Chow forms and complete intersections in the projective space

Abstract

We prove that every Hodge cohomology class of bidegree (q,q) on a projective manifold X can be recovered from its image by the Chow transformation restricted to a suitable irreducible algebraic component of the space Cq−1(X) of effective algebraic cycles in X of dimension q−1. An application to the problem of the approximation by algebraic cycles is given. In the case of the cohomology class of an effective algebraic cycle, this injectivity at the cohomological level is a consequence of the inversion formula for the Chow transform of a conormal. When X=PN, the inversion formula for the conormal is extended to the case of the conormal of any closed positive (q,q)-current. An inversion formula for the Radon transform, defined on the Grassmannian, of smooth functions is involved and is also used to obtain a characterization of Chow forms of complete intersections in the projective space, expressed by means of the Capelli differential operators.