发表机构
Universidade Federal da Integração Latino-Americana; Universidade Federal do Rio de Janeiro（拉丁美洲联邦大学; 里约热内卢联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究复射影空间上余维一全纯叶状结构的可积性，在不变超平面上奇异除子为最大次数不可约时，证明其具有有理首次积分，并给出刚性定理及反例说明假设的尖锐性。
AI 中文摘要
设 $\F$ 为 $\PP^n$（$n\geq3$）上次数为 $d$ 的余维一全纯叶状结构，且具有一个不变超平面 $H$。我们研究极值情形：$S=(H\cap\Sing(\F))_{\rm red}$ 是 $H$ 上次数为 $d+1$ 的不可约超曲面。当 $d+1$ 为素数幂时，我们证明在适当的齐次坐标（其中 $H=(t=0)$）下，有 \\[ \Omega=Q\\,dt-\frac{t}{d+1}\\,dQ, \\] 其中 $Q$ 是次数为 $d+1$ 的齐次多项式。因此 $Q/t^{d+1}$ 是一个有理首次积分。证明将 Frobenius 方程归结为平面截面上的扭曲闭性方程，并利用 Zariski 关于不可约平面曲线 Alexander 多项式的定理。我们还证明了关于任意光滑不变超曲面 $D\subset\PP^n$ 的一个互补刚性定理：若 $D$ 上的约化奇异除子光滑、不可约且具有最大次数，则同样的正规形现象成立，且对其次数无需算术假设；在低权值范围内，奇异除子的光滑性假设可以去掉。最后，我们证明主要假设是尖锐的。去掉最大次数条件会产生一个具有不可约约化奇异支撑且无非平凡有理首次积分的族。对于每个非素数幂的 $d+1$，我们构造一个具有不可约最大次数奇异支撑的全局反例，而最后一个族表明约化支撑的不可约性也确实是本质必要的。
英文摘要
Let $\F$ be a codimension-one holomorphic foliation of degree $d$ on $\PP^n$, $n\geq3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\cap\Sing(\F))_{\rm red}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homogeneous coordinates with $H=(t=0)$, \[ Ω=Q\,dt-\frac{t}{d+1}\,dQ , \] where $Q$ is homogeneous of degree $d+1$. Thus $Q/t^{d+1}$ is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski's theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface $D\subset\PP^n$: if the reduced singular divisor on $D$ is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every $d+1$ which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.
