## 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

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