AI 中文总结
研究复平面\(\mathbb{C}^2\)中沿特定亚纯铅笔纤维同调平凡的多项式全纯\(1\)-形式，通过建立同调特征和积分原理，应用于尖点叶状结构，证明全局化定理，给出标准形式和第一积分，在多方面架起桥梁。
AI 中文摘要
我们研究复平面\(\mathbb{C}^2\)中沿形如\(\phi = \frac{f^p}{g^q}\)的亚纯铅笔纤维同调平凡的多项式全纯\(1\)-形式，其中\(f,g\)是处于一般位置的全纯函数（可能是多项式）且\((p,q)=1\)。首先建立相对精确性的同调特征：若多项式\(1\)-形式\(\Omega\)沿\(\phi_c\)中每条闭路径周期为零，则\(\Omega\)可分解为\(\Omega = a\omega_0 + dh\)，其中\(\omega_0 = p gdf - q f dg\)。在齐次情形下，次数限制使\(a\)为常数。然后将此积分原理应用于保持尖点型平面曲线奇点\(f^p + g^q = 0\)不变的叶状结构。在自然一般性（莫尔斯型）条件下，证明一个全局化定理，表明沿相关铅笔的同调平凡性意味着\(\Omega\)是多项式尖点基本形式\(\Omega = d(f^p + g^q) + \lambda (p gdf - q fdg)\)，\(\lambda \in \mathbb{C}\)。特别地，此类叶状结构允许超几何型的刘维尔第一积分。我们的结果在相对上同调、有理铅笔几何和尖点叶状结构的解析结构之间架起桥梁，在同调假设下给出明确的标准形式和第一积分。
英文摘要
We study polynomial holomorphic $1$-forms in $\mathbb{C}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $ ϕ= \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 $Ω$ has vanishing periods along every closed path contained in the fibers $ϕ_c$, then $Ω$ decomposes as $Ω= aω_0 + dh,$ where $ω_0 = p gdf - q f dg,$ 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 \[ f^p + g^q = 0. \] Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that $Ω$ 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.
