Algebraic Geometry: Salt Lake City 2015

Cover
Title page
Contents
Preface
Scientific program
Part 1
Wall-crossing implies Brill-Noether: Applications of stability conditions on surfaces
1. Introduction
2. The heart of the matter
3. Geometric stability
4. Moduli spaces of stable objects
5. Brill-Noether and the moduli space of torsion sheaves
6. Hitting the wall
7. Conclusion
8. Geometry of the Brill-Noether locus and birational geometry of the moduli space
9. Birational geometry of moduli spaces of sheaves: a quick survey
References
Kähler–Einstein metrics, canonical random point processes and birational geometry
1. Introduction
2. Outline of the proof of Theorem 1.1 and relations to statistical mechanics
3. Analytic setup
4. The LDP for temperature deformed determinantal point processes
5. Canonical random point processes on varieties of positive Kodaira dimension and log pairs
6. Fano manifolds and Gibbs stability
7. Appendix: Probabilistic background
Acknowledgments
References
Hall algebras and Donaldson-Thomas invariants
1. Introduction
2. Hall algebras
3. Integration map
4. Generalized DT invariants
5. Framed invariants and tilting
References
The Cremona group
1. An introduction based on examples
2. Algebraic subgroups of \Cr_{?}(\bfk)
3. Generating sets and relations
4. An infinite dimensional hyperbolic space
5. The Cremona group is thin
6. Finitely generated subgroups
7. Small cancellation and normal subgroups
8. Zimmer program
9. Growths
References
Mori dream spaces and blow-ups
1. Introduction
2. Mori dream spaces
3. Examples
4. Structure theory
5. Blow-ups of surfaces of Picard number one
6. Blow-ups of weighted projective planes
7. Blow-ups of higher dimensional toric varieties
8. Blow-ups of toric surfaces
References
The space of arcs of an algebraic variety
1. Introduction
2. The space of arcs
3. Arcs through the singular locus
4. Dimension one
5. Dimension two
6. Higher dimensions
7. The Nash problem in the analytic topology
8. The Nash problem in positive characteristics
References
Stability of algebraic varieties and Kähler geometry
1. Background
2. Proofs of cases of the YTD conjecture
3. Kähler-Einstein metrics on Fano manifolds
4. Concluding discussion
References
Syzygies of projective varieties of large degree: Recent progress and open problems
Introduction
1. Non-vanishing for asymptotic syzygies
2. Veronese Varieties
3. Betti numbers
4. Asymptotic ?_{?,1} and the gonality conjecture
References
Stable gauged maps
1. Introduction
2. Scaled curves
3. Mumford stability
4. Kontsevich stability
5. Mundet stability
6. Applications
References
Uniformisation of higher-dimensional minimal varieties
1. Introduction
2. Notation
Part OT1OT1cmrcmrmmnncmrcmrI. Techniques
3. Reflexive differentials
4. Existence of maximally quasi-étale covers
5. Nonabelian Hodge theory
6. Higgs sheaves on singular spaces
Part OT1OT1cmrcmrmmnncmrcmrII. Proof of the main results
7. Characterisation of torus quotients
8. Proof of the Miyaoka-Yau inequality
9. Characterisation of singular ball quotients
References
Boundedness of varieties of log general type
1. Introduction
1.1. Semi log canonical models
1.2. Main Theorems
1.3. Boundedness of canonical models
2. Preliminaries
2.1. Notation and conventions
2.2. Volumes
2.3. Non Kawamata log terminal centres
2.4. Minimal models
2.5. DCC sets
2.6. Good minimal models
2.7. Log birational boundedness
3. Pairs with hyperstandard coefficients
3.1. The DCC for volumes of log birationally bounded pairs
3.2. Adjunction
3.3. DCC of volumes and birational boundedness
4. Birational boundedness: the general case
4.1. Boundedness of the anticanonical volume
4.2. Birational boundedness
4.3. ACC for numerically trivial pairs
4.4. ACC for the log canonical threshold
5. Boundedness
References
Θ-stratifications, Θ-reductive stacks, and applications
1. The Harder-Narasimhan problem
2. Θ-reductive stacks
3. Derived Kirwan surjectivity
4. Applications of derived Kirwan surjectivity
5. Non-abelian virtual localization theorem
References
Bimeromorphic geometry of Kähler threefolds
1. Introduction
2. Brief review of the algebraic case
3. Kähler spaces and the generalised Mori cone \NAX
4. MMP for Kähler threefolds
5. Abundance for Kähler threefolds
6. Applications
7. Outlook
References
Moduli of stable log-varieties—an update
1. Introduction
2. Canonical polarization
3. Demi-normal schemes, slc singularities, and stable varieties
4. Viehweg’s functor versus Kollár’s functor
5. Moduli of stable log-varieties
References
Enumerative geometry and geometric representation theory
1. Introduction
2. Basic concepts
3. Roots and braids
4. Stable envelopes and quantum groups
5. Further directions
References
A calculus for the moduli space of curves
1. Introduction
2. Tautological classes on \M_{?}
3. Faber-Zagier relations on \M_{?}
4. ? classes on \M_{?,?}^{\ccc}
5. Pixton’s relations on \overline{\M}_{?,?}
6. Double ramification cycles
Acknowledgments
References
Frobenius techniques in birational geometry
1. Introduction
2. Setup
3. Basic notions - fundamental results
4. Newer methods - finding sections
5. Applications to higher dimensional algebraic geometry
References
Singular Hermitian metrics and positivity of direct images of pluricanonical bundles
1. Introduction
2. Singular Hermitian metrics on vector bundles
3. Metric properties of direct images
4. Further results
References
Positivity for Hodge modules and geometric applications
1. Introduction
2. Preliminaries on Hodge modules
2.1. Background on \Dmod-modules
2.2. Background on Hodge modules
3. Vanishing and positivity package for Hodge modules
3.1. Vanishing theorems
3.2. Weak positivity
4. Generic vanishing
4.1. Generic vanishing on smooth projective varieties
4.2. Generic vanishing on compact Kähler manifolds
5. Families of varieties
5.1. A Hodge module construction for special families
5.2. Zeros of holomorphic one-forms
5.3. Families of varieties of general type over abelian varieties
5.4. Families of maximal variation and Viehweg-Zuo sheaves
6. Hodge ideals
6.1. Motivation
6.2. Alternative definition and local properties
6.3. Examples
6.4. Vanishing and applications
7. Applications of Hodge modules in other areas
References
Notes on homological projective duality
1. Introduction
2. Projective bundles and blow ups
3. Homological projective duality I
4. Homological projective duality II
5. Examples
References
Non-commutative deformations and Donaldson-Thomas invariants
1. Introduction
2. Non-commutative deformation theory of sheaves
3. Examples from 3-fold flopping contractions
4. Relation to the DT type invariants
5. Global NC structures
References
Nakamaye’s theorem on complex manifolds
1. Introduction
2. Basic results
3. The main theorem
4. Applications
5. Ideas from the proof
6. The problem of effectivity
References
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