Geometric Arveson–Douglas Conjecture-Decomposition of Varieties

In this paper, we prove the Geometric Arveson–Douglas Conjecture for a special case that allows 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 Suarez, 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 \(\mathbb {B}_n\).