arXiv subjects

Kamtila Kari

Publications and source records attributed to Kamtila Kari.

3 recordsLinked to original sources

arXiv2026

On Logarithmic Poisson and De Rham Cohomology Groups of a Class of Inhomogeneous Divisors

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.

math.AG↗

arXiv2026

On logarithmic Poisson cohomology of a degenerate Poisson bivector in affine plane

In this paper, we show that for a given degenerate bivector $π= y^n\partial_x \wedge \partial_y$ with $n>1$, the classical Poisson cohomology group and the logarithmic Poisson cohomology group along the ideal $\mathcal{I}=y^n\mathbb{F}[x,y] $ are isomorphics in every dégrée. This result follows from determination of the logarithmic Hamiltonian operator and the logarithmic Poisson cochain complexe in order to compute the cohomological invariants associated to $π$. $\mathbb{F}$ is the field of characteristic 0.

math.AG↗

arXiv2026

On examples of duals Saito's basis of some inhomogeneous divisors, and application

We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for the module of logarithmic vector fields $Der(logD)$. This construction is then applied to the setting of logarithmic Poisson geometry. Focusing on the example defined by $h=xy+x^{2}y^{2}+x^3y^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y], \{-,-\}_{h})$, where the Poisson bracket is induced by the bivector $π= h\partial x\wedge\partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $π$ endows the sheaf of logarithmic 1-forms $Ω^{1}(log D )$ with a Lie-Rinehart algebra structure. Furthermore, we introduce and provide explicit descriptions for the resulting cohomology theory, which we term the logarithmic Poisson cohomology $H_{log}^{\bullet} $ of $\{-,-\}_{h}$. As a related and foundational computation, we also calculate the corresponding logarithmic De Rham cohomology $H^{\bullet}_{DR}$ for the divisor $D$ and we make a generalization in dimension 2.

math.DG↗