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K3 Surfaces

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

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

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 Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.Less

S‐Duality in HyperkäHler Hodge Theory

TamáS Hausel

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

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

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 bundles on a Riemann surface?Less

SUSY GAUGE THEORIES

John Terning

in Modern Supersymmetry: Dynamics and Duality

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

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 finally the super Higgs mechanism. There are exercises at the end of the chapter.Less

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)

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

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 liftable, holomorphic mapping that is an integral manifold of the canonical differential ideal. The moduli space of Γ‎-equivalence classes of polarized Hodge structures is also considered, along with a generic point for the VHS and the monodromy group of the VHS. Associated to a VHS is its Mumford-Tate group. The chapter proceeds by discussing the structure theorem for VHS, where S is a quasi-projective algebraic variety, referred to as global variations of Hodge structure. It concludes by describing an application of Mumford-Tate groups, along with the Noether-Lefschetz locus.Less

Moduli Space

Benson Farb and Dan Margalit

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

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

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, conformal classes of Riemannian metrics on S, biholomorphism classes of complex structures on S, and isomorphism classes of smooth algebraic curves homeomorphic to S. The chapter first considers the moduli space as the quotient of Teichmüller space before discussing the moduli space of the torus. It then examines the theorem (due to Fricke) that Mod(S) acts properly discontinuously on Teich(S), with a finite-index subgroup of Mod(S) acting freely such that M(S) is finitely covered by a smooth aspherical manifold. The chapter also looks at Mumford's compactness criterion, which describes what it means to go to infinity in M(S), and concludes by showing that M(Sɡ) is very close to being a classifying space for Sɡ-bundles.Less

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

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

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 nilpotent cone of GLn-Higgs bundles and of moduli of systems of Hodge bundles on curves. As it does not impose coprimality restrictions, it can apply this to prove connectedness for moduli spaces of U(p, q)-Higgs bundles.Less

Moduli spaces of shtukas

Peter Scholze and Jared Weinstein

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

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

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 analogues of the moduli spaces of global equal-characteristic shtukas introduced by Varshavsky. It may be helpful to briefly review the construction in the latter setting. The ingredients are a smooth projective geometrically connected curve X defined over a finite field Fq and a reductive group G/Fq. Each connected component is a quotient of a quasi-projective scheme by a finite group. From there, it is possible to add level structures to the spaces of shtukas, to obtain a tower of moduli spaces admitting an action of the adelic group. The cohomology of these towers of moduli spaces is the primary means by which V. Lafforgue constructs the “automorphic to Galois” direction of the Langlands correspondence for G over F.Less

Mappings and moduli

Simon Donaldson

in Riemann Surfaces

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

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

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 Brauer group of the smooth locus of these varieties.Less

Introduction

Peter Scholze and Jared Weinstein

in Berkeley Lectures on p-adic Geometry

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

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 of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.Less

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

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

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 bundle with an element α‎ of order 2 in the Jacobian of the curve, combined with multiplication of the Higgs field by ±1. It describes the fixed points of these involutions in terms of the Prym variety of the covering of X defined by α‎, and gives an interpretation in terms of the moduli space of representations of the fundamental group.Less

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

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

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 covered in the original work on the Hitchin connection. In fact, this construction provides a Hitchin connection which is a partial connection on the space of all compatible complex structures on an arbitrary but fixed prequantizable symplectic manifold which satisfies a certain Fano-type condition. The subspace of the tangent space to the space of compatible complex structures on which the constructed Hitchin connection is defined is of finite codimension if the symplectic manifold is compact. It also proves uniqueness of the Hitchin connection under a further assumption. A number of examples show that this Hitchin connection is defined in a neighbourhood of the natural families of complex structures compatible with the given symplectic form which these spaces admit.Less

Local Shimura varieties

Peter Scholze and Jared Weinstein

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

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

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 triple (G, b, µ) consisting of a reductive group G over Qp, a conjugacy class µ of minuscule cocharacters. Rapoport-Zink spaces are moduli of deformations of a fixed p-divisible group. After reviewing these, the chapter shows that the diamond associated with the generic fiber of a Rapoport-Zink space is isomorphic to a moduli space of shtukas of the form with µ minuscule. It then extends the results to general EL and PEL data.Less

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

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

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 certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular, the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.Less

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)

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

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 foundational theorems about vector bundles on the Fargues-Fontaine curve. The first is the semicontinuity of the Newton polygon. The second theorem of Kedlaya-Liu concerns the open locus where the Newton polygon is constant 0. For the applications to the moduli spaces of shtukas, one needs to generalize the results to the case of G-torsors for a general reductive group G. The chapter then identifies the classification of G-torsors. It also looks at the semicontinuity of the Newton point.Less

Seiberg–Witten invariants

Selman Akbulut

in 4-Manifolds

Study of Seiberg witten invariants or 4-manifolds and applications