A DOLBEAULT ISOMORPHISM FOR COMPLETE INTERSECTIONS IN INFINITE-DIMENSIONAL PROJECTIVE SPACE

Abstract

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.