Distinguished varieties in the polydisc and dilation of commuting contractions

A distinguished variety in the polydisc Dn is an affine complex algebraic variety that intersects Dn and exits the domain through the n-torus Tn without intersecting any other part of the topological boundary of Dn. We find two different characterizations for a distinguished variety in the polydisc Dn in terms of the Taylor joint spectrum of certain linear matrix-pencils and thus generalize the seminal work due to Agler and McCarthy [Acta Math., 2005] on distinguished varieties in D2. We show that a distinguished variety in Dn is a part of an affine algebraic curve which is a set-theoretic complete intersection. We also show that if (T1,⋯,Tn) is commuting tuple of Hilbert space contractions such that the defect space of T=∏i=1nTi is finite dimensional, then (T1,⋯,Tn) admits a commuting unitary dilation (U1,⋯,Un) with U=∏i=1nUi being the minimal unitary dilation of T if and only if some certain matrices associated with (T1,⋯,Tn) define a distinguished variety in Dn.