## Positivity in algebraic geometry 2This two volume work on "Positivity in Algebraic Geometry" contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Whereas Volume I is more elementary, the present Volume II is more at the research level and somewhat more specialized. Both volumes are also available as hardcover edition as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete". |

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### Contents

Ample and Nef Vector Bundles | 7 |

61A Definition and First Properties | 8 |

61B Cohomological Properties | 11 |

61C Criteria for Amplitude | 15 |

61D Metric Approaches to Positivity of Vector Bundles | 18 |

62 QTwisted and Nef Bundles | 20 |

62B Nef Bundles | 24 |

63 Examples and Constructions | 27 |

93C Monomial Ideals | 170 |

93D Analytic Construction of Multiplier Ideals | 176 |

93E Adjoint Ideals | 179 |

93F Multiplier and Jacobian Ideals | 181 |

93G Multiplier Ideals on Singular Varieties | 182 |

94 Vanishing Theorems for Multiplier Ideals | 185 |

94A Local Vanishing for Multiplier Ideals | 186 |

94B The Nadel Vanishing Theorem | 188 |

63B Ample Cotangent Bundles and Hyperbolicity | 36 |

63C Picard Bundles | 44 |

63D The Bundle Associated to a Branched Covering | 47 |

63E Direct Images of Canonical Bundles | 51 |

63F Some Constructions of Positive Vector Bundles | 53 |

64 Ample Vector Bundles on Curves | 56 |

64A Review of Semistability | 57 |

64B Semistability and Amplitude | 60 |

Notes | 64 |

Geometric Properties of Ample Bundles | 65 |

71B Theorem of Bloch and Gieseker | 68 |

71C A BarthType Theorem for Branched Coverings | 71 |

72 Degeneracy Loci | 74 |

72B Proof of Connectedness of Degeneracy Loci | 78 |

72C Some Applications | 82 |

72D Variants and Extensions | 87 |

73 Vanishing Theorems | 89 |

73B Generalizations | 95 |

Notes | 98 |

Numerical Properties of Ample Bundles | 101 |

81A Chern Classes for QTwisted Bundles | 102 |

81B Cone Classes | 104 |

81C Cone Classes for QTwists | 110 |

82 Positivity Theorems | 111 |

82B Positivity of Cone Classes | 114 |

83 Positive Polynomials for Ample Bundles | 117 |

84 Some Applications | 125 |

84B NonEmptiness of Degeneracy Loci | 127 |

84C Singularities of Hypersurfaces Along a Curve | 129 |

Notes | 132 |

Introduction to Part Three | 135 |

Multiplier Ideal Sheaves | 139 |

91 Preliminaries | 140 |

91B Normal Crossing Divisors and Log Resolutions | 142 |

91C The KawamataViehweg Vanishing Theorem | 147 |

92 Definition and First Properties | 151 |

92A Definition of Multiplier Ideals | 152 |

92B First Properties | 158 |

93 Examples and Complements | 162 |

93B Invariants Arising from Multiplier Ideals | 165 |

94C Vanishing on Singular Varieties | 191 |

94D Nadels Theorem in the Analytic Setting | 192 |

94E NonVanishing and Global Generation | 193 |

95 Geometric Properties of Multiplier Ideals | 195 |

95B Subadditivity | 201 |

95C The Summation Theorem | 204 |

95D Multiplier Ideals in Families | 210 |

95E Coverings | 213 |

96 Skodas Theorem | 216 |

Statements | 221 |

Proofs | 226 |

96D Variants | 228 |

Notes | 230 |

Some Applications of Multiplier Ideals | 233 |

101B Singularities of Theta Divisors | 235 |

101 C A Criterion for Separation of Jets of Adjoint Series | 238 |

102 Matsusakas Theorem | 239 |

103 Nakamayes Theorem on Base Loci | 246 |

104 Global Generation of Adjoint Linear Series | 251 |

104A Fujitas Conjecture and AngehrnSiu Theorem | 252 |

104B Loci of LogCanonical Singularities | 254 |

104C Proof of the Theorem of Angehrn and Siu | 258 |

105 The Effective Nullstellensatz | 262 |

Notes | 267 |

Asymptotic Constructions | 269 |

111 Construction of Asymptotic Multiplier Ideals | 270 |

111B Graded Systems of Ideals and Linear Series | 276 |

112 Properties of Asymptotic Multiplier Ideals | 282 |

112B Global Results | 285 |

112C Multiplicativity of Plurigenera | 292 |

113 Growth of Graded Families and Symbolic Powers | 293 |

114 Fujitas Approximation Theorem | 299 |

114B Proof of Fujitas Theorem | 305 |

114C The Dual of the Pseudoeffective Cone | 307 |

115 Sius Theorem on Plurigenera | 312 |

Notes | 320 |

323 | |

Glossary of Notation | 357 |

363 | |

### Other editions - View all

Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and ... R.K. Lazarsfeld No preview available - 2004 |

Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and ... R.K. Lazarsfeld No preview available - 2004 |