# Geometirc Arveson-Douglas Conjecture - Decomposition of Varieties

## TL;DR

This paper proves the Geometric Arveson-Douglas Conjecture for certain varieties with singularities by decomposing them into well-behaved parts and applying localization techniques, extending previous results to more complex cases.

## Contribution

It establishes the conjecture for varieties decomposable into parts with nice boundary interactions, even with some boundary singularities, using localization and linear subspace intersections.

## Key findings

- Proved the conjecture for a special class of varieties with boundary singularities.
- Applied localization techniques to extend previous results.
- Reduced the problem to intersections of linear subspaces with the unit ball.

## Abstract

In this paper, we prove the Geometric Arveson-Douglas Conjecture for a special case which allow some singularity on $\partial{\mathbb{B}_n}$. More precisely, we show that if a variety can be decomposed into two varieties, each having nice properties and intersecting nicely with $\partial\mathbb{B}_n$, then the Geometric Arveson-Douglas Conjecture holds on this variety. We obtain this result by applying a result by Su\'arez, which allows us to "localize" the problem. Our result then follows from the simple case when the two varieties are intersection of linear subspaces with $\mathbb{B}_n$.