|x|<1$, thus creating the desired singularity at $x_0$. Call the resulting curve $\Gamma''$. By construction, the normalization of $\Gamma''$ is exactly $\Gamma'$, and $\Gamma''$ has one single singular point, whose singularity type is plane and monomial of type $(a,b)$ (for an excellent and very elementary discussion around this subject we refer the reader to \cite[Chapter III, Section 2]{Mir95}).

Next, we embed $\Gamma''$ in some large projective space, and then we project it to $\mathbb P^2$, in such a way that the singular point is left untouched and outside it we create at most a finite number of nodes (\textsl{i.e.} plane monomial singularity of type $(2,2)$). Call the resulting projective plane curve $C_0$, whose normalization is of course again $\Gamma'$. Observe that the normalization map $\nu\colon\Gamma'\to C_0$ is an immersion outside the (single) preimage of the first singular point we created. On the other hand, at this point it has multiplicity $a$.

In order to obtain the desired surface $S$, we select then $a$ so that $a-1>2g-2$, \textsl{i.e.} $a\ge 2g$. Such a surface $S$ then does contain a curve which violates the criterium given in Theorem \ref{algcrit}, and we are done.

Take a (reduced) homogeneous polynomial equation $P_0(z_0,z_1,z_2)=0$ for $C_0$ in $\mathbb P^2$. Then, we necessarily have $d=\deg P_0\ge 4$, since otherwise $C_0$ would be normalized by a rational or an elliptic curve. Next, complete $P_0$ into a basis $\{P_0,P_1,\dots,P_N\}$ of the space $H^0(\mathbb P^2,\mathcal O(d))$ of homogeneous polynomials of degree $d$ in three variables, and consider the corresponding universal family 
$$
\mathcal U=\bigl\{\bigl([z_0:z_1:z_2],[\alpha_0:\cdots:\alpha_N]\bigr)\in\mathbb P^2\times\mathbb P^N\mid\sum_{j=0}^N\alpha_j\,P_j(z)=0\bigr\}\subset\mathbb P^2\times\mathbb P^N,
$$
of curves of degree $d$ in $\mathbb P^2$, together with the projection $\pi\colon\mathcal U\to\mathbb P^N$. Our starting curve $C_0$ is then the fiber $U_{[1:0:\cdots:0]}$ over the point $[1:0:\cdots:0]\in\mathbb P^N$. Now, we embed the first curve $\Gamma$ into $\mathbb P^N$ (this is of course possible since $N\ge 3$) in such a way that $[1:0:\cdots:0]\in\Gamma$. The desired fibration $S\to\Gamma$ will be obtained as the pull-back family
$$
\xymatrix{S=\mathcal U\times_{\mathbb P^N}\Gamma \ar@{->}[r]\ar@{->}[d]& \mathcal U \ar@{->}[d] \\ \Gamma \ar@{^{(}->}[r]& \mathbb P^N.}
$$
Of course, we have to select carefully the embedding of $\Gamma$ into $\mathbb P^N$, so that $S$ will be non singular, and in such a way that we have a good control of the singular fibers out of $U_{[1:0:\cdots:0]}$.

In order to do so, the first observation is that ---as it is well-known--- the locus $Z$ in $\mathbb P^N$ which corresponds to singular curve is an algebraic hypersurface and, moreover, the locus $Z'\subset Z$ which corresponds to curves which have not only one node in their singularity set is of codimension $2$ in $\mathbb P^N$. In particular, by possibly moving $\Gamma$ with a generic projective automorphism of $\mathbb P^N$ leaving fixed $[1:0:\cdots:0]$, we can suppose that $\Gamma\cap Z'=\{[1:0:\cdots:0]\}$, so that all the fibers of $S$, except from $C_0$, are either smooth, or with a single node. If such an $S$ were non singular, we would be done. Indeed, by PlŸcker's formula, the smooth fibers have genus $(d-1)(d-2)/2\ge 3$, $U_{[1:0:\cdots:0]}$ has genus $g\ge 2$ by construction, and the other singular fibers have genus $(d-1)(d-2)/2-1\ge 2$, since they have only one node. Therefore, $S$ is a fibration onto a hyperbolic Riemann surface with all hyperbolic fibers and is then hyperbolic (and hence with ample canonical bundle), with a fiber which contradicts Theorem \ref{algcrit}.

So we are left to check the smoothness of $S$, knowing that we can possibly use again generic automorphisms of $\mathbb P^N$ leaving fixed $[1:0:\cdots:0]$ to move $\Gamma$. Thus, since $\Gamma$ is embedded in $\mathbb P^N$, we can think at $S$ as included in $\mathcal U$, and since $\mathcal U$ is smooth, Bertini's theorem immediately implies that $S$ can be chosen non singular outside $U_{[1:0:\cdots:0]}$. Now, what about points along $U_{[1:0:\cdots:0]}$? Fix such a point $([z_0:z_1:z_2],[1:0:\cdots:0])\in U_{[1:0:\cdots:0]}$, and suppose, just to fix ideas, that $z_0\ne 0$. Take the corresponding affine coordinates, say $((z,w),(a_1,\dots,a_N))$ around this point, set $p_j(z,w)=P_j(1,z,w)$ to be the dehomogenization of the $P_j$'s, and let $f_1(a),\dots,f_r(a)$ be affine equations of the curve $\Gamma$. Then, we have to check the rank of the following Jacobian matrix at the point $\bigl((z,w),(0,\dots,0)\bigr)$, the affine equation for $\mathcal U$ being $p_0(z,w)+\sum_{j=1}^N a_j\,p_j(z,w)=0$:
$$
\begin{pmatrix}
1
\end{pmatrix}
$$

\end{proof}