AI 中文总结
该论文研究复空间上对数叶状结构的相对上同调，在合适假设下得到其分解式，建立亚纯延拓定理，处理奇异对数叶状结构，并将结果应用于推导解析可积形变的形式正规形。
AI 中文摘要
我们研究对数微分形式及关于关联对数叶状结构相对闭的亚纯形式的相对上同调。在合适的丢番图、几何与拓扑假设下，我们得到分解式，该分解式可表示为定义对数形式的亚纯倍数、恰当亚纯形式及具常留数的对数形式的和。我们建立了从合适二维截面出发的亚纯延拓定理，并证明了相对上同调分解的局部、多项式及齐次版本；整体多项式结果通过环境叶状延拓与亚纯延拓在任意维数下得证。我们运用奇点解消、非节点饱和及和乐粘合处理奇异对数叶状结构。作为应用，我们推导了解析可积形变的形式正规形，包括二维的多分量结果及高维的二分量推广。
英文摘要
We study relative cohomology for logarithmic differential forms and meromorphic forms that are relatively closed with respect to the associated logarithmic foliation. Under suitable Diophantine, geometric, and topological hypotheses, we obtain decompositions into a meromorphic multiple of the defining logarithmic form, an exact meromorphic form, and a logarithmic form with constant residues. We establish a meromorphic extension theorem from suitable two-dimensional sections and prove local, polynomial, and homogeneous versions of the relative-cohomology decomposition; the global polynomial result is proved in arbitrary dimension by ambient leafwise continuation and meromorphic extension. Resolution of singularities, non-nodal saturation, and holonomy gluing are used to treat singular logarithmic foliations. As an application, we derive formal normal forms for analytic integrable deformations, including a several-component result in dimension two and a two-component extension in higher dimensions.
