Relative Cohomology and Deformations of Foliations with Holomorphic First Integrals

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 η ω ₀ = dh ω ₀, η ∧ ω 0 = d h ∧ ω 0 , where ω ₀=p y\,dx+q x\,dy ω 0 = p y d x + q x d y admits the holomorphic first integral φ =xᵖ yq ϕ = x p y q , with p,q∈ N p , q ∈ N , in local coordinates ( x , y ) near the origin of C² C 2 . We prove the existence of a local canonical form for a given meromorphic 1-form η η in terms of ω ₀ ω 0 and φ ϕ , provided that η 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 η η over the first homology of the fibers \φ =c\ { ϕ = c } . In the global setting, we prove polynomial versions of the local results on C² C 2 for fibrations