## Classification of Higher Dimensional Algebraic VarietiesHigher Dimensional Algebraic Geometry presents recent advances in the classification of complex projective varieties. Recent results in the minimal model program are discussed, and an introduction to the theory of moduli spaces is presented. |

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### Contents

CHAPTER 1 Introduction | 3 |

CHAPTER 2 Preliminaries | 17 |

CHAPTER 3 Singularities | 27 |

Part II Recent advances in the minimal model program | 46 |

CHAPTER 4 Introduction | 49 |

CHAPTER 5 The main result | 51 |

CHAPTER 6 Multiplier ideal sheaves | 67 |

CHAPTER 7 Finite generation of the restricted algebra | 79 |

Part III Compact moduli spaces of canonically polarized varieties | 103 |

CHAPTER 11 Moduli problems | 105 |

CHAPTER 12 Hilbert schemes | 111 |

CHAPTER 13 The construction of the moduli space | 116 |

CHAPTER 14 Families and moduli functors | 133 |

CHAPTER 15 Singularities of stable varieties | 141 |

CHAPTER 16 Subvarieties of moduli spaces | 148 |

Part IV Solutions and hints to some of the exercises | 172 |

### Common terms and phrases

ˇ ˇ admissible families algebraic ample line bundle ample Q-divisor assume blow-up boundedness canonical bundle canonical models canonical singularities Cartier divisor codimension complex components cone contains no non-kit curves of genus DB singularities deﬁned deﬁnition dimension dimX divisorial contraction dlt pair exceptional divisor EXERCISE exists extremal ray f W X f W Y Fano varieties ﬁber ﬁeld ﬁnite type finitely ﬁrst ﬁxed flipping ﬂips follows Gorenstein hence Hilbert polynomial Hilbert scheme integer irreducible isomorphism Kodaira dimension Lemma Let f log canonical log resolution log terminal minimal model program moduli functor moduli space morphism of normal Msmoothh non-kit centers prime divisor projective morphism projective variety Proof pseudo-effective Q-Cartier Q-factorial quasi-projective varieties rational map rational singularities REMARK resolution of singularities resp scheme semiample Shafarevich’s conjecture sheaf Show simple normal crossings smooth variety strict transform subscheme subvariety surface vanishing theorems Xcan Xmin