3
$\begingroup$

Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : Firstly, the metric $h$ induces a Fubini-study metric on the fibres of $\mathbb{P}(E)$ (just take local orthonormal frames to get a smooth fibre bundle, use the usual Fubini-study metric and then pullback to the holomorphic bundle). Secondly, the Chern connection of $h$ induces a splitting $T\mathbb{P}(E) = T\mathbb{C}\mathbb{P}^{r} \oplus TX$. Now put the direct sum metric. My question is: This seems like a very natural construction. Has this been studied before (in the sense of curvature properties etc)? If so, I'd be most grateful if a reference is pointed out.

$\endgroup$
1
$\begingroup$

I have seen a description of this metric in Andrei Teleman's paper Families of holomorphic bundles. Commun. Contemp. Math. 10 (2008), no. 4, 523–551.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.