On the Dolbeault cohomology of projective varieties and locally residual currents

Abstract

Let $X$ be a projective manifold. Let $Y_1,...,Y_{p+1}$ be $p+1$ ample hypersurfaces in complete intersection position on $X$, each defined by the global section of an ample Cartier divisor. We show in this note that for $i\le p+1$, the cohomology groups $H^i(\Omega^q)$ can be computed as the $i-$th cohomology groups of some complex of global sections of locally residual currents on $X$. We could also compute the cohomology of the subsheaves ${\tilde\Omega}^q\subset \Omega^q$ of $\partial-$closed holomorphic forms by the corresponding subsheaves of $\partial-$closed locally residual currents. We deduce like this that any cohomology class of bidegree $(i,i)$ has an element which is a $d-$closed locally residual current with support in $Y_1\cap >...\cap Y_i$. We also show that any locally residual current $T$ of bidegree $(q,i-1)$ with support in $Y_1\cap ... Y_{i-1}$ can be written as a global residue $T=Res_{Y_1,...,Y_{i-1}}{\Psi}$ of some meromorphic form with pole in $Y_1\cup...\cup Y_{i}$. We can avoid $Y_{i}$ iff the current in $\bar\partial-$exact; we deduce as corollaries a theorem of Hererra-Dickenstein-Sessa. We give as a conclusion a new formulation of the Hodge conjecture.