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Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvexdomain with a $C^2$-smooth boundary. We study the compactness of compositionoperators on the Bergman spaces of smoothly bounded convex domains. We give apartial characterization of compactness of the composition operator (withsufficient regularity of the symbol) in terms of the behavior of the Jacobianon the boundary. We then construct a counterexample to show the converse of thetheorem is false.

Compactness and Singular Points of Composition Operators on Bergman spaces