Subelliptic estimates for the $\bar{\partial}$-problem on complex algebraic surfaces with isolated singularities

TL;DR

This paper establishes subelliptic estimates for the $ar{ ext{d}}$-problem on complex algebraic surfaces with isolated singularities, linking Sobolev norms to weighted $L^2$ norms near singularities.

Contribution

It introduces new subelliptic estimates for the $ar{ ext{d}}$-problem on singular surfaces using weighted norms that account for singularities.

Findings

Sobolev norms estimated via weighted $L^2$ norms of $ar{ ext{d}}f$ and $ar{ ext{d}}^{ ext{*}}f$

Weighted norms vanish or blow up at singularities, capturing their influence

Results applicable to complex algebraic surfaces with isolated singularities

Abstract

We obtain subelliptic estimates for the $\bar{\partial}$-problem on complex algebraic surfaces embedded in $\mathbb{C}^n$ with isolated singularities. $W^{\epsilon}$ Sobolev norms of a form, $f$, for $0< \epsilon < 1$ are estimated in terms of weighted $L^2$ norms of $\bar{\partial} f$ and $\bar{\partial}^{\ast}f$, with weights which vanish at the singularities, as well as weighted $L^2$ norms of $f$, with weights which blow up at the singularities.