Lemma 9 ([Lifting and separating a birational action]). Let (V,B_V) be as in Lemma 8, of dimension n>0 . Choose a product cover P=∏_i∈I P_i→V as in Input, and let f be a B-birational self-map with f^*θ=cθ . There are normal integral …

A cover admitting a lift. Put U=V_ sm . The connected \'etale cover P_U→U corresponds to a finite-index subgroup of π_1^ top (U) . This fundamental group is finitely generated, so it has only finitely many subgroups of any fixed index. Their intersection is a finite-index characteristic subgroup contained in the subgroup defining P_U . Riemann existence gives its connected algebraic \'etale cover. Normalize V in its function field to obtain a finite quasi-\'etale cover R→V factoring through P . The pseudo-automorphism f identifies two open subsets of U with complements of codimension at least two. Removing such subsets does not change the fundamental group of a complex manifold, by transversality. Characteristicity therefore lifts this isomorphism to the chosen cover. Riemann existence makes the lift algebraic, and it induces a pseudo-automorphism g of R . If H_R and θ_R denote the pulled-back boundary and form, then g^*θ_R=cθ_R, and g preserves H_R in codimension one. No degree bound for R→V is required. Preserving the factor directions. Over the product of the smooth loci of the P_i , the map R→P is \'etale. The tangent directions of the factors give a splitting there, whose reflexive extension is T_R=_ i∈I E_i. Let T_R(- H_R) be the reflexive sheaf of derivations preserving the reduced ideal of every boundary component, a condition tested at codimension-one points. Consider the finite-dimensional -algebra A= u∈H^0(R,( T_R)): u( T_R(- H_R)) ⊆ T_R(- H_R) . Conjugation by g acts on A : both the tangent sheaf and the logarithmic condition are determined on a big open set. Every member of A has zero entries between different E_i . To see this, fix general smooth points in the complementary factors and slice in one factor direction. Each connected component over that factor's smooth locus extends by normalization to a connected normal finite quasi-\'etale cover of the factor. The omitted set has codimension at least two, since a divisorial point of a finite cover lies over a divisorial, hence smooth, point of the normal factor. The ambient cover is \'etale on this open slice, so restricted entries are regular and extend reflexively across its complement. On the slice its tangent directions are E_i , while complementary summands are trivial. An entry in either direction involving a nonabelian boundary-free factor therefore gives reflexive one-forms or vector fields on its cover, and these vanish by Input. For the only remaining case, an abelian and a rationally connected factor, slice in the rationally connected direction. An entry towards the abelian summand gives one-forms; an entry in the reverse direction gives vector fields tangent to the boundary: that boundary is pulled back from the rationally connected factor, so the ideal-preservation condition restricts at every divisorial point of the slice. These vanish as well. The slices cover a dense open of R , proving the assertion. Consequently every block projection is a central idempotent of A . The primitive central idempotents give nonzero direct summands of the rank- n tangent sheaf, so there are at most n of them. Conjugation by g permutes these idempotents. Therefore h=g^ n! fixes every primitive central idempotent. Every central idempotent is a sum of primitive central idempotents, so h fixes every block projection and preserves each factor direction. Recovering the factor fields. Let K_i be the relative algebraic closure of (P_i) in (R) , and let R_i be the normalization of P_i in K_i . Equivalently, R→R_i→P_i is the Stein factorization of the projection. At the generic point of R , the directions complementary to i span Der_ (P_i) (R) . In characteristic zero their common field of constants is exactly K_i . Preservation of these directions thus gives a birational self-map h_i of R_i . The map ψ:R→∏_iR_i is proper and has finite fibres, because its composite to P is finite. It is finite and, by dimension, surjective. The maps R_i→P_i are quasi-\'etale. Indeed, since (P)/(P_i) is a regular extension, the product R_i×∏_ j≠i P_j is an intermediate normal finite cover of P . Ramification over a prime of P_i would persist in this product and then in a prolongation to R , contradicting quasi-\'etaleness of R→P . The same ramification calculation in the tower proves that ψ is quasi-\'etale. In particular the pulled-back pairs (R_i,H_i) are klt. The relation between h and the h_i is rational equivariance of ψ ; no regularity assertion about the h_i is needed. Comparing the forms. On each boundary-free R_i , choose a generator θ_i of its trivial canonical sheaf. For the rationally connected factor, restrict the degree- m trivialization pulled back from V to that factor of P , using smooth complementary points and the external product of canonical lines. The resulting form has divisor -mH on that factor; pull it back to R_i . This constructs the forms of degree p_i in the statement. Their product, in any fixed order, satisfies θ_R=a\,ψ^*(_iθ_i^ ⊗m/p_i ) (a∈^*). Here the wedge denotes external product of pluricanonical lines. Indeed both sides have divisor -mH_R , so their ratio is a global unit on the normal projective integral variety R . Write u_i=h_i^*θ_i/θ_i∈(R_i)^* . Pulling back by h shows that the pullback of ∏_i u_i^ m/p_i is the constant c^ n! . Since ψ is dominant, the product is constant on ∏_iR_i . Fixing general points in all complementary factors shows that every u_i^ m/p_i , and hence every u_i , is constant. Denote it by d_i . This proves; proportionality of the forms proves that each h_i is B-birational.