On the Dolbeault cohomology of projective varieties and locally residual currents

الأصل · EN

Let X be a projective manifold. Let Y₁,...,Yₚ₊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≤ p+1, the cohomology groups Hⁱ(Ωq) 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 Ωq⊂ Ωq of ∂-closed holomorphic forms by the corresponding subsheaves of ∂-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 Y₁∩ >...∩ Yᵢ. We also show that any locally residual current T of bidegree (q,i-1) with support in Y₁∩... Yᵢ₋₁ can be written as a global residue T=ResY₁,...,Yᵢ₋₁Ψ of some meromorphic form with pole in Y₁∪...∪ Yᵢ. We can avoid Yᵢ iff the current in ∂-exact; we deduce as corollaries a theorem of Hererra-Dickenstein-Sessa. We give as a conclusion a new formulation of the Hodge conjecture.