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%----------------------------------------------------------------------------

\title[On the hyperbolicity of very general hypersurfaces]{Proof of the Kobayashi conjecture on the hyperbolicity of very general hypersurfaces}

\author{Jean-Pierre Demailly}
\date{November 24, 2014, revised on February 13, 2015}
\begin{document}

\noindent
Let $\cX\to S$ be a submersive proper map between complex manifolds,
with smooth projective fibers. We use the following facts about vector 
fields on directed manifolds. Let $(\cX_k,\cV_k)$ be the ``relative'' 
Semple tower of $(\cX,T_{\cX/S})$ and $(\cX^a_k,\cV^a_k)$ be the ``absolute''
Semple tower of $(\cX,T_\cX)$.

\claim Lemma|
{\plainitemindent=7mm
\plainitem{\rm (a)} Every local holomorphic vector field $\zeta$ on an open
set $U\subset \cX$ has a natural lifting $\zeta^{(k)}$ to the open set
$(\pi^a_{k,0})^{-1}(U)$ in the total space of the absolute Semple 
bundle $\cX_k^a$. 
\vskip3pt
\plainitem{\rm (b)} 
In particular, if one assumes that $\zeta$ is 
in~$H^0(U,T_{\cX/S})\subset H^0(U,T_{\cX})$, the lifted flow leaves 
invariant the relative Semple tower
$\cX_k\subset\cX_k^a$ in $\pi_{k,0}^{-1}(U)$, thus 
$\smash{\zeta^{(k)}_{|\cX_k}}$ is tangent to~$\cV_k$. 
\vskip3pt
\plainitem{\rm (c)} 
Every local holomorphic vector field $\sigma\in H^0(\Omega,T_S)$ has a 
$($nonunique$\,)$ lifting $\tau^{(k)}$ to the open set
$(\pi\circ\pi^a_{k,0})^{-1}(\Omega)$ in the total space of the absolute
Semple bundle $\cX^a_k$, and the flow of $\smash{\tau^{(k)}}$ induces
a local biholomorphism $(\cX_k,\cV_k)\to(\cX_k,\cV_k)$ of the relative Semple
tower. These local flows preserve the exact sequences $(5.1.1-5.1.5)$ and
we again conclude that $\smash{\tau^{(k)}_{|\cX_k}}$ is tangent to~$\cV_k$.
\vskip6pt
\plainitem{\rm (d)} If a holomorphic vector field $\zeta\in 
H^0(\cX\smallsetminus|A|,T_X)$ extends as a meromorphic section in
$H^0(\cX,T_\cX\otimes\cO(pA))$ with poles of multiplicity${}\le p$ along $A$, 
then 
$$\zeta^{(k)}\in H^0(\cX^a_k,T_{\cX^a_k}\otimes(\pi^a_{k,0})^*\cO((p+k)A)).$$
\plainitem{\rm (e)} If a global vector field $\sigma$ on $S$ is lifted as 
a meromorphic section $\tau\in H^0(\cX,T_\cX\otimes\cO(pA))$, then we get a 
restriction $\tau^{(k)}_{|\cX_k}\in H^0(\cX_k,T_{\cX_k}\otimes
\pi_{k,0}^*\cO((p+k)A))$.\vskip0pt
}
\endclaim

\proof (a) Every local holomorphic vector field $\zeta$ on $\cX$ generates a
flow of local biholomorphisms on open subsets of $\cX$, and 
we can apply the fonctoriality property~1.10 to lift it to a flow 
on $(\cX_k^a,\cV_k^a)$. The differentiation of the lifted flow gives 
back what we define to be the lifted vector field $\zeta^{(k)}$ 
on~$\cX_k^a$.\vskip3pt

\noindent(b) If $\zeta$ lies in $\cV_0=T_{\cX/S}$, the 
lifted flow acts similarly on the relative tower $(\cX_k,\cV_k)$ since
the fibers $X_{t,k}$ over $X_t=\pi^{-1}(t)$ are preserved. Therefore
$\smash{\zeta^{(k)}_{|\cX_k}}$ is tangent to $\cV_k$, since the relative
Semple tower is the absolute Semple tower of the fibers~$X_t$.
\vskip3pt


\noindent(c) Since $\cX\to S$ 
is a holomorphic submersion, $\cX$~is locally holomorphically
trivialized as a product $S\times X_t\,$; if $\sigma$ is a local vector 
field on~$S$, it can thus be lifted (possibly not uniquely) as a 
local vector field $\tau$ on $\cX$. The resulting flow of 
$\tau$ on $\cX$ commutes with the flow of $\sigma$ on $S$, 
i.e.\ acts by (local) biholomorphisms $(\cX,T_{\cX/S})\to(\cX,T_{\cX/S})$ of
directed varieties. From there we conclude, again by
fonctoriality, that the lifting of the flow to $\cX_k^a$
preserves $(\cX_k,\cV_k)\subset (\cX^a_k,\cV^a_k)$, in particular 
the exact sequences are preserved. Moreover the differential
$\smash{\tau^{(k)}}$ is tangent to~$T_{\cX_k}$ (but, already for $k=0$,
it is not tangent to~$\cV_k$).\vskip3pt

\noindent(d) The question is merely to compute the poles of
$\smash{\zeta^{(k)}}$ along $A$. By our assumption, we can write
$\zeta=\eta/s_A^p$ near any point in the support of $A$, where $\eta$ is
a local section of $T_\cX$ and $s_A=0$ is a local holomorphic
equation for $|A|$ of divisor equal to~$A$. The flow of $\tau$ on $\cX$
is the solution $\bC\ni t\mapsto\Phi_t(x)\in\cX$ of the
ordinary differential equation
$$
\frac{d}{dt}\Phi_t(x)=\zeta(\Phi_t(x)),\quad\Phi_0(x)=x.
$$
The corresponding lifted flow on $\cX_1^a$ can be expressed as
$$
\Phi_{[1],t}(x,[v])=\big(\Phi_t(x),[d\Phi_t(x)\cdot v]\big),\qquad
v\in T_{\cX,x}\quad(v\ne 0).
$$
By taking the derivative in $t$ and writing $\Phi_t(x)=x+t\zeta(x)+O(t^2)$, 
we find
$$
\eqalign{
\zeta^{(1)}(x,[v])=\frac{d}{dt}_{|t=0}\Phi_{[1],t}(x,[v])
&=\big(\zeta(x), d\zeta(x)\cdot v~\mod~\varepsilon\big)\cr
&=\big(s_A^{-p}\eta(x),s_A^{-p}d\eta(x)\cdot v-
p\eta(x)s_A^{-p-1}ds_A(x)\cdot v~\mod~\varepsilon\big),\cr}
$$
where $\varepsilon=\sum v_j\partial/\partial v_j$ is the Euler vector field
on $T_{\cX,x}$. As a consequence, we find 
that $\zeta^{(1)}\in H^0(\cX^a_1,T_{\cX^a_1}\otimes\pi_{1,0}^*\cO(-(p+1)A))$.
The statement for $\zeta^{(k)}$ follows by induction, by iterating $k$~times
the $1$-jet fonctor. Property (e) is a special case, if we take (c) into 
account and restrict the vector field to $\cX_k$.\qed\vskip5pt

\end{document}
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