As is well known, every Kaehler manifold can canonically be given the structure of a symplectic manifold. Is it naive to assume that holomorphic vector bundles over a Kaehler manifold can be given the structure of a symplectic vector bundle? 



Symplectic and Holomorphic Vector Bundles
This seems to have nothing to do with Kahler manifolds, at least not how you have stated it. Any $C^{\infty}$-complex vector bundle over a paracompact smooth manifold admits a Hermitian metric, by employing a partition of unity. The imaginary part of this Hermitian metric is a skew-symmetric, non-degenerate bilinear form on each fiber which makes that complex vector bundle into a symplectic vector bundle. 


