It is well known that a complex Finsler structure F on a holomorphic vector bundle E over a compact K?hler manifold (M; g) defines a bundle-like pseudo-K?hler metric of Sasaki type on the total space of the associated projectivized bundle PE. Then using an argument inspired from the theory of vanishing theorems for complex analytic foliations, we obtain a vanishing cohomology theorem of Nakano's type for base-like forms with values in a foliated line bundle over PE.
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