Curvature for tautological bundle of projectivation

Question

I'm trying to compute locally the curvature of $\mathscr{O}_{\mathbb{P}(E)}(-1)$, where $E\to X$ is a holomorphic bundle. The covering is given by $U_i\times \mathbb{P}(E)_j$, where $\{U_i\}$ is a covering of manifold $X$ and $E$ is a trivialization on it. $\mathbb{P}(E)_j=\{[z]\in \mathbb{P}(E)|z_j\not =0\}$. Thus the metric of tautological bundle is $$h(\phi_j,\phi_j)=\frac{1}{z_j \bar{z}_j}\sum_l z_l\bar{z}_l.$$ Thus the Fubini-Study form is the curvature form. Here $\phi_j$ is the section of bundle by unit vector. Am I right in this computation? I'm not sure if this metric is unrelavent with the original manifold.

Answer 1

$\newcommand{\Proj}{\mathbf{P}}\newcommand{\Reals}{\mathbf{R}}$Not sure this answers your question, but it's too long for a comment.

Generalities: Let $p:(E, h) \to X$ be an Hermitian holomorphic vector bundle of rank $k$, $\pi:\Proj(E) \to X$ the projectivized bundle, and $(\tau, h)$ the tautological subbundle of $\pi^{*}E \to \Proj(E)$, equipped with the ambient Hermitian structure from $E$. The curvature form of $(\tau, h)$ is $$ \omega = -i\partial\bar{\partial} \log h. $$ The Leray-Hirsch theorem implies $H^{*}\bigl(\Proj(E), \Reals\bigr)$ is generated, as a $H^{*}(X, \Reals)$-module, by $\zeta = [\omega] = 2\pi c_{1}(\tau)$, the $2$-dimensional cohomology class represented by $\omega$, subject to the relation $$ 0 = \sum_{j=0}^{k} (-1)^{j}c_{j}(E) \zeta^{k-j}. \tag{1} $$

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In your situation, it appears you're assuming $E$ is equipped with an Hermitian structure $$ h(z, z) = \sum_{\ell=1}^{k} z_{\ell} \bar{z}_{\ell}, $$ i.e., the Euclidean metric (possibly up to scaling) in each fibre. Restricting to a trivializing neighborhood for $\tau$ (a set of the form $\{z_{j} \neq 0\}$ in a trivializing neighborhood for $E$), the local holomorphic section $\phi_{j}$ of $\tau = \mathscr{O}_{\Proj(E)}(-1)$ has squared norm $$ h(\phi_{j}, \phi_{j}) = \frac{1}{z_{j} \bar{z}_{j}} \sum_{\ell} z_{\ell} \bar{z}_{\ell}, \tag{2} $$ as you say. However, dependence of the Hermitian structure on the base has been suppressed throughout. Consequently, the curvature form of the Hermitian metric (2) is a Fubini-Study form in the fibre directions, but in general has "horizontal" components (compare (1)) arising because $h$ depends on the coordinates of $X$.