M. Herrera

De Rham theorems on semianalytic sets

Semianalytic sets are subsets of real analytic manifolds locally defined by inequalities of real analytic functions. We refer to [5] for precise definitions and properties about them. Let M be a semianalytic set in the real analytic manifold X. We consider in this note the complexes of differential forms and currents induced on M by the smooth forms and currents of X, and relate them to the real cohomology and homology of M. The following version of de Rhams theorems holds: there is an epimorphism from the cohomology of the forms on M onto the cohomology of M, and there is a monomorphism from the homology of M into the homology of the currents on M. In general these maps are not isomorphisms, even on algebraic sets in i?. These results answer a question posed by Norguet [7]. We show in the first section that homology and cohomology classes of M can be represented by semianalytic chains and cochains. This is used in the second section to prove de Rhams theorems. In the third section an example is given in which the above maps are not isomorphisms, together with some particular remarks on the Poincaré lemma. I t is supposed throughout this note that X is paracompact and that M is closed in X and has dimension p. Then the set M* of the regular points of M is an analytic submanifold of X and the singular set dM = M— M* is semianalytic in X with dimension dim dM<p. If N is semianalytic and locally closed in X with dim N^q, then bN=N — N is semianalytic and closed in X with dim bN<q. Unless stated otherwise, K is a principal ideal domain. If 0 is a family of supports on the locally compact space Y and

Algebraic cycles as residues of meromorphic forms

is a sheaf of X-modules on F, then H*(Y; Ç)(H

On the global lifting of meromorphic forms

(Y;

Residues and Principal Values on Complex Spaces.

)) denotes the BorelMoore homology of Y with coefficients in

De Rham cohomology of an analytic space

and closed supports (supports in </>) [ l ] . If F(Z Y is closed, there is an exact sequence

Integration on a semianalytic set

According to a classical result of Weil [15,1, a divisor ~ of a smooth n-dimensional projective variety X is homologous to zero if and only if it is the residue of a closed meromorphic 1-form on X. Griffiths proved recently [9, pp. 3-8,1 that a 0-cycle ~t of X is homologous to zero if and onty if it is the Grothendieck residue of a meromorphic n-form & on X having poles in the union of a family of complex hypersurfaces Yl . . . . . II, of X, such that (~ Y~ is 0-dimensional and contains the support of ~. We show in this paper (Theorem 3.7) that, in fact, any q-dimensional algebraic cycle ~ of X, 0< q < n, is the analytic residue of a semimeromorphic (n-q)-form on X, having poles in the union of a family ~ = {Y1 . . . . . Yn-q} of hypersurfaces in X such that N ~ contains the support of ~. The form & is not closed, in general, but its differential verifies

Duality and the deRham cohomology of infinitesimal neighborhoods

We characterize the existence of global liftings of local data of meromorphic forms, satisfying a family of compatibility conditions on their polar sets, in terms of obstructions belonging to the cohomology H.(X, Ω.), where X is the ambient manifold. These obstructions are constructed canonically, using residue-principal value operators.