Relative Cohomology and Deformations of Foliations with Holomorphic First Integrals

Abstract This work is devoted to the study and computation of relative cohomology for 1-forms defining foliations with holomorphic first integrals. We study both the local and global cases. In the local setting, we study solutions h of the cohomological equation $$ \eta \wedge \omega _0 = dh \wedge \omega _0, $$ η ∧ ω 0 = d h ∧ ω 0 , where $$\omega _0=p y\,dx+q x\,dy$$ ω 0 = p y d x + q x d y admits the holomorphic first integral $$\phi =x^p y^q$$ ϕ = x p y q , with $$p,q\in \mathbb {N}$$ p , q ∈ N , in local coordinates ( x , y ) near the origin of $$\mathbb {C}^2$$ C 2 . We prove the existence of a local canonical form for a given meromorphic 1-form $$\eta $$ η in terms of $$\omega _0$$ ω 0 and $$\phi $$ ϕ , provided that $$\eta $$ η has only simple poles contained in the coordinate axes $$(xy=0)$$ ( x y = 0 ) . We then characterize relative exactness by the vanishing of the periods of $$\eta $$ η over the first homology of the fibers $$\{\phi =c\}$$ { ϕ = c } . In the global setting, we prove polynomial versions of the local results on $$\mathbb {C}^2$$ C 2 for fibrations