Intégration des classes de \(d'd''\)-cohomologie sur les cycles analytiques. Solution du problème de l’injectivité. (Integration of \(d'd''\)-cohomology classes on analytic cycles. Solution of the problem of injectivity). (French) Zbl 0621.32020

Ofman, Salomon

J. Math. Pures Appl., IX. Sér. 68, No. 1, 73-94 (1989).

Let \(X\) be a \((n-q-1)\)-submanifold of a complex algebraic \(n\)-manifold \(Z\), \(Y=Z-X\) and \[ V^{*,*}(Y)= \left[Ker\left[A^{*,*}(Y) \overset{\partial{\bar\partial}} \longrightarrow A^{*,*}(Y)\right]\right] \partial A^{*,*}(Y)\oplus {\bar\partial}A^{*,*}(Y) \] where \(A^{r,s}(Y)\) is the space of \({\mathcal C}^{\infty}(r,s)\)-differential forms on Y; then \(V^{q,q}(Y)\) is an infinite dimensional space.
Let \(\rho _ 0\) be the application: \[ V^{q,q}(Y)\to H^ 0(C_ q(Y),{\mathcal H})\text{ defined by: }\rho_ 0\theta(c)= \int_{c}\Phi \] where \(\Phi\) is a \(\partial{\bar\partial}\)-closed differential form representing the \(\partial{\bar\partial}\)-cohomology class \(\theta\) in \(V^{q,q}(Y)\), \(C_ q(Y)\) the space of compact analytic cycles of dimension \(q\), and \({\mathcal H}\) the sheaf of germs of pluriharmonic functions; it is proved here that its kernel, which is a finite dimensional space, is contained in the restriction of \(V^{q,q}(Z)\) to \(V^{q,q}(Y)\) and is linked to the topological properties of the compact manifold \(Z\).

MSC:

32C30	Integration on analytic sets and spaces, currents
32F10	\(q\)-convexity, \(q\)-concavity
32C35	Analytic sheaves and cohomology groups

Keywords:

integration on analytic cycles; cohomology; injectivity; differential forms