Geodesic
Quasi-Log Varieties
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Birational geometry: Linear systems and finitely generated algebras, Tome 240 (2003), pp. 220-239

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We extend the Cone and Contraction Theorems of the Log Minimal Model Program to log varieties with arbitrary singularities. 
@article{TM_2003_240_a6,
     author = {F. Ambro},
     title = {Quasi-Log {Varieties}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {220--239},
     publisher = {mathdoc},
     volume = {240},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2003_240_a6/}
}
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F. Ambro. Quasi-Log Varieties. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Birational geometry: Linear systems and finitely generated algebras, Tome 240 (2003), pp. 220-239. http://geodesic.mathdoc.fr/item/TM_2003_240_a6/