Vanishing of the first Dolbeault cohomology group of holomorphic line bundles on complete intersections in infinite-dimensional projective space

We consider a complex submanifold $X$ of finite codimension in an infinite-dimensional complex projective space $\bp$ and prove that the first Dolbeault cohomology group of all line bundles $\mathcal{O}_X(n)$, $n \in \mathbb{Z}$, vanishes when $X$ is a complete intersection and $\bp$ admits smooth partitions of unity.

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