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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


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


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


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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