Finite generation of the restricted algebra

Abstract

Theorem 7.1. Assume Theorem 5.60 in dimension ≤ n - 1. Let π: X → Z be a projective morphism from a smooth quasi-projective variety of dimension n to a normal affine variety. Let Δ S + A + B be a ℚ-divisor such that S = [Δ] is irreducible and smooth, A ≥ 0 is a general ample ℚ-divisor, B ≥ 0, (X,Δ) is plt, (S, Ω +A_S is canonical where Ω = (A + B)|_S, and B(K _X + Δ) does not contain S. For any sufficiently divisible integer m > 0 let

$$ F_m = Fix(|m(K_X + \Delta )|s)/m and F = \lim F_{m!} . $$

$$ Then \Theta = \Omega - \Omega \wedge F is rational and if k \Delta and k \Theta are Cartier, then $$

$$ R_S (k(K_X + \Delta )) \simeq R(k(K_S + \Theta )). $$