Quasi-log varieties

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We extend the Cone Theorem of the Log Minimal Model Program to log varieties with arbitrary singularities.

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Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, methodology): “Thus this completes the proof by f∗OY ′′( A − Y ′) ≃ IX ′ obtained in the proof of (i).”
Log canonical singularities are Du Bois (2009, background): “[ Amb03 , 4.7]) that each f -qlc stratum is normal.”
Letters of a Bi-rationalist. VII Ordered termination (2006, background): “Otherwise by (11) there exists an extremal contraction X → Y/Z negative with respect to KX +BX [ Am , Theorem 2]. We claim that the contraction is birational, does not contract components D of BX with positive multiplicities, and does not touch LCS(X,B), that is, an isomorphism in a neighborhood of ”
Fundamental Theorems for the Log Minimal Model Program (2009, background): “The first half of Theorem 1.1 is the main result of [ A1 ].”
On existence of log minimal models II (2009, result): “1 (the same is true in general for lc pairs by the results of Ambro [1] and Fujino [6], Theorem 1.”
Existence of log canonical flips and a special LMMP (2011, result): “For each i, j there is a rational number bi,j such that KX + Bi ≡ bi,j(KX + Bj)/Y By the cone theorem for lc pairs proved by Ambro [2] and Fujino [14], KX + Bi ∼Q bi,j(KX + Bj)/Y Therefore, there is a morphism X+ → Y which gives the flip of KX + Bi for every i.”