de Rham cohomology of $H^1_{(f)}(R)$ where $V(f)$ is a smooth hypersurface in $\mathbb{P}^n$

TL;DR

This paper computes the de Rham cohomology modules of certain local cohomology modules associated with smooth hypersurfaces in projective space, revealing detailed algebraic structures in a graded Weyl algebra setting.

Contribution

It provides explicit calculations of de Rham cohomology for local cohomology modules of smooth hypersurfaces, extending understanding of their algebraic and geometric properties.

Findings

Computed de Rham cohomology modules for smooth hypersurfaces

Established holonomicity of local cohomology modules in graded Weyl algebra

Analyzed the structure of local cohomology in the context of isolated singularities

Abstract

Let $K$ be a field of characteristic zero, $R = K[X_1,\ldots,X_n]$. Let $A_n(K) = K<X_1,\ldots,X_n, \partial_1, \ldots, \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $\deg X_i = \omega_i $ and $\deg \partial_i = -\omega_i$ for $i =1,\ldots,n$ (here $\omega_i$ are positive integers). Set $\omega = \sum_{k=1}^{n}\omega_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we compute the de Rham cohomology modules $H^j(\mathbb{\partial}; H^1_{(f)}(R))$ for $j \leq n-2$ when $V(f)$ is a smooth hypersurface in $\mathbb{P}^n$ (equivalently $A = R/(f)$ is an isolated singularity).