BRST Anomaly and Superspace Constraints of the Pure Spinor Heterotic String in a Curved Background

The pure spinor heterotic string in a generic super Yang-Mills and supergravity background is considered. We determine the one-loop BRST anomaly at the cohomological level. We prove that it can be absorbed by consistent corrections of the classical constraints due to Berkovits and Howe, in agreement with the Green-Schwarz cancelation mechanism.