Algebraic Geometry: Salt Lake City 2015

Cover
Title page
Contents
Preface
Scientific program
Part 2
Betti Geometric Langlands
1. Introduction
2. Two toy models
3. Spectral side
4. Automorphic side
References
Specializing varieties and their cohomology from characteristic 0 to characteristic ?
1. Introduction
2. Perfect and perfectoid rings
3. Some almost mathematics
4. Framed algebras and input from perfectoid geometry
5. The decalage functor
6. The complex ̃Ω_{\frakX}
7. The complex ?Ω_{\frakX}
8. Global results
References
How often does the Hasse principle hold?
1. Introduction
2. Châtelet surfaces
3. Degree 4 del Pezzo surfaces
4. Cubic surfaces
5. Principal homogeneous spaces of tori
References
Tropical methods in the moduli theory of algebraic curves
1. Introduction and Notation
2. Tropical curves
3. From algebraic curves to tropical curves
4. Curves and their Jacobians
5. Torelli theorems
6. Conclusions
References
A graphical interface for the Gromov-Witten theory of curves
1. Introduction
2. Preliminaries
3. Correspondence theorem for tropical descendant GWI
4. Tropical GW/Hurwitz equivalence
5. Fock spaces and Feynman diagrams
References
Some fundamental groups in arithmetic geometry
1. Acknowledgments
2. Deligne’s conjectures: ℓ-adic theory
3. Deligne’s conjectures: crystalline theory
4. Malčev-Grothendieck’s theorem, Gieseker’s conjecture, de Jong’s conjecture
References
From local class field to the curve and vice versa
Introduction
1. The curve
2. Vector bundles
3. The curve compared to ℙ¹
4. ?-bundles on the curve ([4])
5. Archimedean/?-adic twistors
6. The fundamental class of the curve is the fundamental class of class field theory ([4])
7. Conjectures: ramified local systems and coverings
8. Speculations: Fourier transform and ?-adic local Langlands correspondence
References
Intrinsic mirror symmetry and punctured Gromov-Witten invariants
Introduction
1. Punctured invariants
2. The construction of mirrors
References
Diophantine and tropical geometry, and uniformity of rational points on curves
1. Introduction
2. The method of Chabauty–Coleman
3. Berkovich curves and skeletons
4. Theories of ?-adic Integration
5. Uniformity results
6. Other directions
References
On categories of (?,Γ)-modules
1. The original category of (?,Γ)-modules
2. Interlude on perfectoid fields
3. Slopes of ?-modules
4. From ?-modules to (?,Γ)-modules
5. Cohomology of (?,Γ)-modules
6. The cyclotomic deformation
7. Iwasawa cohomology and the cyclotomic deformation
8. Coda: beyond the cyclotomic tower
References
Principal bundles and reciprocity laws in number theory
1. Principal bundles and their moduli
2. Some fundamental groups
3. Reciprocity laws
4. Explicit reciprocity laws on curves
5. Analogies to gauge theory
Acknowledgments
References
Bi-algebraic geometry and the André-Oort conjecture
1. Introduction
2. The André-Oort conjecture
3. Special structures on algebraic varieties
4. Bi-algebraic geometry
5. O-minimal geometry and the Pila-Wilkie theorem
6. O-minimality and Shimura varieties
7. The hyperbolic Ax-Lindemann conjecture
8. The two main steps in the proof of the André-Oort conjecture
9. Lower bounds for Galois orbits of CM-points
10. Further developments: the André-Pink conjecture
References
Moduli of sheaves: A modern primer
1. Introduction
1.1. The structure of this paper
1.2. Background assumed of the reader
1.3. Acknowledgments
Part 1. Background
2. A mild approach to the classical theory
2.1. The \Quot scheme
2.2. The Picard scheme
2.3. Sheaves on a curve
2.4. Sheaves on a surface
2.5. Guiding principles
3. Some less classical examples
3.1. A simple example
3.2. A more complex example
3.3. A stop-gap solution: twisted sheaves
4. A catalog of results
4.1. Categorical results
4.2. Results related to the geometry of moduli spaces
4.3. Results related to non-commutative algebra
4.4. Results related to arithmetic
Part 2. A thought experiment
5. Some terminology
5.1. The 2-category of \simplespaces
5.2. Sheaves on \simplespaces
6. Moduli of sheaves: Basics and examples
6.1. The basics
6.2. Example: almost Hilbert
6.3. Example: invertible 1-sheaves on an elliptic merbe
6.4. Example: sheaves on a curve
6.5. Example: sheaves on a surface
6.6. Example: sheaves on a K3 \simplespace
7. Case studies
7.1. Period-index results
7.2. The Tate conjecture for K3 surfaces
References
Geometric invariants for non-archimedean semialgebraic sets
1. Introduction
2. The motivic volume of Hrushovski-Kazhdan
3. Tropical computation of the motivic volume
4. Application: refined Severi degrees
References
Symplectic and Poisson derived geometry and deformation quantization
Introduction
1. Shifted symplectic structures
2. Shifted Poisson structures
3. Deformation quantization
References
Varieties that are not stably rational, zero-cycles and unramified cohomology
1. Rational, unirational and stably rational varieties
2. Specialization method and applications
3. Unramified Brauer group and fibrations in quadrics
References
On the proper push-forward of the characteristic cycle of a constructible sheaf
References
The ?-adic Hodge decomposition according to Beilinson
1. Introduction
1.1. The Hodge decomposition over \C
1.2. Algebraization
1.3. The case of a ?-adic base field
1.4. Beilinson’s method
1.5. Overview of the present text
2. The cotangent complex and the derived de Rham algebra
2.1. The cotangent complex of a ring homomorphism
2.2. First-order thickenings and the cotangent complex
2.3. The derived de Rham algebra
3. Differentials and the de Rham algebra for ?-adic rings of integers
3.1. Modules of differentials for ?-adic rings of integers
3.2. The universal ?-adically complete first order thickening of \OCK/\OK
3.3. Derived de Rham algebra calculations
3.4. The ?-completed derived de Rham algebra of \OCK/\OK.
4. Construction of period rings
4.1. Construction and basic properties of \Bdr
4.2. Deformation problems and period rings
4.3. The Fontaine element
5. Beilinson’s comparison map
5.1. Sheaf-theoretic preliminaries
5.2. Preliminaries on logarithmic structures
5.3. The geometric side of the comparison map
5.4. The arithmetic side of the comparison map
6. The comparison theorem
6.1. Proof of the comparison isomorphism
6.2. Proof of the Poincaré lemma
A. Appendix: Methods from simplicial algebra
A.1. Simplicial methods
A.2. Associated chain complexes
A.3. Bisimplicial objects
A.4. Simplicial resolutions
A.5. Derived functors of non-additive functors
A.6. Application: derived exterior powers and divided powers
A.7. Cohomological descent
A.8. Hypercoverings
References
Specialization of ℓ-adic representations of arithmetic fundamental groups and applications to arithmetic of abelian varieties
0. Introduction
1. Uniform open image theorems (joint work with Anna Cadoret)
2. Specialization of first cohomology groups (joint work with Mohamed Saïdi)
3. A local-global principle for first cohomology groups
Acknowledgments
References
Rational points and zero-cycles on rationally connected varieties over number fields
1. Introduction
2. Over number fields: general context
3. Methods for rational and rationally connected varieties
References
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