Algebraic geometry : Salt Lake City 2015 : 2015 summer research institute, July 13-31, 2015, University of Utah, Salt Lake City, Utah / Tommaso de Fernex [and five others], editors.

Format:

Book

Conference/Event

Contributor:

De Fernex, Tommaso, 1970- editor.

Conference Name:

American Mathematical Society Summer Institute on Algebraic Geometry (2015 : University of Utah)

Series:

Proceedings of symposia in pure mathematics ; Volume 97.2.

Proceedings of symposia in pure mathematics ; Volume 97.2

Language:

English

Subjects (All):

Geometry, Algebraic--Congresses.

Geometry, Algebraic.

Physical Description:

1 online resource (658 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society : Clay Mathematics Institute, [2018]

Summary:

This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic p and p-adic tools, etc. The resulting articles will be important references in these areas for years to come.

Contents:

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

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

How often does the Hasse principle hold?

2. Châtelet surfaces

3. Degree 4 del Pezzo surfaces

4. Cubic surfaces

5. Principal homogeneous spaces of tori

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

A graphical interface for the Gromov-Witten theory of curves

2. Preliminaries

3. Correspondence theorem for tropical descendant GWI

4. Tropical GW/Hurwitz equivalence

5. Fock spaces and Feynman diagrams

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

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

Intrinsic mirror symmetry and punctured Gromov-Witten invariants

1. Punctured invariants

2. The construction of mirrors

Diophantine and tropical geometry, and uniformity of rational points on curves

2. The method of Chabauty-Coleman

3. Berkovich curves and skeletons

4. Theories of -adic Integration

5. Uniformity results

6. Other directions

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

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

Bi-algebraic geometry and the André-Oort conjecture

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

Moduli of sheaves: A modern primer

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

Geometric invariants for non-archimedean semialgebraic sets

2. The motivic volume of Hrushovski-Kazhdan

3. Tropical computation of the motivic volume

4. Application: refined Severi degrees

Symplectic and Poisson derived geometry and deformation quantization

1. Shifted symplectic structures

2. Shifted Poisson structures

3. Deformation quantization

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

On the proper push-forward of the characteristic cycle of a constructible sheaf

The -adic Hodge decomposition according to Beilinson

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

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

References.

Rational points and zero-cycles on rationally connected varieties over number fields

2. Over number fields: general context

3. Methods for rational and rationally connected varieties

Back Cover.

Notes:

"Clay Mathematics Institute."

Includes bibliographical references.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-4680-4

OCLC:

1041035506