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

Abstract

Let $K$ be a field of characteristic zero, $R = K[X_1,\ldots,X_n]$. Let $A_n(K) = K $ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i = ω_i $ and $°\partial_i = -ω_i$ for $i =1,\ldots,n$ (here $ω_i$ are positive integers). Set $ω= \sum_{k=1}^{n}ω_k$. 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).