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2013•Annals of MathematicsOpen access

On the birational automorphisms of varieties of general type

Christopher D. Hacon, James McKernan, Chenyang Xu

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Abstract

We show that the number of birational automorphisms of a variety of general type X is bounded by c • vol(X, KX ), where c is a constant that only depends on the dimension of X. Contents 1. Introduction 1078 1.1.Sketch of the proof of Theorem 1.4 1082 2. Preliminaries 1086 2.1.Notation and conventions 1086 2.2.Log pairs 1088 2.3.The volume 1088 2.4.Bounded pairs 1091 3. Birationally bounded pairs 1092 4. Deformation invariance of log plurigenera 1095 5. DCC for the volume of bounded pairs 1100 6. Birational geometry of global quotients 1106 7. Proof of (1.4) and (1.1) 1108 References 1109

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We show that the number of birational automorphisms of a variety of general type X is bounded by c • vol(X, KX ), where c is a constant that only depends on the dimension of X. Contents 1. Introduction 1078 1.1.Sketch of the proof of Theorem 1.4 1082 2. Preliminaries 1086 2.1.Notation and conventions 1086 2.2.Log pairs 1088 2.3.The volume 1088 2.4.Bounded pairs 1091 3. Birationally bounded pairs 1092 4. Deformation invariance of log plurigenera 1095 5. DCC for the volume of bounded pairs 1100 6. Birational geometry of global quotients 1106 7. Proof of (1.4) and (1.1) 1108 References 1109

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Available abstract

We show that the number of birational automorphisms of a variety of general type X is bounded by c • vol(X, KX ), where c is a constant that only depends on the dimension of X. Contents 1. Introduction 1078 1.1.Sketch of the proof of Theorem 1.4 1082 2. Preliminaries 1086 2.1.Notation and conventions 1086 2.2.Log pairs 1088 2.3.The volume 1088 2.4.Bounded pairs 1091 3. Birationally bounded pairs 1092 4. Deformation invariance of log plurigenera 1095 5. DCC for the volume of bounded pairs 1100 6. Birational geometry of global quotients 1106 7. Proof of (1.4) and (1.1) 1108 References 1109

Key concepts: Mathematics, Automorphism, Birational geometry, Dimension (graph theory), Type (biology), Pure mathematics, Bounded function, Variety (cybernetics)

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