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\title[Curvature of Metrics on Demailly-Semple Tower]{Curvature of Metrics on Demailly-Semple Tower}
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\author{Mohammad Reza Rahmati}
\thanks{}
\address{ Centro de Investigacion en Matematicas, CIMAT
\hfill\break 
\hfill\break \\
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\email{mrahmati@cimat.mx}


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\begin{document}

\begin{abstract}
We compute the curvature of several metrics on Demailly-Semple projective tower on a projective variety. The holomorphic Morse inequalities will then provide estimates to prove the Green-Griffiths-Lang conjecture. Our work is along the generalization and completion of the work of Demailly on this conjecture. 

\end{abstract}

\maketitle

%\noindent Version: 

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\section{Introduction}

Demailly-Semple tower associated to a pair $(X,V)$ where $X$ is a complex projective manifold (may be singular) and $V$ is a holomorphic subbundle of $T_X$ the tangent bundle of $X$ of rank $r$, is a prolongation sequence of projective bundles 

\begin{equation}
\mathbb{P}^{r-1} \to X_k=P({V_{k-1}}) \to X_{k-1}, \qquad k \geq 1
\end{equation} 

\noindent
obtained inductively making $X_k$ a weighted projective bundle over $X$. The sequence provides a tool to study the locus of nonconstant holomorphic maps $f:\mathbb{C} \to X$ such that $f'(t) \in V$. It is a conjecture due to Green-Griffiths-Lang that the total image of all these curves is included in a proper subvariety of $X$; provided $(X,V)$ is of general type. By general type we mean $K_{V}$ the canonical bundle of $V$ is big. 

\vspace{0.5cm}

\noindent
\textbf{Conjecture: (Green-Griffiths-Lang)} \cite{GG}

\noindent 
Let $(X,V)$ be a pair where $X$ is a projective variety of general type, and $V$ is a holomorphic subbundle of the tangent bundle $T_X$. Then there should exists an algebraic subvariety $Y \subsetneqq X$ such that every nonconstant entire curve $f:\mathbb{C} \to X$ tangent to $V$ is contained in $Y$. It is a basic fact by J. Semple that the map $f$ can be lifted to maps $f_k:\mathbb{C} \to X_k$. 


\vspace{0.5cm}


\noindent
An alternative definition is to consider the vector bundles $E_{k,m}^{GG}V^*$ of germs of weighted homogeneous polynomials along the fibers in $X_k \to X_{k-1} \to ... \to X$. These polynomials can be definitely considered in the germs of indeterminates 
$z=f,\ \xi_1=f',\ ...,\ \xi_k=f^{(k)}$, where $f^{(k)}$ is the k-th derivative of $f$. Then a motivation to approach the above conjecture is the following.

\begin{theorem} (Demailly vanishing Theorem) \cite{D1}
Any section of $P \in H^0(X,E_{k,m}^{GG}V^* \otimes L^{-1})=H^0(X_k,\mathcal{O}_{X_k}(m) \otimes \pi_{0k}^*L^{-1})$ with $L$ being an ample line bundle on $X$, satisfies an algebraic differential equation $P(f,f',...,f^{(k)})=0$. 
\end{theorem}


\begin{definition}
A smooth $k$-jet metric $h_k$ on $(X,V)$ is a hermitian metric on the line bundle $\mathcal{O}_{X_k}(-1)$ over $X_k$ in the Green-Griffiths tower. Say $h_k$ has negative jet curvature along $V_k$, if 

\begin{equation}
\Theta_{h_k^{-1}}(\xi) \geq \epsilon \mid \xi \mid_{w_k}^2, \qquad \exists \epsilon >0 , w_k \ \text{on} \ T_{X_k}
\end{equation}

\end{definition}
\noindent
The $k$-jet metrics can be constructed inductively on the Green-Griffiths bundles, starting from some metric $h$ on $ V \subset T_X$.
A basic example is, 

\begin{equation}
\mid (z;\xi)\mid =(\sum_s  \parallel \xi_s \parallel_h^{2p/s})^{1/p}
\end{equation}

\noindent
where $p$ should be chosen as a multiple of $lcm(1,2,...k)$

\begin{equation}
\parallel \xi_s \parallel^2 =\sum_{\alpha} \mid \xi_{s\alpha}\mid^2 +\sum_{ij\alpha \beta} c_{ij\alpha \beta}z_i \bar{z}_j \xi_{s\alpha}\bar{\xi}_{s\beta}
\end{equation} 

\noindent
Here $p$ should be chosen as a multiple of $lcm(1,2,...k)$. It gives the Fubini-Study metric with its curvature in level $s$ of the jet bundle. Denote the second term in (5) by $\Theta_s^{FS}$. Therefore 

\begin{equation}
\begin{split}
\log \mid (z,\xi) \mid=\frac{1}{p} \log \big( \sum_s (\parallel \xi_s \parallel^2 +\Theta_s^{FS})^{p/s}
& =\frac{1}{p} \log \big( \sum_s \parallel \xi_s \parallel^2 (1+\frac{\Theta_s^{FS}}{\parallel \xi_s \parallel^2})^{p/s} \big)\\
& =\frac{1}{p} \log \big( \sum_s \parallel \xi_s \parallel^2 (1+\frac{p}{s}.\frac{\Theta_s^{FS}}{\parallel \xi_s \parallel^2}+...) \big)\\
& =\frac{1}{p} \log \big( \sum_s \parallel \xi_s \parallel^2 +  \big( \sum_s \parallel \xi_s \parallel^2 \frac{p}{s}.\frac{\Theta_s^{FS}}{\parallel \xi_s \parallel^2} \big)
\end{split}
\end{equation}

\noindent
Now Apply the operator $\dfrac{i}{2\pi}\partial \bar{\partial}$ to obtain

\begin{equation}
\Theta= w_{a,r,p}(\xi)+\frac{i}{2\pi} \sum_s \frac{1}{s}\frac{\mid \xi_s\mid^{2p/s}}{\sum_t \mid \xi_t \mid^{2p/t}} \sum c_{ij\alpha \beta} \frac{\xi_{s\alpha}\bar{\xi}_{s\beta}}{\mid \xi_s \mid^2} dz_i \wedge d\bar{z}_j+...
\end{equation}

\noindent
(We have used the identity $\log(f+g)=\log(f) +(g/f)-....$). 
The next motivation due to Demailly is the holomorphic Morse inequalities;

\begin{theorem} (Holomorphic Morse inequalities-J. Demaily)
If $E \to X$ is a holomorphic vector bundle over a compact complex manifold $X$, and $(L,h)$ a hermitian line bundle, then 

\begin{equation}
\sum_{0 \leq j \leq q}(-1)^{q-j}h^j(X,E \otimes L^k) \leq r\frac{k^n}{n!} \int_{X(L,h,\leq q)}(-1)^q\Theta_{L,h}^n +o(k^n)
\end{equation}

\noindent
where $\Theta$ denotes the curvature of the k-jet metric associated to $h$, and $X(q,L)=\{x \in X | \ \frac{i}{2 \pi} \Theta(L) \ \text{has exactly \textit{q} negative eigenvalues}\ \}$. 
\end{theorem}


\noindent
For example, from the theorem one deduces that

\begin{equation}
h^0(X,E \otimes A^p) \geq h^0(X,E \otimes A^p)-h^1(X,E \otimes A^p) \geq r \frac{k^n}{n!}\int_{X(A,h, \leq 1)}\Theta_{A,h}^n
\end{equation}

\noindent
If we could show that

\begin{equation}
\int_{X (\leq 1, L)}(\frac{i}{2\pi} \Theta(L))^n >0
\end{equation}

\noindent
then some high power of $L$ twisted by $E$ has many non-zero sections. An example of this application is the following.

\begin{theorem}\cite{D1}
Lets $(X,V)$ be a directed projective variety such that $K_V$ is big, and let $A$ be an ample divisor. Then for $k>>1$ and $\delta \in \mathbb{Q}_+$ small enough, and $\delta \leq c (\log k)/k$, the number of sections $h^0(X,E_{k,m}^{GG} \otimes \mathcal{O}(-m \delta A))$ has maximal growth, i.e is larger than $c_km^{n+kr-1}$ for some $m \geq m_k$, where $c,c_k >0, n=\dim (X), r= \text{rank}(V)$. In particular entire curves $f:\mathbb{C} \to X$ satisfy many algebraic differential equations.
\end{theorem}


\noindent
The next motivation toward the GGL-conjecture is the singularity locus of the $k$-jet metrics which we denote by $\Sigma_{h_k}$. A general fact about this locus proved in \cite{D1} is,

\begin{equation}
\Sigma_{h_k} \subset \pi_k^{-1}(\Sigma_{h_{k-1}}) \cup D_k
\end{equation}

\noindent
where $D_k=P(T_{X_{k-1}/X_{k-2}}) \subset X_k$. The divisors $D_k$ are the singularity locus of the projective jet bundle $X_k$ and their relation with the singularity of the the k-jet metric is   

\begin{equation}
\mathcal{O}_{X_k}(1)=\pi_k^*\mathcal{O}_{X_{k-1}}(1) \otimes \mathcal{O}(D_k)
\end{equation}

\noindent
There are canonical inclusions $\mathcal{O}_{X_k}(-1) \subset \pi_k^* V_{k-1} \subset \pi_k^*  \mathcal{O}_{X_{k-1}}(-1)$ which admit $D_k$ as a zero divisor. 


\begin{theorem} (J. P. Demaily \cite{D1}, \cite{D2})
Let $(X,V)$ be a compact directed manifold. If $(X,V)$ has a $k$-jet metric $h_k$ with negative jet curvature, then every entire curve $f:\mathbb{C} \to X$ tangent to $V$ satisfies $f_k(\mathbb{C}) \subset \Sigma_{h_k}$, where $\Sigma_{h_k}$ is the singularity locus of $h_k$.
\end{theorem}



\noindent
Assume $k,m$ be large enough such that $H^0(X_k,\mathcal{O}_{X_k}(m) \otimes \pi_{0k}^*L^{-1})$ has non-zero sections $\sigma_1,...,\sigma_N$ with base locus $Z$. Assume $L$ has a smooth metric $h_L$ with positive curvature. Then we can define a singular metric $h'_k$ on $\mathcal{O}_{X_k}(-1)$ by

\begin{equation}
h'_k(w,\xi)=(\sum_j |\sigma_j(w) . \xi^m|^2)^{1/m}, \qquad w \in X_k, \ \xi \in \mathcal{O}_{X_k}(-1)_w
\end{equation}

\noindent
and $\Sigma_{h'_k}=Z$. The calculation shows 

\begin{equation}
\frac{i}{2 \pi}\partial \overline{\partial} \log h'_k =\Theta_{h_k'^{-1}} (\mathcal{O}_{X_k}(-1)) \geq \frac{1}{m} \pi_{0k}^*\Theta_L
\end{equation}

Summarizing all we obtain;

\begin{corollary}(J. P. Demaily \cite{D1}, \cite{D2}) 
Assume that there exists $k,m >0$ and an ample line bundle $L$ on $X$ such that $H^0(X_k,\mathcal{O}_{X_k}(m) \otimes \pi_{0k}^*L^{-1})=H^0(X,E_{k,m}(V^*) \otimes L^{-1})$ has non-zero sections $\sigma_1,...,\sigma_n$. Let $Z \subset X_k$ be the base locus of these sections. then every entire curve tangent to $V$ is such that $f_k(\mathbb{C}) \subset \Sigma_{h_k}$. In other words, for every global $G_k$invariant polynomial differential operator $P$ of weighted degree $m$ and values in $L^{-1}$, every entire curve $f$ must satisfy $P(f)=0$.
\end{corollary}



In case of use the line bundle $L$ is defined via $\mathcal{O}_{X_k}(1)$ by a rational twist denoted by $\delta$ in Theorem 1.1. It can  specifically be given by

\begin{equation}
L_k=\mathcal{O}_{X_k}(1) \otimes \pi_k^* \mathcal{O}(\frac{1}{kr}(1+\frac{1}{2}+...+\frac{1}{k})L^{-1})
\end{equation}

\noindent
In \cite{D1} Demailly uses the $k$-jet metric 

\begin{equation}
\mid (\xi)\mid =(\sum_s \epsilon_s \parallel \xi_s \parallel_h^{2p/s})^{1/p}=( \sum_s (\sum_{\alpha} \mid \xi_{s\alpha}\mid^2 +\sum_{ij\alpha \beta} c_{ij\alpha \beta}z_i \bar{z}_j \xi_{s\alpha}\bar{\xi}_{s\beta})^{p/s})^{1/p} +...
\end{equation}

\noindent
on the Green-Griffiths bundle of jets (here the metric is written on a orthonormal frame of $V$; see the Remark below). The exponent $p$ should be chosen as a multiple of $lcm(1,2,...k)$ to obtain integer exponents, and $1 >> \epsilon_1 >> \epsilon_2 >> ... >>\epsilon_k >0$ are scaled according to a partition of unity on $X$ such that as $\epsilon_s/\epsilon_{s-1} \to 0$ the metric converges to (3). The scaling with the epsilons also guarantees the orthogonality in the components $(\xi_1,...,\xi_k)$ in the limit (see the reference). The curvature of this metric is calculated in \cite{D1} as (5) and (6). Then for the volume estimation, we need to look at the integral 

\begin{equation}
\int_{X_k^{GG},q}\Theta^{n+kr-1}=\frac{(n+kr-1)!}{n!(kr-1)!}\int_X \int_{P(1^r,...,k^r)}w_{a,r,p}^{kr-1}(\xi)1_{\gamma_k,q}(z,\xi)\gamma_k(z,\xi)^n
\end{equation}

\noindent
The factor $w_{a,r,p}^{kr-1}$ does not depend to $z \in X$ and only depend universally on fibers. The expression of the curvature can be made simpler by re-scaling $x_s=|\xi_s|_h^{2p/s}, \ u_s=\xi_s/|\xi_s|_h$. The curvature finds the following formula,

\begin{equation}
\Theta_{L_k,h_k}=w_{FS,p,k}(\xi)+\frac{i}{2\pi}\sum_s\frac{1}{s}\sum c_{ij\alpha \beta}u_{s\alpha}\bar{u}_{s\beta}dz_i\wedge d\bar{z}_j
\end{equation}

\noindent
where $w_{FS,p,k}(\xi)$ is positive definite on $\xi$. The other terms are weighted average of the values of the curvature $\Theta_{V,h}$ on vectors $u_s$ in the sphere bundle $S(V)$. The weighted projective space is a quotient of the pseudo-sphere $\sum |\xi_s|^{2p/s}=1$. Thus the second sum in the curvature is of the form $\sum \frac{1}{s} \gamma(u_s)$ where $u_s$ are essentially random variables on the sphere. Because $\gamma$ is quadratic 

\begin{equation}
\int_{u \in S(V)} \gamma(u)^n=(\sum_s \frac{1}{s})^nTr(\gamma)
\end{equation}

\noindent
The integral becomes 

\begin{equation}
\frac{(n+kr-1)!}{n!(k!)^r(kr-1)!}\frac{1}{(kr)^n}(1+\frac{1}{2}+...+\frac{1}{k})^n \int_X 1_{\eta,q}\eta^n +...
\end{equation}

\noindent
where $\eta=\Theta_{det(V^*), h^*}+\Theta_{L,h_L}$. 

\begin{remark}
If $X$ is a projective variety and $E$ a hermitian vector bundle on $X$  with hermitian metric $h$, such that in an orthonormal frame $(e_{\lambda})$ around $x \in X$ the metric can be written as

\begin{equation}
h(e_{\lambda},e_{\mu})= \delta_{\lambda,\mu}- \sum_{j,k}c_{jk\lambda \mu}z_j\overline{z}_k+ O(|z^3|)
\end{equation}

\noindent
This fact has been used in (17). We will also employ this formula for the test metrics on invariant jets in the next section.
\end{remark}

\begin{remark}
Forgetting about the complications in the metrics in order to define positive $k$-jet metrics on Green-Griffiths bundle one can proceed as follows. It is possible to choose the metric $h$ on $X$ such that the curvature of the canonical bundle $K_V=\bigwedge^r V^*$ is given by a Kahler form, i.e.

\begin{equation}
\Theta_{K_V,det(h^*)}=w>0
\end{equation} 

\noindent
Then $\Theta_{det(V),det(h)}=-w$. In the fibration sequence 

\begin{equation}
\begin{array}{cccc}
& \mathcal{O}_{X_1}(-1) & \hookrightarrow & \pi_1^*V \\
& \downarrow & &\\
& X_1 & & \\
& \downarrow & &\\
& X &  &
\end{array}
\end{equation}

\noindent
One can consider a metric with curvature $\epsilon_1\Theta_{\mathcal{O}_{X}(-1)}+\pi_1^*w>0$ for $\epsilon_1<<1$ on $\mathcal{O}_{X_1(-1)}$. Repeating this argument inductively, one obtains a metric on $\mathcal{O}_{X_k}(-1)$ with curvature

\begin{equation}
\Theta_{\mathcal{O}_{X_k}(-1), \vartheta}=\sum_j\epsilon_1...\epsilon_j\pi_{jk}^*\Theta_{\mathcal{O}_{X_j}(-1)}+\pi_{0k}^*w>0\ , \qquad ...<<\epsilon_2 <<\epsilon_1<<1
\end{equation} 

\noindent
However the metric $\vartheta$ is hard to analyze, because of adding the curvature to the metric at each step. 
\end{remark}

\vspace{0.5cm}

\section{Invariant Jets vs Invariant metrics}

\vspace{0.5cm}

J. P. Demaily \cite{D2} develops the ideas in \cite{GG} and considers the jets of differentials that are also invariant under change of coordinate on $\mathbb{C}$. 
In fact if we consider the bundle $J_k \to X$ of germs of parametrized curves in $X$. Its fiber at $x \in X$ is the set of equivalence classes of germs of holomorphic maps $f:(\mathbb{C},0) \to (X,x)$ with equivalence relation $f^{(j)}(0)=g^{(j)}(0), \ 0 \leq j \leq k$. By choosing local holomorphic coordinates around $x$, the elements of the fiber $J_{k,x}$ can be represented by the Taylor expansion 

\begin{equation}
f(t)=x+tf'(0)+\frac{t^2}{2!}f''(0)+...+\frac{t^k}{k!}f^{(k)}(0)+O(t^{k+1})
\end{equation}

\noindent
Setting $f=(f_1,...,f_n)$ on open neighborhoods of $0 \in \mathbb{C}$, the fiber is 

\begin{equation}
J_{k,x}=\{(f'(0),...,f^{(k)}(0))\} = \mathbb{C}^{nk}
\end{equation}

Lets $G_k$ be the group of local reparametrizations of $(\mathbb{C},0)$

\begin{equation}
t \longmapsto \phi(t)=\alpha_1t+\alpha_2t^2+...+\alpha_kt^k, \qquad \alpha_1 \in \mathbb{C}^* 
\end{equation}

\noindent
Its action on the $k$-jets is given by the following matrix multiplication

\begin{equation}
[f'(0), f''(0)/2!, ..., f^{(k)}(0)/k!].
\left[ \begin{array}{ccccc}
a_1 & a_2 & a_3 & ... & a_k \\
0 & a_1^2 & 2a_1a_2 & ... & a_1a_{k-1}+...a_{k-1}a_1\\
0 & 0 & a_1^3 & ... & 3a_1^2a_{k-2}+...\\
. & . & . & ... & .\\
0 & 0 & 0 & ... & a_1^k 
\end{array} \right ]
\end{equation}

\noindent
Let $E_{k,m}$ be the Demaily-Semple bundle whose fiber at $x$ consists of $U_k$-invariant polynomials on the fiber coordinates of $J_k$ at $x$ of weighted degree $m$. Set $E_k=\bigoplus_m E_{k,m}$, the Demaily-Semple bundle of graded algebras of invariants. Then one needs to work out the aforementioned calculations with an invariant metric. Toward this we examine the following metrics

\vspace{0.5cm}

\textbf{Test 1:} We replace the metric in (17) or (18) by 

\begin{equation}
\mid (z;\xi)\mid =(\sum_s \epsilon_s \parallel \mathbf{W}_s \parallel_h^{2p/s(s+1)})^{1/p}=( \sum_s (\sum_{l=1}^s \mid W_l \mid^2 +\sum_l \sum_{ij\alpha \beta} c_{ij\alpha \beta}z_i \bar{z}_j \xi_{l\alpha}\bar{\xi}_{l\beta})^{p/s(s+1)})^{1/p} 
\end{equation}

\noindent
where $\parallel \mathbf{W}_s \parallel=\sum_{l=1}^s \mid W_l \mid^2$. We have used the Remark 1.4 for the replacement of the curvature at each stage and the formula ${W}_l=\xi_1 \wedge ...\wedge \xi_l$. Taking the logarithm and using the identity $\log(f-g)=\log f + f/g +...$ and applying $\frac{i}{2\pi} \partial \overline{\partial}$ similar to the former case we obtain

\begin{equation}
\Theta=  \sum_s \mid \mathbf{W}_s \mid^{p/s(s+1)}+\frac{i}{2\pi} \sum_s \frac{1}{s(s+1)}\frac{\mid \mathbf{W}_s\mid^{2p/s(s+1)}}{\sum_t \mid \mathbf{W}_t \mid^{2p/t(t+1)}}\sum_l \sum c_{ij\alpha \beta} \xi_{s\alpha}\bar{\xi}_{s\beta} dz_i \wedge d\bar{z}_j
\end{equation}

\noindent
In this case one estimates 

\begin{equation}
\int_{X_k^{GG},q}\Theta^{n+k(r-1)}=\frac{(n+k(r-1))!}{n!(k(r-1))!}\int_X \int_{P(1^r,...,k^r)}Q(\xi)1_{\gamma_k,q}(z,\xi)\gamma_k(z,\xi)^n
\end{equation}

\noindent
where $\gamma$ is the second term in the sum above and $Q$ is function depending only to $\xi$. Doing the same change of coordinates and re-scaling one receives to consider expected values for 

\begin{equation}
\int \gamma_k^n=\frac{1}{kr}\sum_s \frac{1}{s(s+1)} \frac{i}{2\pi}\sum_{l=1}^s\sum_{ij\alpha}c_{ij\alpha \alpha} (z)dz_i d\bar{z}_j
\end{equation} 

\noindent
which gives similar estimates as before. We formulate it below.

\begin{corollary}
The lifts of the entire curve locus for projective directed $(X,V)$ along the Demaily-Semple bundles are contained in the zero set of all Wronskians 

\begin{equation}
W_1=W_2=...=W_k=...=0
\end{equation}
\end{corollary}
\begin{proof}
The theorem follows from Theorem 1.2 and the convergence in (32), followed by the methodology in \cite{D1} which is exactly the same as that of Demailly metric in Section 1.
\end{proof}

\vspace{0.5cm}

\textbf{Test 2:} We test the following metric

\vspace{0.5cm}

\begin{equation}
\sum_{s=1}^k \epsilon_s (\ \sum_{\alpha} \mid P_{\alpha}(\xi) \mid ^2)^{p/w(P_{\alpha})}\ )^{1/p}
\end{equation}

\noindent
where the $P_{\alpha}$ are a set of invariant polynomials in jet coordinates. The effect of this is then, the Demailly-Semple locus of the lifts of entire curves should be contained in 

\begin{equation}
P_{\alpha}=0 , \qquad \forall \alpha
\end{equation} 

\noindent
For instance a choice of $P_{\alpha}$'s could be the Wronskians (Test 1). However we slightly try to do some better choice. First lets make some correspondence between invariant jets with non-invariant ones. Lets consider a change of coordinates on the $\mathbb{C}$ by

\begin{equation}
\xi=(f_1,...,f_r) \longmapsto (f_1 \circ f_1^{-1},...,f_r \circ f_1^{-1})=(t,g_2,...,g_r)=\eta
\end{equation} 

\noindent
This makes the first coordinate to be the identity and the other components to be invariant by any change of coordinates on $\mathbb{C}$. If we differentiate in the new coordinates successively, then all the resulting fractions are invariant of degree $0$

\begin{equation}
g_2'=\frac{f_2'}{f_1'^2}, \qquad g_2''=\frac{P_2=f_1'f_2''-f_2'f_1''=W(f_1,f_2)}{f_1'^3},...
\end{equation} 

\noindent
We take the $P_{\alpha}$'s to be all the polynomials that appear in the numerators of the components when we successively differentiate (36) with respect to $t$. An invariant metric in the first coordinates corresponds to a usual metric in the second one subject to the condition that we need to make the average under the unitary change of coordinates in $V$. This corresponds to change of coordinates on the manifold $X$ as 

\begin{equation}
(\psi \circ f)^{(k)}(0)= \psi'(0). f^{(k)}+\text{higher order terms according to epsilons }
\end{equation} 

\noindent
That is the effect of the change of variables in $X$ has only effect as the first derivative by composition with a linear map, up to the epsilon factors. Therefore the above metric becomes similar to the metric used in \cite{D1} in the new coordinates produced by $g$'s, 

\begin{equation}
\mid (z;\xi)\mid \sim (\sum_s  \epsilon_s \parallel \eta_s . (\eta_{11})^{2s-1}\parallel_h^{p/(2s-1)})^{1/p} = (\sum_s  \epsilon_s \parallel \eta_s \parallel_h^{p/(2s-1)})^{1/p}\mid \eta_{11} \mid
\end{equation}

\noindent
where the weight of $\eta_s$ can be seen by differentiating (36) to be equal $(2s-1)$ inductively. We need to modify the metric in (39) slightly to be invariant under hermitian transformations of the vector bundle $V$. In fact the role of $\eta_{11}$ can be done by any other $\eta_{1i}$ or even any other non-sero vector. To fix this we consider 

\begin{equation}
\mid (z;\xi)\mid = \int_{\parallel v \parallel_1=1} \sum_s  \epsilon_s \parallel \eta_s \parallel_h^{p/(2s-1)})^{1/p}\mid <\eta_{1}.v> \mid^2
\end{equation}

\noindent
where the integration only affects the last factor making average over all vectors in $v \in V$. This will remove the the former difficulty. 
The curvature is the same as for the metric in (17) but only an extra contribution from the last factor, 

\begin{equation}
\gamma_k(z,\eta)=\frac{i}{2\pi}\big(w_{r,p}(\eta)+\sum_{lm\alpha}b_{lm\alpha}(\int_{\parallel v \parallel_1=1} v_{\alpha})dz_l \wedge d\bar{z}_m+\sum_{s} \frac{1}{s}\frac{\mid \eta_s\mid^{2p/s}}{\sum_t \mid \eta_t \mid^{2p/t}} \sum c_{ij\lambda \mu}\frac{ \eta_{s\alpha}\bar{\eta}_{s\beta}}{\mid \eta_s \mid^2} dz_i \wedge d\bar{z}_j\big)
\end{equation}


The contribution of the factor $\mid \eta_{11} \mid$ can be understood as the curvature of the sub-bundle of $V$ which is orthogonal complement to the remainder. Thus 

\begin{equation}
b_{lm\alpha}=c_{lm\alpha \alpha}
\end{equation} 

\noindent
where $c_{lm11}$ is read from the coefficients of the curvature tensor of $(V,w^{FS})$ the Fubini-Study metric on $V$ (the second factor in (40). Then we need to look at the integral 

\begin{equation}
\int_{X_k,q}\Theta^{n+k(r-1)}=\frac{(n+k(r-1))!}{n!(k(r-1))!}\int_X \int_{P(1^r,...,k^r)}w_{a,r,p}^{k(r-1)}(\eta)1_{\gamma_k,q}(z,\eta) \gamma_k(z,\eta)^n
\end{equation}

\noindent
In the course of evaluating with the Morse inequalities the curvature form is replaced by the trace of the above tensor in raising to the power $n=\dim X$, then if we use polar coordinates

\begin{equation}
x_s=\parallel \eta_s \parallel^{2p/s}, \qquad u_s=\eta_s/\parallel \eta_s \parallel
\end{equation}

\noindent
Then the value of the curvature when integrating over the sphere yields the following

\begin{equation}
\gamma_k=\frac{i}{2\pi}\big(\sum_{lm}b_{lm\alpha} dz_l \wedge d\bar{z}_m+\sum_{s} \frac{1}{s}\sum c_{ij\lambda \lambda} u_{s\lambda}\bar{u}_{s \lambda}dz_i \wedge d\bar{z}_j\big)
\end{equation}

\noindent
Because the first term is a finite sum with respect to $s$, the estimates for this new form would be essentially the same as those in \cite{D1}. Therefore one expects

\begin{equation}
\int_{X_k,q}\Theta^{n+k(r-1)}=\frac{(\log k)^n}{n!(k!)^r}( \int_X 1_{\gamma,q}\gamma^n +O((\log k)^{-1}))
\end{equation}

\noindent
similar to \cite{D1}, for non-invariant case. We have proved the following.

\vspace{0.3cm}

\begin{corollary}
The analogue of Theorem 1.4 holds for the bundle $E_{k,m}^{GG}$ replaced by $E_{k,m}$.
\end{corollary}


\section{Existence of global dual differential operators} 


\vspace{0.5cm}

In \cite{M} J. Merker proves the Green-Griffiths-Lang conjecture for a generic hypersurface in $\mathbb{P}^{n+1}$. He proves for $X \subset \mathbb{P}^{n+1}(\mathbb{C})$ of degree $d$ as a generic member in the universal family

\begin{equation}
\mathfrak{X} \subset \mathbb{P}^{n+1} \times \mathbb{P}^{(\frac{n+1+d)!}{(n+1)!d!}-1}
\end{equation}

\noindent
parametrizing all such hypersurfaces, the GGL-conjecture holds. His method uses a theorem of Y. T. Siu, as the following.

\begin{theorem} (Y. T. Siu (2004) \cite{S}, \cite{D})
Let $X$ be a general hypersurface in $\mathbb{P}^{n+1}$ as explained above. Then, there are two constants $c_n \geq 1$ and $c_n' \geq 1$ such that the twisted tangent bundle 

\begin{equation}
T_{J_{\text{vert}}^n(\mathfrak{X})} \otimes \pi_1^*\mathcal{O}_{\mathbb{P}^{n+1}}(c_n) \otimes \pi_2^* \mathcal{O}_{\mathbb{P}^{(\frac{n+1+d)!}{(n+1)!d!}-1}}(c_n')
\end{equation}

\noindent
where $\pi_1, \ \pi_2$ are the projections, is generated at every point by its global sections.
\end{theorem}

The proof by Merker, the above Theorem is established outside a certain exceptional algebraic subset $\Sigma \subset J_{\text{vert}}^n(\mathfrak{X})$ defined by vanishing of certain Wronskians. In order to give a similar proof of GGL conjecture for general $X$ one may use the following generalization.

\vspace{0.3cm}

\textbf{Question:}
If $X \subset \mathbb{P}^{n+1}$ be a generic member of a family $\mathfrak{X}$ of projective varieties, then there are constants $c_n$ and $c_n'$ such that


\begin{equation}
T_{J_{\text{vert}}^n(\mathfrak{X})} \otimes \mathcal{O}_{\mathfrak{X}_k}(c_n) \otimes \pi_{0k}^* L^{c_n'}
\end{equation}

\noindent
is generated at every point by its global sections, where $L$ is an ample line bundle on $\mathfrak{X}$. By the analogy between microlocal differential operators and formal polynomials on the symmetric tensor algebra it suffices to show 

\begin{equation}
H^0(X_k, Sym^{\leq m'}\tilde{V}_k \otimes \mathcal{O}_{X_k}(m) \otimes \pi_{0k}^*B) \ne 0, \qquad m'>>m >>k
\end{equation}

\noindent
where $\tilde{V}_k$ is the in-homogenized $V_k$ as acting as differential operators in first order. We also wish to work over the Demailly-Semple bundle of invariant jets. To this end by a similar procedure as the former case one may check the holomorphic Morse estimates applied to the following metric on the symmetric powers.


\vspace{0.3cm}
 

\begin{equation}
\vert (z,\xi) \vert= \Big( \sum_{s=1}^k \epsilon_s \big( \sum_{u_i \in S^sV^*} \vert W_{u_1,...,u_s}^s \vert^2 + \sum_{iju_{\alpha}u_{\beta}}C_{iju_{\alpha}u_{\beta}} z_i\bar{z}_j u_{\alpha}\bar{u}_{\beta}\big)^{p/s(s+1)} \Big)^{1/p}
\end{equation} 

\noindent
where $W_{u_1,...,u_s}^s$ is the Wronskian 

\begin{equation}
W_{u_1,...,u_s}^s=W(u_1 \circ f,...,u_s \circ f)
\end{equation}

\noindent
and we regard the summand front the $\epsilon_s$ as a metric on $S^sV^*$. The coefficient $C_{iju_{\alpha}u_{\beta}}$ we are going to compute. Moreover the frame $\langle u_i \rangle$ is chosen of monomials to be holomorphic and orthonormal at $0$ dual to the frame $\langle e^{\alpha}=\sqrt{l!/\alpha!}e_1^{\alpha_1}...e_r^{\alpha_r}\rangle$. 

The scaling of the basis in $S^lV^*$ is to make the fame to be orthonormal and are calculated as follows;

\begin{equation}
\langle e^{\alpha}, e^{\beta} \rangle=\langle \sqrt{l!/\alpha!}e_1^{\alpha_1}...e_r^{\alpha_r},\sqrt{l!/\alpha!}e_1^{\beta_1}...e_r^{\beta_r} \rangle = \sqrt{1/\alpha! \beta!}\langle \prod_{i=1}^le_{\eta(i)}, \sum_{\sigma \in S_l} \prod_{i=1}^le_{\eta \circ \sigma(i)} \rangle 
\end{equation}

\noindent
via the embedding $S^lV^* \hookrightarrow V^{\otimes l}$ and the map $\eta:\{1....l\} \to \{1....r\}$ taking the value $i$ the $\alpha_i$ times. 

To compute the coefficients of the curvature tensor we proceed as follows. Because the frame $\langle e_{\lambda} \rangle$ of $V$ were chosen to be orthonormal at the given point $x \in X$, substituting 

\begin{equation}
\langle e_{\lambda} , e_{\mu} \rangle=\delta_{\lambda \mu}+ \sum_{ij\lambda \mu}c_{ij\lambda \mu} z_i\bar{z}_j+...
\end{equation} 

\noindent
It follows that 

\begin{equation}
\langle e^{\alpha}, e^{\beta} \rangle= \sqrt{1/\alpha! \beta!} (\delta_{\alpha \beta} + \sum_{\eta \circ \sigma(i)=\eta(i)} c_{ij\alpha_{\eta(i)}\beta_{\eta(i)}}z_i\bar{z}_j+...
\end{equation}

\noindent
which explains the scalars $C_{iju_{\alpha}u_{\beta}}$ in terms of curvature of the metric on $V$. In the estimation of the volume 

\begin{equation}
\int_{X_k} \Theta^{n+k(r-1)} = \int_{X_k} (\Theta_{\text{vert}}+\Theta_{\text{hor}})^{n+k(r-1)}= \frac{(n+k(r-1))!}{n!(k(r-1))!}\int_X \int_{\mathbb{P}(1^{[r]},...,k^{[r]})} \Theta_{\text{vert}}^{k(r-1)}\Theta_{hor}^n
\end{equation}

\noindent
However the calculations with $\Theta$ involves more complicated estimates. The above metric reflects some ideas of D. Brotbek \cite{B}, on Wronskian ideal sheaves. It would be interesting to check out if the Monte Carlo process would converge with this metric. 

Lets see what is the conclusion of our discussions and the aforementioned metrics. As was mentioned the Theorem 1.1 has been stated and proved in \cite{D1} and \cite{D2} as an application of holomorphic Morse inequalities using the metric (18). Our methodology of using invariant metrics proves the same result for the invariant metrics on Demailly-Semple bundles. The convergence of the Monte-Carlo integrals in (32) and (45) implies the possibility to apply holomorphic  Morse inequalities in the new case. The conclusion is that, the Theorem 1.1 is also true for Demailly-Semple jet differentials, i.e with $E_{k,m}^{GG}$ replaced by $E_{k,m}$ (see \cite{D1} for definitions). 

\begin{thebibliography}{99}

\bibitem[B]{B} D. Brotbek, On the hyperbolicity of general hypersurface, preprint 2016

\bibitem[D1]{D1}  J. P. Demailly, Hyperbolic algebraic varieties and holomorphic differential equations, Acta Math. Vietnam, 37(4) 441-512, 2012

\bibitem[D2]{D2}  J. P. Demailly, Holomorphic Morse inequalities and Green-Griffiths-Lang conjecture, preprint

\bibitem[GG]{GG} M. Green, P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The
Chern Symposium 1979. (Proc. Intern. Sympos., Berkeley, California, 1979) 41-74, Springer, New York, 1980.

\bibitem[M]{M} J. Merker, low pole order frameson vertical jets of the universal hypersurface, Ann. Institut Fourier (Grenoble), 59, 2009, 861-933 

\bibitem[S]{S} Y. T. Siu, Hyperbolicity in complex geometry, The legacy of Niels Henrick Abel, Springer, Berlin, 2004, 543-566


\end{thebibliography}


 
\end{document}
