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Singularities of the Minimal Model Program

Book Description

This book gives a comprehensive treatment of the singularities that appear in the minimal model program and in the moduli problem for varieties. The study of these singularities and the development of Mori's program have been deeply intertwined. Early work on minimal models relied on detailed study of terminal and canonical singularities but many later results on log terminal singularities were obtained as consequences of the minimal model program. Recent work on the abundance conjecture and on moduli of varieties of general type relies on subtle properties of log canonical singularities and conversely, the sharpest theorems about these singularities use newly developed special cases of the abundance problem. This book untangles these interwoven threads, presenting a self-contained and complete theory of these singularities, including many previously unpublished results.

Table of Contents

  1. Cover
  2. Cambridge Tracts in Mathematics
  3. Cambridge Tracts in Mathematics
  4. Title Page
  5. Copyright
  6. Table of Contents
  7. Preface
  8. Introduction
  9. 1 Preliminaries
    1. 1.1 Notation and conventions
      1. Divisors and ℚ-divisors
      2. Pairs
      3. Normal crossing conditions
      4. Resolution of singularities
    2. 1.2 Minimal and canonical models
    3. 1.3 Canonical models of pairs
      1. Existence of minimal and canonical models
    4. 1.4 Canonical models as partial resolutions
      1. Canonical and log canonical modifications
      2. Cornucopia of dlt models
      3. Non-normal cases
    5. 1.5 Some special singularities
  10. 2 Canonical and log canonical singularities
    1. 2.1 (Log) canonical and (log) terminal singularities
      1. Crepant morphisms and maps
    2. 2.2 Log canonical surface singularities
      1. Terminal and canonical pairs
      2. The reduced boundary
      3. The different
    3. 2.3 Ramified covers
    4. 2.4 Log terminal 3-fold singularities
      1. Canonical singularities with KX Cartier
      2. Group actions on canonical singularities
    5. 2.5 Rational pairs
      1. CM sheaves
      2. CM criteria and the ω -dual
      3. Vanishing theorems
      4. Rational pairs
      5. Rational pairs in characteristic zero
  11. 3 Examples
    1. 3.1 First examples: cones
      1. Auxiliary results on cones
    2. 3.2 Quotient singularities
    3. 3.3 Classification of log canonical surface singularities
      1. First examples of log canonical singularities
      2. Plan of the classification
      3. Cyclic quotients
      4. One fork case
      5. List of log canonical surface singularities
      6. Twisted quotients
      7. Boundary with coefficients ≥ 1/2
    4. 3.4 More examples
      1. Singularities with prescribed exceptional divisors
      2. Construction of log canonical 3-fold singularities
      3. Construction of terminal 4-fold singularities
    5. 3.5 Perturbations and deformations
      1. Perturbing the boundary
      2. Non-lc deformations
  12. 4 Adjunction and residues
    1. 4.1 Adjunction for divisors
    2. 4.2 Log canonical centers on dlt pairs
      1. Log canonical centers in the dlt case
    3. 4.3 Log canonical centers on lc pairs
    4. 4.4 Crepant log structures
      1. Connectedness theorems
      2. ℙ1-linked lc centers
    5. 4.5 Sources and springs of log canonical centers
  13. 5 Semi-log canonical pairs
    1. 5.1 Demi-normal schemes
    2. 5.2 Statement of the main theorems
    3. 5.3 Semi-log canonical surfaces
    4. 5.4 Semi-divisorial log terminal pairs
    5. 5.5 Log canonical stratifications
    6. 5.6 Gluing relations and sources
    7. 5.7 Descending the canonical bundle
  14. 6 Du Bois property
    1. 6.1 Du Bois singularities
      1. The Deligne-Du Bois complex of a pair
      2. Vanishing of the top cohomology
      3. Cohomology with compact support
      4. DB pairs and the DB defect
      5. General hyperplane cuts and taking roots
      6. A DB criterion
    2. 6.2 Semi-log canonical singularities are Du Bois
  15. 7 Log centers and depth
    1. 7.1 Log centers
    2. 7.2 Minimal log discrepancy functions
    3. 7.3 Depth of sheaves on slc pairs
  16. 8 Survey of further results and applications
    1. 8.1 Ideal sheaves and plurisubharmonic funtions
      1. Connections with complex analysis
    2. 8.2 Log canonical thresholds and the ACC conjecture
    3. 8.3 Arc spaces of log canonical singularities
    4. 8.4 F-regular and F-pure singularites
    5. 8.5 Differential forms on log canonical pairs
    6. 8.6 The topology of log canonical singularities
    7. 8.7 Abundance conjecture
    8. 8.8 Moduli spaces for varieties
    9. 8.9 Applications of log canonical pairs
  17. 9 Finite equivalence relations
    1. 9.1 Quotients by finite equivalence relations
      1. Finite equivalence relations
      2. Basic properties
      3. Stratified equivalence relations
      4. General quotient theorems
      5. Almost group actions
    2. 9.2 Descending seminormality of subschemes
    3. 9.3 Descending line bundles to geometric quotients
      1. Seifert bundles
    4. 9.4 Pro-finite equivalence relations
  18. 10 Ancillary results
    1. 10.1 Birational maps of 2-dimensional schemes
      1. Rational singularities of surfaces
    2. 10.2 Seminormality
      1. Basic definitions
      2. Seminormal reducible schemes
      3. Seminormality of subschemes
    3. 10.3 Vanishing theorems
    4. 10.4 Semi-log resolutions
    5. 10.5 Pluricanonical representations
    6. 10.6 Cubic hyperresolutions
  19. References
  20. Index