Birational Geometry, Kähler–Einstein Metrics and Degenerations: Moscow, Shanghai and Pohang, April–November 2019

Preface
Contents
Classification of Exceptional Complements: Elliptic Curve Case
	1 Introduction
		1.1 The Dual Graph of a Configuration
		1.2 Types of Singularities on C
	2 Elliptic Curve Case
		2.1 Reduction to mathbbF2
		2.2 The Search for Exceptions
	References
Cylinders in Del Pezzo Surfaces of Degree Two
	1 Introduction
	2 Del Pezzo Surfaces with Singularities of Type A1
	3 Smooth Del Pezzo Surfaces
		3.1 P is an Intersection Point of Two (-1)-Curves
		3.2 P is Not an Intersection Point of Two (-1)-Curves
	References
Finiteness of Real Structures on KLT Calabi–Yau Regular Smooth Pairs of Dimension 2
	1 Introduction
	2 Preliminaries: Geometric Actions on CAT(0) Spaces
	3 Finiteness Theorem
	4 Two Examples
	5  Appendix
	References
Anticanonical Volumes of Fano 4-Folds
	1 Introduction
	2 Proof of Results
	References
Constant Scalar Curvature Sasaki Metrics and Projective Bundles
	1 The Projective Bundles
	2 The Orbifold Boothby-Wang Construction
	3 The Topology of the Orbifolds
		3.1 The Orbifold Cohomology Groups
		3.2 The Cohomology of the Branched Covers
		3.3 The Regular Case
		3.4 The General Case; Branched Covers
		3.5 When M7 is an S3w-Join
	4 Admissible Projective Bundles
		4.1 Orbifolds
		4.2 Connection with S3w-Joins
		4.3 Connection with Yamazaki's Fiber Joins
		4.4 Ricci Solitons and Kähler-Einstein
		4.5 Extremal and CSC Kähler Metrics
		4.6 Weighted Extremal Metrics
	References
Toric Sarkisov Links of Toric Fano Varieties
	1 Introduction
	2 The Context
		2.1 Overview of the Sarkisov Program
		2.2 Hard Fano Varieties from Easy Ones
		2.3 The Midpoints of Sarkisov Links
	3 Operations on a Simplicial Fan
	4 Extending Prokhorov's Web
		4.1 Links from a Smooth Point
		4.2 Links from the Eighth Toric Mori–Fano 3-Fold
		4.3 Links from 1-Dimensional Centres
		4.4 Bounding Links from a Smooth Centre
		4.5 Higher Rank Fano 3-Folds, Links and Relations
	5 The Web of 4-Dimensional Toric Sarkisov Links
		5.1 Blowups of mathbbP(1,2,3,4,5)
		5.2 Links Between Fake Weighted Projective Spaces
		5.3 Blowups of mathbbP4
	6 Further Related Problems
		6.1 High-Discrepancy Blowups
		6.2 Non-terminal Singularities
		6.3 Running the Sarkisov Program on Toric n-Folds
		6.4 Higher Picard Rank
		6.5 Fano 3-Folds and 4-Folds More Generally
	References
Du Val Singularities
	1 Introduction
	2 Finite Subgroups of SL2(mathbbC)
	3 Description of Du Val Singularities
	4 A1-Singularity
	5 E6-Singularity
	6 Two-Dimensional McKay Correspondence
	References
K-Polystability of Two Smooth Fano Threefolds
	1 Introduction
	2 Smooth Divisor in mathbbP1timesmathbbP1timesmathbbP2 of Degree (1,1,1)
	3 Blow up of a Complete Intersection of Two Quadrics in a Conic
	References
A Note on Families of K-Semistable Log-Fano Pairs
	1 Introduction
	2 Basis Type Divisors, Delta Invariants and K-Stability
	3 Harder–Narasimhan Filtration and Lift of Basis Type Divisors
	4 Nefness Threshold
	5 Semipositivity
	6 Bound on the Multiplicity of the Fibers
	References
The Yau–Tian–Donaldson Conjecture for Cohomogeneity One Manifolds
	1 Introduction
	2 Recollections
		2.1 Cohomogeneity One Manifolds and Spherical Varieties
		2.2 On Rank One Spherical Varieties
		2.3 On Uniform K-Stability
		2.4 On K-Stability of Spherical Varieties
	3 Uniform K-Stability of Rank One Spherical Varieties
	4 Examples
		4.1 An Example of Kähler Class with No Extremal Kähler Metrics
		4.2 Strong Calabi Dream Manifolds of Cohomogeneity One, and an Answer to a Question of Kanemitsu
	References
Fibrations by Affine Lines on Rational Affine Surfaces with Irreducible Boundaries
	1 Introduction
	2 Preliminaries
		2.1 Notations and Basic Definitions
		2.2 Models of Smooth Affine Surfaces with Irreducible Rational Boundaries
	3 Families of mathbbA1-Fibrations of Complete Type
	4 Equivalence Classes of mathbbA1-Fibrations of Affine Type
		4.1 Special Pencils of Rational Curves and Associated mathbbA1-Fibrations of Affine Type
		4.2 Some Classes of mathbbA1-Fibrations of Affine Type on Surfaces mathbbFnB
		4.3 Proof of Theorem2
	References
On Fano Threefolds of Degree 22 After Cheltsov and Shramov
	1 Introduction
	2 Log Fano Pairs
	3 On the Volume Functions
	4 On Quasi-Log Schemes
	5 Proof of Theorem1.2
	References
Lagrangian Skeleta, Collars and Duality
	1 Skeleton to Collar Duality
	2 Lagrangian Skeleton of T*mathbbPn
	3 Potentials on the Cotangent Bundle
	4 Birational Maps within the Skeleton
	5 Duality for Multiplicity n Singularities
	6 Vector Bundles on Local Surfaces
	7 Deformations
	References
Quot-Scheme Limit of Fubini–Study Metrics and Its Applications to Balanced Metrics
	1 Introduction
	2 Summary of the Methods Developed in ch14HK1,ch14HK2
		2.1 Fubini–Study Metrics
		2.2 Quot-Scheme Limit of Fubini–Study Metrics
		2.3 The Non-Archimedean Donaldson Functional
		2.4 Summary of Results in ch14HK1,ch14HK2
	3 Comparison to the Case of the Yau–Tian–Donaldson Conjecture
		3.1 Dictionary Between Vector Bundles and Manifolds
		3.2 Comments on the Deligne Pairing
	4 Bergman 1-Parameter Subgroups as Subgeodesics
	5 Gieseker Stability and Balanced Metrics
		5.1 Variational Formulation of the Problem
		5.2 Main Result and Proof
	6 Towards Effective Results and an Algorithm for Computing Hermitian–Einstein Metrics
	References
Existence of Canonical Models for Kawamata Log Terminal Pairs
	1 Introduction
	2 Preliminaries
		2.1 Notations and Definitions
		2.2 Iitaka Dimension and Iitaka Fibration
		2.3 Klt-trivial Fibrations
	3 Existence of Canonical Models
	References
Generalized Thomas–Yau Uniqueness Theorems
	1 Introduction
	2 Geometric Gradings
	3 Maslov Forms
	4 HF* Nonvanishing Theorems
	5 Hamiltonian Perturbation Theorem
	6 Conclusions
	References
Birationally Rigid Complete Intersections of Codimension Three
	1 Statement of the Main Result
	2 Multi-quadratic Singularities
	3 The Regularity Conditions
	4 Birational Superrigidity
	5 Codimension of the Complement
	6 Concluding Remarks
	References
Characterizing Terminal Fano Threefolds with the Smallest Anti-canonical Volume
	1 Introduction
	2 Reid's Riemann–Roch Formula
	3 Proofs
	References
Rationality of Quotients by Finite Heisenberg Groups
	1 Introduction
	2 Proof of Theorem 1.2
	3 Miscellany
	References
Interpretations of Spectra
	1 Introduction
	2 Noncommutative Spectra
		2.1 Definitions of Quantum and Nc Spectra
		2.2 Dimension Theory
		2.3 Some Computational Tools
		2.4 New Nonrationality Results
	3 Application to Arithmetics
	4 Low Dimensional Topology Invariants
		4.1 Spectra and WRT
		4.2 Topological Invariants of Plane Curve Singularity
	5 Generalization of Spectra
		5.1 Splitting of a Potential
		5.2 Category Filtrations
	6 Spectrum, Orbifoldization and Conformal Field Theory
	References
On Singular Del Pezzo Hypersurfaces of Index 3
	1 Introduction
	2 Preliminaries
		2.1 α-Invariant of Tian
		2.2 δ-Invariant
		2.3 β-Invariant
	3 Small α-Invariants
	4 K-stable Singular del Pezzo Hypersurfaces
	5 Non K-semistable Singular del Pezzo Hypersurfaces
	6 Table
	References
Blow-ups of Three-dimensional Toric Singularities
	1 Preliminary Results and Facts
	2 Toric Blow-ups
	3 Three-dimensional Blow-ups. Case of Curve
	4 Toric Log Surfaces
	5 Non-toric Three-dimensional Blow-ups. Case of Point
	6 Main Theorems. Case of Point
	References
Automorphisms of Hyperkähler Manifolds and Groups Acting on CAT(0) Spaces
	1 Introduction
	2 Preliminaries
		2.1 Hyperkähler Manifolds
		2.2 The Kawamata–Morrison Conjecture
	3 The CAT(0) Space
		3.1 Hyperbolic Space and its Isometries
		3.2 Construction of a CAT(0) Space
		3.3 Explanation of Step 1
		3.4 Explanation of Step 4 (cf. [Lemma 2.10]ch23Benzerga)
	4 Applications
		4.1 Tits' Alternative
		4.2 Some Applications to Dynamics
		4.3 Structure of Centralizers
		4.4 Cohomological Properties
		4.5 Infinite Automorphism Groups of Hyperkähler Manifolds
	References
On Generalized Büchi Surfaces
	1 Introduction
	2 Generalized Büchi Surfaces
	3 Proof of Theorem 1
	4 Proof of Theorem 2
	5 The Quotient by the Group of Even Sign Changes
	6 Hyperelliptic Curves and Generalized Büchi Surfaces
		6.1 Rationals Points
	References
Fano Visitors, Fano Dimension and Fano Orbifolds
	1 Introduction
	2 Preliminaries
		2.1 Fano Visitor Problem
		2.2 Coarse Moduli Spaces of Deligne-Mumford Stacks
		2.3 Semiorthogonal Decomposition
	3 Cayley's Trick and Weighted Complete Intersections
		3.1 Cayley's Trick
		3.2 Complete Intersections in Projective Space
		3.3 A Generalization
		3.4 Weighted Complete Intersections
	4 Fourier-Mukai Transforms and an Embeddability Criterion
	5 Curves and their Jacobians
		5.1 Hyperelliptic Curves
		5.2 Low Genus Curves
		5.3 Jacobians of Curves
	6 Surfaces
		6.1 κ= -infty Case
		6.2 κ= 0 Case
		6.3 κ= 1 Case
		6.4 κ= 2 Case
	7 Discussions
		7.1 Phantom Categories
		7.2 Noncommutative Varieties
		7.3 Applications and Perspectives
	References
K-stability and Fujita Approximation
	1 Introduction
	2 Preliminaries
		2.1 Non-archimedean Metrics Associated to Models
		2.2 Restricted Volumes and Positive Intersection Products
	3 Positive Intersection Formula
	4 First Riemann-Roch Coefficients of Big Line Bundles  and Fujita Approximations
	5 Conjecture 4.7 for Nakayama's Examples Without Birational Zariski Decomposition
	References
On a Conjecture of Fulton on Isotropic Grassmannians
	1 Introduction and Preliminaries
	2 Proof of the Positive Conjecture for IGr(2,2n) with nge3
	References
On Locally Nilpotent Derivations of Danielewski Domains
	1 Introduction
	2 Definitions, Notations and Technical Lemmas
	3 Results and Proofs
	4 Conclusion
	References
Action of the Automorphism Group on the Jacobian of Klein's Quartic Curve
	1 Introduction
	2 Klein's Group H168, Its Double G336 and the Invariant Lattice Λ
	3 Fixed Loci of Elements of G336 Acting on mathcalJ=mathbbC3/Λ
	4 Orbits with Elliptic Stabilizers
	5 Parabolic Orbits and Singularities of mathcalJ/G336
	References
Some Observations on the Dimension of Fano K-Moduli
	1 Main Statement
	2 Proof of Proposition 1.2
	3 Some Final Comments
	References
Okounkov Bodies and the Kähler Geometry of Projective Manifolds
	1 Introduction
		1.1 Main Results
		1.2 The Big Case
		1.3 Related Work
	2 Okounkov Bodies and Domains
	3 Torus-Invariant Kähler Forms and Moment Maps
	4 Kähler Embeddings of Domains
	5 Big Line Bundles
	References
Fano Shimura Varieties with Mostly Branched Cusps
	1 Introduction
	2 Main Results and Proofs
		2.1 Convention and Notation
		2.2 Special Reflective Modular Forms
		2.3 Main General Results and Proofs
		2.4 Modular Varieties with Big Anti-Canonical Classes
	3 Examples of Fano and K-ample Cases
		3.1 Siegel Modular Cases
		3.2 Orthogonal Modular Cases, Part I
		3.3 Orthogonal Modular Cases, Part II
		3.4 Preparation for Unitary Case—Hermitian Lattice
		3.5 Unramifiedness of Unitary Modular Varieties
		3.6 Unitary Modular Cases, Part I—Fano Cases
		3.7 Unitary Modular Cases, Part II—with Ample Canonical Class
		3.8 More Examples
	References
Simply Connected Sasaki-Einstein 5-manifolds: Old and New
	1 Introduction
		1.1 Sasaki-Einstein 5-Manifold
		1.2 Kähler-Einstein Orbifold
		1.3 Tools for α-Invariant
	2 Link of a Quasi-Homogeneous Hypersurface Singularity
		2.1 From Kähler-Einstein to Sasaki-Einstein
		2.2 Smale 5-Manifold 3Minfty#3M2
		2.3 Smale 5-Manifolds 2Minfty#4M2
		2.4 Smale 5-Manifolds 2Minfty#3M2
	3 Sasaki-Einstein 5-Manifolds: Old and New
		3.1 Known Sasaki-Einstein 5-Manifolds
		3.2 Remaining Candidates for Sasaki-Einstein 5-manifolds
	References
Singularities of Pluri-Fundamental Divisors on Gorenstein Fano Varieties of Coindex 4
	1 Introduction
	2 Pluri-fundamental Divisors on Singular Fano Varieties
	3 Pluri-fundamental Divisors on Smooth Fano Varieties
	References
A Database of Group Actions on Riemann Surfaces
	1 Introduction
	2 Background on Riemann Surfaces
	3 Breuer's Code
	4 New Additions
		4.1 Full Actions
		4.2 Special Properties
		4.3 Equivalence Relations
		4.4 Summary
		4.5 Organization of the Data on LMFDB
	5 Future Work
		5.1 Better Representation of Groups
		5.2 Equivalence Relations
		5.3 Higher Genus Data
		5.4 Other Topics
	References
A 1-Dimensional Component of K-Moduli of del Pezzo Surfaces
	1 Introduction
	2 Proof
		2.1 Deformations of 14(1,1)
		2.2 The Surface X
		2.3 The 3-Fold Y and the Proof of Theorem 1.1
		2.4 Proof of Proposition 2.3(A)
		2.5 Proof of Proposition 2.3(B)
	3 Mirror Symmetry
		3.1 Combinatorial Avatars of Connected Components of Moduli of Del Pezzo Surfaces
		3.2 Classical Period
		3.3 Quantum Period
		3.4 Equality of Periods
	References
A Non-standard Bezout Theorem for Curves
	1 Etale Morphisms and Algebraic Multiplicity
	2 Zariski Multiplicity
	3 Analytic Methods
	4 Families of Curves in P2(L)
	References
Embeddings of the Symmetric Groups to the Space Cremona Group
	1 Introduction
	2 Preliminaries
	3 Main Reduction
	4 Non-gorenstein Fano Threefolds
	5 Proof of Proposition 1.1
	6 Examples
	7 del Pezzo Threefolds
	References
On Hodge-Riemann Cohomology Classes
	1 Introduction
		1.1 Comparison with Previous Work
		1.2 Organization of the Paper
	2 Notation and Convention
	3 Derived Schur Classes
	4 Cone Classes
	5 Fulton-Lazarsfeld Positivity
	6 Hodge-Riemann Classes
	7 Schur Classes Are in overlineHR
	8 The Kähler Case
	9 Combinations of Derived Schur Classes
	10 Inequalities
		10.1 Hodge-Index Type Inequalities
		10.2 Khovanskii-Tessier-Type Inequalities
		10.3 Lorentzian Property of Schur Polynomials
	References
On Large Deviation Principles and the Monge–Ampère Equation (Following Berman, Hultgren)
	1 Introduction
	2 Monge–Ampère and Probability
	3 Large Deviation Principles
	4 Moment Generating Functions
		4.1 Cramér's Theorem
		4.2 Sanov's Theorem
		4.3 Properties of the Moment Generating Function
	5 Proof of the Gärtner–Ellis Theorem
		5.1 The Upper Bound for Compact Sets
		5.2 Exponential Tightness
		5.3 The Lower Bound
	6 LDP Without Moment Generating Functions
	7 Optimal Transport
		7.1 From the Discrete Problem to the General One
		7.2 Dual Formulation
		7.3 The Legendre Transform of Wasserstein Distance
		7.4 Rate Function for Monge–Ampère
	8 Moment Generating Function for Monge–Ampère
		8.1 Finite-dimensional Approximations of Wasserstein Distance
		8.2 An Alternative Proof—Zero Temperature Approach
	References
On Birational Boundedness of Some Calabi–Yau Hypersurfaces
	1 Introduction
	2 Finiteness of Anticanonical Calabi–Yau Surfaces in a 3-fold
	3 Birational Bounded Family of Du Val K3 Surfaces which are Unbounded
	4 Some Results in Higher Dimensional Case
	References
Abelian Varieties, Quaternion Trick and Endomorphisms
	1 Introduction
	2 Polarizations and Isogenies
	3 Base Change
	4 Quaternion Trick
	References
On the Cheltsov–Rubinstein Conjecture
	1 Introduction
	2 Examples and Counterexamples in Dimension 2
		2.1 Preliminaries
		2.2 Basic Setup and Notation
		2.3 Blowing up Two Special Points
		2.4 Counterexamples
	3 Higher Dimensional Counterexamples and Further Discussion
		3.1 Product Spaces
		3.2 K-stability of the Base
	References