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Let $E\to X$ be a holomorphic vector bundle over a compact Kahler manifold $X$ with Kahler form $\omega_{X}$. For a given hermitian metric on $E$, let $\omega_{E}$ be the Chern form of the line bundle $\mathcal{O}(1)$ over the projective bundle $\mathbb{P}(E)$. Then $\mathbb{P}(E)$ admits a family of 'canonical' Kahler metrics with Kahler forms $\omega_{E}+\lambda\pi^{*}\omega_{X}$ for $\lambda>>0$ where $\pi:\mathbb{P}(E)\to X$ is the bundle projection.

My question: Is there any 'rigidity theorem' such as 'If a symplecitc manifold $(M,\omega_{M})$ is diffeomorphic to $\mathbb{P}(E)$, then $(M,\omega_{M})$ must defomation equivalent to $\mathbb{P}(E)$ with the canonical Kahler form as above.'

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  • $\begingroup$ Why is $\omega_{E}+\lambda\omega_{X}$ a Kähler form for $\lambda \gg 0$? $\endgroup$
    – abx
    Nov 20, 2014 at 6:58
  • $\begingroup$ I'm sorry. It should be $\omega_{E}+\lambda\pi^{*}\omega_{X}$ with $\pi:\mathbb{P}(E)\to X$ be the projection and $X$ should be compact. $\endgroup$
    – Jiang
    Nov 20, 2014 at 8:57
  • $\begingroup$ Still, I do not believe your statement without any assumption on $E$. $\endgroup$
    – abx
    Nov 20, 2014 at 10:38
  • $\begingroup$ As I wrote, $E$ is a holomorphic vector bundle. This is not my statement. It is Proposition 3.18 in Vosin's book 'Hodge theory and complex algebraic geometry I',p.78. $\endgroup$
    – Jiang
    Nov 20, 2014 at 11:08
  • $\begingroup$ OK, you are right, sorry. $\endgroup$
    – abx
    Nov 20, 2014 at 12:33

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