2005/05/03 by Bruno Fabre, Fabre, Bruno
Mathematics · #14H05 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H05
paper · pdf · doi:10.48550/arxiv.math/0505061
A preliminiary version of a future paper, maybe with Sebastien Boucksom
arxiv created 2005/05/03 · arxiv updated 2009/12/01
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=Res_Y1,...,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.