\section{Motivations from birational geometry}

We take the opportunity here to reproduce a quite standard argument in order to see how to reduce the Kobayashi conjecture on the ampleness of the canonical bundle of compact projective hyperbolic manifolds to showing that projective manifolds $X$ with trivial first real Chern class are not hyperbolic. 
The same strategy can also be applied to have a proof, birational in spirit, of the Yau conjecture. All this, provided the abundance conjecture is true. Indeed, a lightly more general statement can be obtained and also the same kind of arguments can be applied to compact KŠhler manifolds, as we shall see.

First of all, if the canonical bundle $K_X$ is nef, then the abundance conjecture predicts that $K_X$ should be semi-ample, \textsl{i.e.} some big tensor power of $K_X$ should be generated by its global sections. This is known in dimension at most three.

So, let $X$ be a smooth Kobayashi hyperbolic projective manifold. By the celebrated criterion of Mori, $K_X$ is nef -- otherwise $X$ would contain a rational curve. Thus, $K_X$ is already in the closure of the ample cone. Suppose now, and for the rest of the section, that the abundance conjecture holds true (in particular all we are saying hold in dimension at most three, unconditionally). Thus, we have that $K_X$ is semi-ample. We are therefore able to use the following for the canonical bundle.

\begin{theorem}[Semiample Iitaka fibrations {\cite[Theorem 2.1.27]{Laz04}}]
Let $X$ be a normal projective variety and $L\to X$ a semi-ample line bundle on $X$. Then, there is an algebraic fiber space (\textsl{i.e.} a projective surjective mapping with connected fibers)
$$
\phi\colon X\to Y,
$$
with $\dim Y=\kappa(L)$, having the property that, for all sufficiently big and divisible integers $m$, it coincides with the map associated to the complete linear system $|L^{\otimes m}|$. Furthermore, there is an integer $f$ and an ample line bundle $A$ on $Y$ such that $L^{\otimes f}\simeq\phi^*A$.
\end{theorem}

Now, since $K_X$ is semi-ample, we get an algebraic fiber space on $X$
$$
\phi\colon X\to Y,
$$
with $\dim Y=\kappa(X)$, and such that some power, say $K_X^{\otimes f}$, of the canonical bundle is the pull-back of an ample divisor $A$ on $Y$. In particular, for every general (hence smooth) fiber $F$ of $\phi$, we have by taking the determinant of the short exact sequence
$$
0\to T_F\to T_X|_F\to\mathcal O_F^{\oplus\kappa(X)}\to 0,
$$
that $K_F\simeq K_X|_F$. But then, $K_F^{\otimes f}\simeq K_X^{\otimes f}|_F\simeq\phi^*A|_F\simeq\mathcal O_F$. Thus, $K_F$ is torsion, and the general fiber has Kodaira dimension zero and trivial first Chern class in real cohomology.

Suppose to be able to show that projective manifolds with trivial real first Chern class are not hyperbolic. We claim that this implies that the Kodaira dimension of a projective Kobayashi hyperbolic manifold $X$ must be maximal, that is, $X$ is of general type. Indeed, if $1\le\kappa(X)<\dim X$, then $\phi$ has positive dimensional fibers and the general ones have zero Kodaira dimension. So we would, by our assumptions, find a non-Kobayashi hyperbolic positive dimensional subvariety of $X$, contradiction.

Now, if $K_X$ is big and there are no rational curves on $X$ it is not difficult to show that $K_X$ is ample (cf. Lemma \ref{lem:ratcurv}), and we are done.

Next, how to prove that a projective manifold $X$ with trivial real first Chern class is not hyperbolic? By the Beauville--Bogomolov decomposition theorem \cite{Bea83}, a compact KŠhler manifold with vanishing real first Chern class is, up to finite Žtale covers, a product of complex tori, Calabi--Yau manifolds and irreducible holomorphic symplectic manifolds. Since complex tori are obviously not Kobayashi hyperbolic, one is reduced to showing that Calabi--Yau manifolds and irreducible holomorphic symplectic manifolds are not Kobayashi hyperbolic (since Kobayashi hyperbolicity is preserved under Žtale covers). Very recently, in the spectacular paper \cite{Ver15}, Verbitsky has shown ---among other things--- that irreducible holomorphic symplectic manifolds with second Betti number greater than three (a condition that should indeed conjecturally hold for every irreducible holomorphic symplectic manifold) are not Kobayashi hyperbolic. 

Thus, one of the main challenge is to show non hyperbolicity of Calabi--Yau manifolds. For such manifolds, much more is expected to be true: they should always contain rational curves! For several results in this direction, at least for Calabi--Yau manifolds with large Picard number, we refer the reader to \cite{Wil89,Pet91,HBW92,Ogu93,DF14,DFM16}, just to cite a few.

Coming back to curvature, of course possessing a KŠhler metric whose holomorphic sectional curvature is negative implies Kobayashi hyperbolicity and thus having ample canonical bundle by the above discussion, provided the abundance conjecture is true. Therefore, this settles Yau's conjecture under the assumptions that abundance conjecture is true.

Now, what about compact KŠhler manifolds with merely non positive holomorphic sectional curvature? Surely, they do not contain any rational curve (cf. Theorem \ref{algcrit}). Thus, if $X$ is projective, we conclude that $K_X$ is nef as before by Mori. If $X$ is merely KŠhler, one needs to work more but the same conclusion of nefness for $K_X$ holds true, thank to a very recent result by Tosatti and Yang \cite{TY15} (which is a slight modification of the original Wu and Yau method \cite{WY16}). Anyway, such a condition is not strong enough in order to obtain positivity of the canonical bundle, as flat complex tori immediately show. A less obvious but still easy counterexample is given by the product (with the product metric) of a flat torus and, say, a compact Riemann surface of genus greater than or equal to two endowed with its PoincarŽ metric. In this example, over each point there are some directions with strictly negative holomorphic sectional curvature but always some flat directions, too (we refer the reader to the recent paper \cite{HLW14} for some nice results about this merely non positive case). 
So, if we look for the weakest condition, as long as the sign of holomorphic sectional curvature is concerned, for which one can hope to obtain the positivity of the canonical bundle, we are led to give the following (standard, indeed) definition.

\begin{definition}
The holomorphic sectional curvature is said to be \emph{quasi-negative} if $\HSC_\omega\le 0$ and moreover there exists at least one point $x\in X$ such that $\HSC_\omega(x,[v])< 0$ for every $v\in T_{X,x}\setminus\{0\}$.
\end{definition}

\begin{remark}
We shall see in the last section a slightly subtler condition on holomorphic sectional curvature, based on the notion of \lq\lq truly flat\rq\rq{} directions, very recently introduced by Heier, Lu, Wong, and Zheng in \cite{HLWZ17}, that should also work for this kind of purposes.
\end{remark}

Now, why should we hope that such a condition would be sufficient? The reason comes again from the birational geometry of complex KŠhler manifolds, and in particular again from the abundance conjecture. Let us illustrate why.

We begin with the following elementary observation.

\begin{proposition}\label{prop:average}
Let $(X,\omega)$ be a compact KŠhler manifold with $\HSC_\omega\le 0$, and suppose there exists a direction $[v]\in P(T_{X,x_0})$ such that $\HSC_\omega(x_0,[v])<0$, for some $x_0\in X$. Then, $c_1(X)\in H^2(X,\mathbb R)$ cannot be zero.
\end{proposition}

\begin{proof}
By Proposition \ref{average}, we know that $\scal_\omega$ is everywhere non positive, and moreover, as an average, it is strictly negative at $x_0$. In particular, the total scalar curvature of $\omega$ is strictly negative. The conclusion follows from Remark \ref{totscalcurv}.
\end{proof}

As a direct consequence, if $X$ is moreover projective and $\operatorname{Pic}(X)$ is infinite cyclic, then $K_X$ must be ample. This gives back (and slightly generalize) a result of \cite{WWY12}.

Now, let $(X,\omega)$ be a compact KŠhler manifold with quasi-negative holomorphic sectional curvature. Then, Proposition \ref{prop:average} implies that $X$ cannot have trivial first real Chern class. Moreover, since being quasi-negative is stronger than being non positive, we saw that, thanks to \cite{TY15}, $K_X$ is nef.

Once again, suppose that the abundance conjecture holds true, but now also for compact KŠhler manifolds. Then, $K_X$ is semi-ample and we can consider exactly as before the semi-ample Iitaka fibration for $K_X$.
Since $X$ has non trivial first real Chern class, we must have that $\kappa(X)>0$, otherwise some power of the canonical bundle would be a pull-back of a (ample) line bundle over point, and thus would be trivial!

If $\kappa(X)=\dim X$, then $X$ would be birational to a projective variety, \textsl{i.e.} would be a Moishezon manifold. By Moishezon's theorem, a compact KŠhler Moishezon manifold is projective. Moreover, $X$ is without rational curves and of general type, and we conclude as before that $K_X$ must be ample.

Next, suppose by contradiction that $1\le\kappa(X)\le \dim X-1$ so that if we call $F$ the general fiber of $\phi$, we have that $F$ is a smooth compact KŠhler manifold of positive dimension and different from $X$ itself. 
Now, on the one hand, the short exact sequence of the fibration shows that
$K_F\simeq K_X|_F$ and therefore it follows that $c_1(F)$ must be zero in real cohomology. On the other hand, the classical Griffiths' formulae for curvature of holomorphic vector bundles imply that the holomorphic sectional curvature decreases when passing to submanifolds, that is for every $x\in F\subset X$
$$
\HSC_{\omega|_F}(x,[v])\le\HSC_{\omega}(x,[v]),
$$
where $v\in T_{F,x}$ and, in the right hand side, $v$ is seen as a tangent vector to $X$.

The quasi-negativity of the holomorphic sectional curvature implies, since $F$ is a general fiber, that there exists a tangent vector to $F$ along which the holomorphic sectional curvature of $\omega|_F$ is strictly negative. Thus, Proposition \ref{prop:average} implies that $F$ cannot have trivial first real Chern class, which is absurd.

As a consequence, me may indeed hope to extend Wu--Yau--Tosatti--Yang theorem to the optimal, quasi-negative case. This is precisely the main contribution of the paper \cite{DT16}.

\begin{theorem}[{\cite[Theorem 1.2]{DT16}}]\label{thm:main}
Let $(X,\omega)$ be a connected compact KŠhler manifold. Suppose that the holomorphic sectional curvature of $\omega$ is quasi-negative. Then, $K_X$ is ample. In particular, $X$ is projective.
\end{theorem}

We shall spend some words on this result in the last section. 

\begin{remark}
To finish this section with, unfortunately, we must confess that we are not aware of any example of a compact KŠhler manifold with a KŠhler metric whose holomorphic sectional curvature is quasi-negative but which does not posses any KŠhler metric with strictly negative holomorphic sectional curvature. In other word, is Theorem \ref{thm:main} a true generalization of Wu--Yau--Tosatti--Yang result? We believe so. Then, such an example, if any, would be urgently needed!
\end{remark}