3
$\begingroup$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern classes and $\alpha$-invariant on $\mathbb{CP}^3$. I would like to know more about the deformation classification of the corresponding $\mathbb{CP}^1$-bundles on $\mathbb{CP}^3$. The precise question is the following:

Let $X$ be the projectivization of a (holomorphic) topologically trivial complex rank two bundle on $\mathbb{CP}^3$. Is $X$ deformation equivalent to $\mathbb{CP}^1\times\mathbb{CP}^3$? If not, how can these complex manifolds be distinguished?

A topologically trivial rank two vector bundle on $\mathbb{CP}^3$ is one whose Chern classes and $\alpha$-invariant are trivial. If I am not mistaken, $X$ as above is diffeomorphic to $\mathbb{CP}^1\times\mathbb{CP}^3$, but I have no idea about the deformation equivalence.

I would also appreciate if anybody has a partial answer to the more general question: What general results exist for the comparison between deformation equivalence and diffeomorphism for manifolds of Kodaira dimension $-\infty$/rational complex varieties?

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.