Geometirc Arveson-Douglas Conjecture - Decomposition of Varieties

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 $partialmathbb{B}_n$, 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 $mathbb{B}_n$.