Generalized Zariski cancellation for Brieskorn--Pham varieties

We establish a generalized Zariski cancellation theorem for Brieskorn–Pham varieties over the field of complex numbers. More precisely, we show that if two complex Brieskorn–Pham varieties become isomorphic after taking a product with an arbitrary separated complex scheme having a smooth point, then they are already isomorphic not merely as complex algebraic varieties but, in fact, as ${\mathbf{C}}^{\ast }$-varieties. The proof combines our general cancellation theorem for complex algebraic varieties with a unique singularity, whose proof relies on the analytic cancellation theorem of Hauser–Müller, with an exponent rigidity theorem for Brieskorn–Pham varieties. The latter asserts that, over any field of characteristic zero, the exponent tuple appearing in the defining equation completely determines the isomorphism class of the corresponding Brieskorn–Pham variety.