Geometric Arveson-Douglas Conjecture and Holomorphic Extension

TL;DR

This paper proves a weaker form of the Geometric Arveson-Douglas Conjecture for certain smooth complex analytic subsets, establishing essential normality and extending holomorphic functions using complex harmonic analysis techniques.

Contribution

It introduces new methods to demonstrate essential normality for quotient modules associated with smooth analytic subsets intersecting transversally with the boundary.

Findings

Projection operator is in the Toeplitz algebra, implying essential normality.

Achieves $p$-essential normality for $p>2d$, where $d$ is the complex dimension.

Results apply to closures of radical polynomial ideals with specific boundary conditions.

Abstract

In this paper we introduce techniques from complex harmonic analysis to prove a weaker version of the Geometric Arveson-Douglas Conjecture for complex analytic subsets that is smooth on the boundary of the unit ball and intersects transversally with it. In fact, we prove that the projection operator onto the corresponding quotient module is in the Toeplitz algebra $\mathcal{T}(L^{\infty})$, which implies the essential normality of the quotient module. Combining some other techniques we actually obtain the $p$-essential normality for $p>2d$, where $d$ is the complex dimension of the analytic subset. Finally, we show that our results apply for the closure of a radical polynomial ideal $I$ whose zero variety satisfies the above conditions. A key technique is defining a right inverse operator of the restriction map from the unit ball to the analytic subset generalizing the result of Beatrous's paper "$L^p$-estimates for extensions of holomorphic functions".