Locally residual currents and Dolbeault cohomology on projective manifolds

Let X be a projective manifold of dimension n, and n   hypersurfaces Yi(1⩽i⩽n) on X, defining ample line bundles, in complete intersection position. After introducing sheaves of locally residual currents, we enunciate the following two main theorems. First, for any positive integer i  , the Dolbeault cohomology group Hi(ΩXq) of the sheaf of holomorphic q-forms on X can be computed as the ith cohomology group of some complex of residual currents on X. We get from this the theorem of [A. Dickenstein, M. Herrera, C. Sessa, On the global liftings of meromorphic forms, Manuscripta Math. 47 (1984) 31–54] that any locally residual current on X   which is ∂¯-exact is globally residual. Secondly, let us assume that Y1∩⋯∩Yp(1⩽p⩽n) are reduced complete intersections. We get another exact sequence computing Hi(ΩXn) by restricting to residual currents obtained from meromorphic forms with simple poles on the YiYi. We deduce from this a reformulation of the main theorem of [B. Khesin, A. Rosly, R. Thomas, A polar De Rham theorem, Topology 43 (2004) 1231–1246], saying that we can compute the cohomology groups Hi(ΩXn) by the cohomology of a complex of principal value currents. We also deduce from this the result from [P. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976) 321–390] that if Y1∩⋯∩YnY1∩⋯∩Yn is a set of distinct points {P1,…,Ps}{P1,…,Ps}, then for any sequence of s   complex numbers ci(1⩽i⩽s), there is a global meromorphic n-form Ψ   with simple poles on each YiYi such that:(∀i,1⩽i⩽s)ResY1,…,YnPiΨ=ci iff ∑i=1sci=0. We give proofs of the theorems by mean of several exact sequences of sheaves of locally residual currents. We conclude by giving an application to the Hodge conjecture, giving some new formulation using our theorems.