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

Abstract

Let $K$ be a field of characteristic zero, $R=K[X_1,\dots ,X_n]$. Let $A_n(K)=K<X_1,\dots ,X_n,{\partial }_1,\dots ,{\partial }_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i={\omega }_i$ and $°{\partial }_i=-{\omega }_i$ for $i=1,\dots ,n$ (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 $i\ge 0$. In this article we compute the de Rham cohomology modules $H^j(\partial ;H_{(f)}^1(R))$ for $j\le n-2$ when $V(f)$ is a smooth hypersurface in ${\mathbb{P}}^n$ (equivalently $A=R/(f)$ is an isolated singularity).

arXiv admin note: text overlap with arXiv:1307.2396