Koppelman formulas on affine cones over smooth projective complete intersections

TL;DR

This paper investigates the regularity and compactness of the Koppelman integral operator on affine cones over smooth projective complete intersections, providing new estimates and homotopy formulas for $ar{ ext{d}}$-operators.

Contribution

It establishes $L^p$- and $C^eta$-estimates, compactness results, and homotopy formulas for $ar{ ext{d}}$-operators on these varieties, especially with small degrees.

Findings

Proves $L^p$- and $C^eta$-estimates for the Koppelman operator.

Shows compactness of the operator under certain conditions.

Derives homotopy formulas for $ar{ ext{d}}$-operators on affine cones.

Abstract

In the present paper, we study regularity of the Andersson-Samuelsson Koppelman integral operator on affine cones over smooth projective complete intersections. Particularly, we prove $L^p$- and $C^\alpha$-estimates, and compactness of the operator, when the degree is sufficiently small. As applications, we obtain homotopy formulas for different $\overline{\partial}$-operators acting on $L^p$-spaces of forms, including the case $p=2$ if the varieties have canonical singularities. We also prove that the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are $C^\alpha$ for $\alpha < 1$.