In the paper by Green and Griffiths Two applications of algebraic geometry to entire holomorphic mappings (Proposition 2.5) it is proved that a jet pseudometric can be constructed on a projective variety, provided that there are sufficient sections. However, I'm not an expert in differential geometry and the proof has a number of obscure points, which I'm going to talk about. As for the notation:
- $P_k(X)$ is the bundle of "projectivized jets", namely the quotient of the bundle of $k$-jets by the "weighted action" of $\mathbb C^*$.
- $\mathcal L^m$ is the sheaf on $P_k(X)$ of "polynomials of weight $m$", see page 46.
The proposition is as follows (I just copy):
Let $X$ be a projective algebraic variety (over the complex numbers), and $E \to P_k(X)$ a very ample line bundle. If $t_0, \ldots, t_M$ is any basis for $H^0(P_k(X), E)$ and $s_0, \ldots, s_N$ any basis for $H^0(P_k(X), \mathcal L^m \otimes E^{-1})$, then for a suitable constant $A > 0$ the jet pseudometric \begin{equation} |j|^2 = A \left( \sum_{i,\alpha} |(s_i \cdot t_\alpha)(j)|^2 \right)^{1/m} \end{equation} has holomorphic sectional curvatures $\leq -1$ on discs.
Question 1: where do we use that $X$ is projective? Is perhaps compactness enough?
The proof goes on as follows: we fix a holomorphic map $f \colon \Delta \to X$ defined on the unit disk and we assume that its image is contained in an open set which is trivializing for both $E$ and $\mathcal L^m$, so that \begin{align} t_\alpha (j_k(f)(z)) = u_\alpha(z), \\ s_i(j_k(f)(z)) = v_i(z), \end{align} where $u_\alpha$ and $v_i$ are holomorphic functions. Then, we compute: \begin{align} i \partial \overline{\partial} \log |j_k(f)|^2 &= \frac{i}{m} \partial \overline{\partial}\log \left( \sum |v_i(z)|^2 \right) + \frac{i}{m} \partial \overline{\partial}\log \left( \sum |u_{\alpha}(z)|^2 \right) \\ &= \frac{\alpha}{m} + \frac{\beta}{m}. \end{align} Question 2: now, why $\alpha$ and $\beta$ are nonnegative?
$\beta$ is described as follows: if $\phi_E \colon P_k(X) \to \mathbb P^N$ is the embedding associated to $E$, and $f_k = j_k(f) \colon \Delta \to P_k(X)$, then $\beta$ is (quite clearly, as I figure) the pullback of the form $\omega$ which is itself the pullback of the Fubini-Study form $\omega_{FS}$ on $\mathbb P^N$: \begin{equation} \beta = f_k^*(\omega). \end{equation} Next, the author claim the following:
$\beta(z) = 0$ if and only if $j_{k+1}(f)(z)$ is a constant jet, if and only if the differential of $f_k$ vanishes at $z$.
Question 3: why is this true, and actually why is it needed in the next part of the proof? How do you compute the "differential of $f_k$"? The target $P_k(X)$ is singular...
The last part of the proof amounts to the following argument:
Now both of the mappings $z \mapsto \beta(z)$ and $z \mapsto |j_k(f)(z)|^2$ are quadratic with respect to reparametrization, and consequently the ratio \begin{equation} \frac{|j_k(f)(z)|^2}{\beta(z)} \end{equation} is locally bounded from above on the projectivized tangent bundle of $P_k(X)$. Since it intrinsic and $X$ is compact, the ratio will everywhere be $\leq B$ for some constant $B$.
Question 4: why does this work? Why is $B$ uniform with respect to the map $f$?
