2026/07/20 by Bruno Scardua
Mathematics · #math.CV
We study polynomial holomorphic 1-forms in ℂ2 that are homologically trivial along the fibers of meromorphic pencils of the form ϕ= (fp)/(gq), where f,g are holomorphic functions (possibly polynomials) in general position and (p,q)=1. We first establish a homological characterization of relative exactness: if a polynomial 1-form Ω has vanishing periods along every closed path contained in the fibers ϕc, then Ω decomposes as Ω= aω0 + dh, where ω0 = p gdf - q f dg, for suitable polynomials a and h. In the homogeneous case, degree constraints force a to be constant. We then apply this integration principle to foliations leaving invariant plane curve singularities of cusp type fp + gq = 0. Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that Ω is a polynomial cusp basic form, Ω= d(fp + gq) + λ(p gdf - q fdg), λ∈ ℂ. In particular, such foliations admit Liouvillian first integrals of hypergeometric type. Our results provide a bridge between relative cohomology, the geometry of rational pencils, and the analytic structure of cusp foliations, yielding explicit normal forms and first integrals under homological hypotheses.