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## K3 Surfaces

*D. Huybrechts*

### in Fourier-Mukai Transforms in Algebraic Geometry

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0010
- Subject:
- Mathematics, Geometry / Topology

After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous ... More

## S‐Duality in HyperkäHler Hodge Theory

*TamáS Hausel*

### in The Many Facets of Geometry: A Tribute to Nigel Hitchin

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199534920
- eISBN:
- 9780191716010
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199534920.003.0016
- Subject:
- Mathematics, Geometry / Topology

This chapter surveys the motivations, related results, and progress made towards the following problem, raised by Hitchin in 1995: What is the space of L2 harmonic forms on the moduli space of Higgs ... More

## SUSY GAUGE THEORIES

*John Terning*

### in Modern Supersymmetry: Dynamics and Duality

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567639
- eISBN:
- 9780191718243
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567639.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter examines SUSY gauge theories. First, it looks at the role of quantum loop corrections on running couplings and masses. It then studies the space of possible vacua (the moduli space), and ... More

## Moduli Space

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0013
- Subject:
- Mathematics, Geometry / Topology

This chapter focuses on the moduli space of Riemann surfaces. The moduli space parameterizes many different kinds of structures on Sɡ, such as isometry classes of hyperbolic structures on S, ... More

## The Mumford-Tate Group of a Variation of Hodge Structure

*Mark Green, Phillip Griffiths, and Matt Kerr*

### in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691154244
- eISBN:
- 9781400842735
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691154244.003.0004
- Subject:
- Mathematics, Analysis

This chapter deals with the Mumford-Tate group of a variation of Hodge structure (VHS). It begins by presenting a definition of VHS, which consists of a connected complex manifold and a locally ... More

##
Irreducibility of Moduli of Semi-Stable Chains and Applications to U(*p*, *q*)-Higgs Bundles

*Steven Bradlow, Oscar García-Prada, Peter Gothen, and Jochen Heinloth*

### in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0018
- Subject:
- Mathematics, Geometry / Topology

This chapter gives necessary and sufficient conditions for moduli spaces of semi-stable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the ... More

## Mappings and moduli

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0014
- Subject:
- Mathematics, Geometry / Topology, Analysis

This chapter analyzes diffeomorphisms of the plane; braids, Dehn twists and quadratic singularities; hyperbolic geometry; and compactification of the moduli space.

## Brauer Group of Moduli of Higgs Bundles and Connections

*David Baraglia, Indranil Biswas, and Laura P. Schaposnik*

### in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0014
- Subject:
- Mathematics, Geometry / Topology

Given a compact Riemann surface X and a semi-simple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X, G). The chapter computes the ... More

## Involutions of Rank 2 Higgs Bundle Moduli Spaces

*Oscar García-Prada and S. Ramanan*

### in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0022
- Subject:
- Mathematics, Geometry / Topology

This chapter considers the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over ℂ, and studies involutions defined by tensoring the vector ... More

## Moduli spaces of shtukas

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0023
- Subject:
- Mathematics, Geometry / Topology

This chapter examines the moduli spaces of mixed-characteristic local G-shtukas and shows that they are representable by locally spatial diamonds. These will be the mixed-characteristic local ... More

## A Hitchin Connection for a Large Class of Families of Kähler Structures

*Jørgen Ellegaard Andersen and Kenneth Rasmussen*

### in Geometry and Physics: Volume I: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802013
- eISBN:
- 9780191840500
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802013.003.0007
- Subject:
- Mathematics, Geometry / Topology

This chapter presents a Hitchin connection constructed in a setting which significantly generalizes the setting covered by the first author, which, in turn, was a generalization of the moduli space ... More

## Introduction

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0001
- Subject:
- Mathematics, Geometry / Topology

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case ... More

## Berkeley Lectures on p-adic Geometry: (AMS-207)

*Peter Scholze and Jared Weinstein*

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory ... More

## Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

*Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada (eds)*

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.001.0001
- Subject:
- Mathematics, Geometry / Topology

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and ... More

## Geometry and Physics: Volume I: A Festschrift in honour of Nigel Hitchin

*Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada (eds)*

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802013
- eISBN:
- 9780191840500
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802013.001.0001
- Subject:
- Mathematics, Geometry / Topology

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and ... More

## Stable Betti Numbers of (Partial) Toroidal Compactifications of the Moduli Space of Abelian Varieties

*Samuel Grushevsky, Klaus Hulek, Orsola Tommasi, and Mathieu Dutour Sikirić*

### in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802020
- eISBN:
- 9780191869068
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802020.003.0024
- Subject:
- Mathematics, Geometry / Topology

This chapter presents an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of Ag, satisfying ... More

## Local Shimura varieties

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0024
- Subject:
- Mathematics, Geometry / Topology

This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a ... More

## Seiberg–Witten invariants

*Selman Akbulut*

### in 4-Manifolds

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198784869
- eISBN:
- 9780191827136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198784869.003.0013
- Subject:
- Mathematics, Geometry / Topology

Study of Seiberg witten invariants or 4-manifolds and applications

##
Vector bundles and *G*-torsors on the relative Fargues-Fontaine curve

*Peter Scholze and Jared Weinstein*

### in Berkeley Lectures on p-adic Geometry: (AMS-207)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0022
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses vector bundles and G-torsors on the relative Fargues-Fontaine curve. This is in preparation for the examination of moduli spaces of shtukas. Kedlaya-Liu prove two important ... More

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