On the Dolbeault cohomology of projective varieties and locally residual currents

Details On the Dolbeault cohomology of projective varieties and locally residual currents On the Dolbeault cohomology of projective varieties and locally residual currents

2005-05-03

arxiv.org/abs/math/0505061v1

Mathematics

Algebraic Geometry

A preliminiary version of a future paper, maybe with Sebastien Boucksom

Scientific paper

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.

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