Locally Residual Currents and Dolbeault Cohomology on Projective Manifolds

COMPLEX ALGEBRAIC GEOMETRY
Topic:	Locally Residual Currents and Dolbeault Cohomology on Projective Manifolds
Speaker:	Bruno Fabre
Affiliation:	Stokholm University and Member, School of Mathematics
Date:	Wednesday, November 1
Time/Room:	1:00pm - 2:00pm/S-101

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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 $j<p$ can be computed by the preceding complex of locally residual currents, by taking sections over $U$; moreover, the sections are "algebraic", in the sense that they extend to the whole $X$ as locally residual currents. For $j=p$, let us give a cohomology class $\alpha\in H^p(U,\Omega^q)$ given by a locally residual current (they give no more all the classes); and assume that the image in $V:=\cup_{t\in U^*}{H_t}\subset P^N$ is $\overline\partial-$exact. Then $\alpha$ is algebraic, in the sense that it comes from a locally residual current on $X$.}