Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Preliminaries
- 2 Canonical and log canonical singularities
- 3 Examples
- 4 Adjunction and residues
- 5 Semi-log canonical pairs
- 6 Du Bois property
- 7 Log centers and depth
- 8 Survey of further results and applications
- 9 Finite equivalence relations
- 10 Ancillary results
- References
- Index
8 - Survey of further results and applications
Published online by Cambridge University Press: 05 July 2013
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Preliminaries
- 2 Canonical and log canonical singularities
- 3 Examples
- 4 Adjunction and residues
- 5 Semi-log canonical pairs
- 6 Du Bois property
- 7 Log centers and depth
- 8 Survey of further results and applications
- 9 Finite equivalence relations
- 10 Ancillary results
- References
- Index
Summary
The aim of this chapter is to give a short summary of further results about the singularities we have been studying, without going into the details of the definitions and results. Some of the topics are covered in recent books or surveys; for these our treatment is especially cursory.
The main topics are
• Linear systems and ideal sheaves.
• Connections with complex analysis.
• Log canonical thresholds.
• The ACC conjecture.
• Arc spaces of lc singularities.
• F-regular and F-pure singularites.
• Differential forms on lc singularities.
• The topology of lc singularities.
• Applications to the abundance conjecture and to moduli problems.
• Unexpected applications of lc pairs.
Assumptions With the exception of Section 8.4, the results in this chapter have been worked out for varieties over fields of characteristic 0. Most of them should hold in more general settings.
Ideal sheaves and plurisubharmonic funtions
In Section 2.1 we defined discrepancies of divisors and related concepts for pairs (X, Δ) where Δ is a ℚ-divisor. It is straightforward and useful to define these notions when divisors are replaced by linear systems or ideal sheaves.
Definition 8.1 (Discrepancies for linear systems) Let X be a normal scheme, ∣Mi∣ linear systems and ci ∈ ℚ. Assume that KX and the Mi are ℚ-Cartier.
Let f: Y → X be a proper morphism with exceptional divisors Ej.
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- Singularities of the Minimal Model Program , pp. 248 - 265Publisher: Cambridge University PressPrint publication year: 2013