A Dolbeault isomorphism for complete intersections in infinite-dimensional projective space

We consider a complex submanifold X of finite codimension in an infinite-dimensional complex projective space P and a holomorphic vector bundle E over X. Given a covering $\mathcal{U}$ of X with Zariski open sets, we define a subcomplex $\mathcal{C}(X, E)$ of the Čech complex corresponding to the vector bundle E and the covering $\mathcal{U}$. For a special class of coverings $\mathcal{U}$, we prove that the complex $\mathcal{C}(X, E)$ is acyclic when X is a complete intersection and P admits smooth partitions of unity.

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