Complex Algebraic Geometry

First we define, for any analytic manifold $X$ of dimension $n$, locally residual currents; $C^{q,p}$ denotes the sheaf of locally residual currents of bidegree $(q,p)$. Then, we have a fundamental resolution of the sheaf of holomorphic $q-$forms $$0\to \Omega^q\to C^{q,0}\to C^{q,1}\to \dots,$$ where the arrows $C^{q,i}\to C^{q,i+1}$ are given by $\overline\partial$. The situation is then the following: we assume that $X$ is irreducible projective in the projective space $P^N$, and let be given $n-p$ hyperplanes in $P^N$ intersecting properly on $X$. For a domain $U^*$ of the grassmannian $G(N-n+p,N)$, we define a domain $U\subset X$ as $\cup_{t\in U^*}{(H_t\cap X)}$. Then we have the following: {\bf Theorem.} {The cohomology $H^j(U,\Omega^q)$ for $j