发表机构
University of Maroua（马里乌大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究仿射平面中一类非齐次自由除子的对数泊松与德拉姆上同调，利用显式基构造复形并计算上同调，证明除次数1外两者自然同构，次数1时为分裂一维扩张。
AI 中文摘要
我们研究了仿射平面中一类非齐次自由除子相关的对数泊松与德拉姆上同调。这些除子作为约化法交叉除子的非齐次形变而出现，并且其对数向量场模（在Saito的意义下）具有显式基。利用这些基，我们构造了相应的对数泊松结构，并在Lie-Rinehart代数框架内显式描述了诱导的对数微分1-形式上的Koszul括号。随后，我们确定了对数泊松上链复形，并计算了所考虑类别的上同调。此外，通过对数Spencer复形，我们识别了相应的对数德拉姆复形并计算了其上同调。这些计算为所考虑的非齐次除子类别提供了显式的上同调不变量，并展示了对数泊松与德拉姆理论如何推广约化法交叉除子的相应构造。另外，我们建立了对数上同调理论之间的显式比较：在任意次数$k \ eq 1$时，它们自然同构；而在次数1时，对数德拉姆上同调群是对数泊松上同调群的一个分裂一维扩张。
英文摘要
We study logarithmic Poisson and de Rham cohomologies associated with a class of inhomogeneous free divisors in the affine plane. These divisors arise as inhomogeneous deformations of reduced normal crossing divisors and admit explicit bases for their modules of logarithmic vector fields in the sense of Saito. Using these bases, we construct the corresponding logarithmic Poisson structures and describe explicitly the induced Koszul bracket on logarithmic differential 1-forms within the framework of Lie-Rinehart algebras. We then determine the logarithmic Poisson cochain complex and compute its cohomology for the class under consideration. Furthermore, by means of the logarithmic Spencer complex, we identify the corresponding logarithmic de Rham complex and compute its cohomology. These computations provide explicit cohomological invariants for the class of inhomogeneous divisors considered and show how logarithmic Poisson and de Rham theories extend the corresponding constructions for reduced normal crossing divisors. In addition, we establish an explicit comparison between the logarithmic cohomological theories: they are naturally isomorphic in any degree $k \neq 1$, whereas in degree 1 the logarithmic de Rham cohomology group is a split one-dimensional extension of the logarithmic Poisson cohomology group.
CommentsMoving beyond the normal-crossing setting, we study logarithmic Poisson and logarithmic de Rham cohomologies for a class of inhomogeneous divisors in the affine plane. We compute these cohomologies explicitly and establish a formal deformation framework connecting the inhomogeneous and normal-crossing cases
