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 $P$ 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 $P$ admits smooth partitions of unity.