VANISHING OF THE FIRST DOLBEAULT COHOMOLOGY GROUP OF HOLOMORPHIC LINE BUNDLES ON COMPLETE INTERSECTIONS IN INFINITE DIMENSIONAL PROJECTIVE SPACE (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2005)
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.