On the Dolbeault cohomology of projective varieties and locally residual currents

Let $X$ be a projective manifold. Let $Y_1,\mathrm{.}\mathrm{.}\mathrm{.},Y_{p+1}$ 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 $i\le p+1$,thecohomologygroups$H^i(\Omega^q)$canbecomputedasthe$i-$th cohomology groups of some complex of global sections of locally residual currents on $X$. We could also compute the cohomology of the subsheaves ${\tilde{\mathrm{\Omega }}}^q\subset {\mathrm{\Omega }}^q$ of $\mathrm{\partial }-$closed holomorphic forms by the corresponding subsheaves of $\mathrm{\partial }-$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$\mathrm{.}Wealsoshowthatanylocallyresidualcurrent$T$ofbidegree$(q,i-1)$withsupportin$Y_1\cap … Y_{i-1}$ can be written as a global residue $T=Re{s}_{Y_1,\mathrm{.}\mathrm{.}\mathrm{.},Y_{i-1}}\mathrm{\Psi }$ of some meromorphic form with pole in $Y_1\cup \mathrm{.}\mathrm{.}\mathrm{.}\cup Y_i$. We can avoid $Y_i$ iff the current in $\bar{\mathrm{\partial }}-$exact; we deduce as corollaries a theorem of Hererra-Dickenstein-Sessa. We give as a conclusion a new formulation of the Hodge conjecture.