International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-09-05 , DOI: 10.1142/s0129167x20500895
Bruno Scárdua

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family ${Ωt}t∈D$ of integrable one-forms $Ωt$ defined in a neighborhood $U⊂ℂn$ of the initial singular point, and parametrized by the disc $D⊂ℂ$. The initial foliation is defined by $Ω0$. The second type, more restrictive, is given by an integrable holomorphic one-form $Ω(x,t)$ defined in the product $U×D⊂ℂn×ℂ$. Then, the initial foliation is defined by the slice restriction $Ω(x,0)$. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by $df=0$ for some germ of holomorphic function $f∈𝒪n$ at the origin $0∈ℂn,n≥3$. We assume that the germ $f$ is irreducible and that the typical fiber of $f$ is simply-connected. This is the case if outside of a dimension $≤n−3$ analytic subset $Y⊂ℂn$, the analytic hypersurface $Xf:(f=0)$ has only normal crossings singularities. We then prove that, if $codsingΩ(x,0)≥2$ then the (germ of the) developing foliation given by $Ω(x,t)=0$ also exhibits a holomorphic first integral. For the general case, i.e. $codsingΩ(x,0)≥1$, we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations ${ℱt}t∈ℂ,0$, of a local pencil $ℱ0:fg=constant$, for $f,g∈𝒪n$. For dimension $n=2,$ we consider $f=x,g=y$. For dimension $n≥3,$ we assume some generic geometric conditions on $f$ and $g$. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type $F̂=f1+λ̂(t)g1+μ̂(t)eĤ(x,y,t)$ with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form $M=fgeP(t)+H(x,y,t)$ with some additional properties, provided that for $n=2$ the axes remain invariant for the foliations $ℱt$.

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