On meromorphic pencils, cusp singularities and holomorphic foliations in the complex plane

Abstract

We study polynomial holomorphic $1$-forms in ${\mathbb{C}}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $\varphi =\frac{f^p}{g^q},$ 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 $\Omega$ has vanishing periods along every closed path contained in the fibers ${\varphi }_c$, then $\Omega$ decomposes as $\Omega =a{\omega }_0+dh,$ where ${\omega }_0=pgdf-qfdg,$ 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 \[ f^p + g^q = 0. \] Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that $\Omega$ is a polynomial cusp basic form, \[ Ω= d(f^p + g^q) + λ(p gdf - q fdg), \qquad λ\in \mathbb{C}. \] 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.