Contents Preface vii Part 1. Liouville Manifolds 1 Introduction 1 Preliminary remarks and notations 6 1. Local Structure of Proper Liouville Manifolds 7 1.1. Liouville manifolds and the properness 7 1.2. Infinitesimal structure at a point in Ms 9 1.3. Local structure around a point in Ms 14 1.4. Proof of Lemma 1.2.7 23 2. Global Structure of Proper Liouville Manifolds 26 2.1. Submanifolds J 26 2.2. Admissible submanifolds 33 2.3. The core of a proper Liouville manifold 42 3. Proper Liouville Manifolds of Rank One 45 3.1. Configuration of zeros and type of cores 45 3.2. Possible cores 49 3.3. Constructing a Liouville manifold from a possible core 52 3.4. Classification 57 3.5. Isomorphisms and isometries 63 3.6. C27r-nietrics 66 Appendix. Simply Connected Manifolds of Constant Curvature 70 A.l. Possible cores 70 A.2. The sphere Sn 71 A.3. The euclidean space Rn 74 A.4. The hyperbolic space Hn 76 Part 2. Kahler-Liouville Manifolds 80 Introduction 80 Preliminary remarks and notations 83 1. Local calculus on M 1 83 2. Summing up the local data 95 3. Structure of M - M 1 96 4. Torus action and the invariant hypersurfaces 106 5. Properties as a toric variety 117 6. Bundle structure associated with a subset of A 126 7. The case where #A = 1 133 8. Existence theorem 139 References 142

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