They can be thought of as quaternionic analogues of kahler manifolds. A twistor sphere of generalized kahler potentials on hyperkahler manifolds. Mar 07, 2020 in this paper, we give a conjectural generalization of these involutions and ktheoretic canonical bases to conical symplectic resolutions which have good hamiltonian torus actions and state several conjectures related to them which we check for toric hyper kahler manifolds. We suggest a way to associate to each lie algebra of type g2, d4, f4, e6, e7, e8 a family of polarized hyperkahler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or quasihomogeneous varieties.
We then give a few applications for the mordellweil group. We construct examples of hyperkahler manifolds of picard number two with automorphisms of positive entropy via derived automorphisms of k3 surfaces of picard number one. An interesting feature of hyperk ahler manifolds is that any family of hyperk ahler manifolds has a dense subset consisting of projective hyperk ahler man. A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies.
Automorphisms of hyperkahler manifolds dipartimento di. Lagrangian submanifolds in hyperkahler manifolds, legendre. Hyperkahler manifolds with torsion hkt manifolds were introduced by p. In many important cases, such as for calabiyau manifolds cat, teich.
Liu, we have proven the injectivity of the period map on the teichmuller space of polarized and marked calabiyau type manifolds. Simple to complex with some numerical computations, was completed by mr. Dancer gave a systematic description of hypertoric varieties. K3 surfaces and their higherdimensional analogs, hyperk ahler manifolds. Presentation mode open print download current view. Kobayashi pseudometric on hyperkahler manifolds core reader.
Papadopoulos hp were much studied in physics literature since then. Besides giving a thorough introduction into kahler geometry, my main aims were cohomology of kahler manifolds, formality of kahler manifolds af. If we view hyperk\ahler manifolds as abelian varieties, then those of jacobian type should be viewed as jacobian varieties. Compact hk manifolds are one of the building blocks for algebraic varieties with trivial first chern class and their role in algebraic geometry has grown immensely over the last 20. K3 surfaces and their higherdimensional analogs, hyperk ahler. Pdf examples of compact holomorphic symplectic manifolds. Hyperk ahler manifolds, moduli spaces, derived categories, holomorphic symplectic manifolds. Here we give a brief introduction to results of this paper. Toric hyperkahler manifolds with quaternionic kahler bases. Toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions by o. We describe how this inequality can be extended to hyperkahler moment maps in some cases, and use morse theory.
Nov 27, 2001 superconformal symmetry and hyperkahler manifolds with torsion. Hence x admits a hyperkahler struc ture by a theorem of yau. Also, by a small birational contraction, we will mean. The connection with the pedersenpoon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation related to eigenfunctions of a certain. In particular, we have focused on projective manifolds of calabiyau type, which is a generalization of calabiyau manifolds. We discuss a category of lagrangian subvarieties and its relationship with the theory of lagrangian intersection. The aim was to describe some background results before describing in detail in subsequent talks the results of arxiv.
Our hyperkahler manifolds have the homotopy type of a union of compact toric varieties intersecting along toric subvarieties. Complex structures on quaternionic manifolds 165 covered by a hyperkahler manifold. Elliptic canonical bases for toric hyperkahler manifolds arxiv. A hyperk ahler manifold is a c1riemannian manifold together with three covari antly constant orthogonal endomorphisms i, jand kof. For a class of hyperkahler manifolds obtained via hyperkahler reduction, we also show that a harmonic spinor, under mild conditions, defines a symplectic structure. Noncommutative automorphism groups of title positive entropy. This satisfactory state of affairs can be used to prove the existence of hyperkahler metrics on many examples of complex manifolds.
First and foremost is my desire to write a readable but rigorous introduction that gets the reader quickly up to speed, to the point where for example he or she can compute. There is, however, an essential difference between kahler and hyperkahler manifolds. We shall use the notation m,i or m i to stress that a particu. Gauging isometries on hyperkahler cones and quaternion kahler manifolds. These are notes of lectures on kahler manifolds which i taught at the university of bonn and, in reduced form, at the erwinschr. Pdf modifying hyperkahler manifolds with circle symmetry. Pdf toward a classification of compact homogeneous complex manifolds with symplectic structure. A hyperk ahler manifold is a c1riemannian manifold together with three covari antly constant orthogonal endomorphisms i, jand kof the tangent bundle which satisfy the. Michael spivak brandeis university calculus on manifolds a modern approach to classical theorems of advanced calculus addisonwesley publishing company the advanced book program reading, massachusetts menlo park, california new york don mills, ontario wokingham, england amsterdam bonn. We first define special manifolds as being the compact kahler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure oil the base being given by the divisor of.
In the case of symmetric products of k3 surfaces our entropy agrees with the black hole entropy of d5d1 system. Finally, we show that if a covariantly constant, generalized spinor satisfies generalized seibergwitten equations, the metric on. In mathematics, the definitions of blablabla manifold are almost always local, because the manifolds are defined in terms of a local atlas, and its natural to define the structures on manifolds in terms of the same atlas. A hyperkahler manifold is a manifold necessarily of dimension a multiple of four which admits an action on tangent vectors of the same i, j and k in a manner which is compatible with a metric. I certify that this is an original project report resulting from the work completed during this period. Verbitsky morrisonkawamata cone conjecture for hyperk. A twistor sphere of generalized kahler potentials on hyperkahler manifolds by malte dyckmanns. This has interest ing implications for the geometry of compact hyperkahler man ifolds and of holomorphic vector bundles over such manifolds. Gbundles on abelian surfaces, hyperkahler manifolds, and. Toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions. Pdf we introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of sun actions. Dec 29, 2004 toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions. Tensor category is an abelian category equipped with the.
Hyperkahler manifolds and birational transformations. Kobayashi pseudometric on hyperkahler manifolds request pdf. A compact hyperkahler or irreducible holomorphic symplectic manifold is a simply connected compact complex manifold of kahler type x such that h0x. Hyperkahler manifolds and nonabelian hodge theory of irregular curves. For an arbitrary variety v, let v reg denote the nonsingular part of v. Superconformal symmetry and hyperkahler manifolds with torsion. You shall have a manifold which is hyperkaehler globally, but not hyperkaehler in a neighbourhood of any point. Pdf superconformal symmetry and hyperkahler manifolds.
Kahlerricci flow, morse theory, and vacuum structure deformation of n 1 supersymmetry in four dimensions gunara, bobby e. These cycles are modeled on the legendrian varieties studied by freudenthal in his geometric approach to the celebrated titsfreudenthal. Hyperkahler manifolds from the titsfreudenthal square. Given a projective hyperkahler manifold with a holomorphic lagrangian fibration, we prove that hyperkahler metrics with volume of the.
Kahler and symplectic structures on 4manifolds and. We extend our previous results on the relation between quaternionkahler manifolds and hyperkahler cones and we describe how isometries, moment maps and scalar potentials descend from the cone to the quaternionkahler space. That page provides downloads for x64 64bit and 32bit versions. With so many excellent books on manifolds on the market, any author who undertakesto write anotherowes to the public, if not to himself, a good rationale. Download the 64bit access database engine from microsofts access database engine 2016 redistributable page. K3 surfaces and over the last ten years or so also their higher dimensional analogues, compact hyperk ahler manifolds, have been studied intensively from various angles. Thus we are able to see relations between certain complex manifolds which are. Dec 26, 2004 nonnilpotent complex geometry of nilmanifolds and heterotic supersymmetry ugarte, luis and villacampa, raquel, asian journal of mathematics, 2014. Riemannian geometry of contact and symplectic manifolds. We will discuss results where this phenomenon re ects interesting geometric aspects, e. Hyperkahler metrics and supersymmetry project euclid. Most users will download and install just the x64 version. The aim was to describe some background results before describing in detail in subsequent talks the results of boa11c related to wild character varieties and irregular mapping class groups. Hyperkahler manifolds are special classes of kahler manifolds.
The point of view will be that of an algebraic or complex geometer. Preliminaries let us recall that a hyper almost complex manifold is a manifold endowed with three almost complex structures, j. To meet both needs, this page provides both newer edge and older official downloads for manifold release 9. We also introduce and study extensively a normalized legendre transformation of lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. An mbm locus on a hyperkahler manifold is the union of all deformations of a minimal rational curve with negative selfintersection. We also propose an elliptic analogue of these bar involutions. Hyperkahler manifolds and nonabelian hodge theory of.
A kahler metric on a given complex manifold can be modified to another. Short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. We determine the mbm loci on a hyperkahler manifold of k3type of low dimension using. Hyperk ahler manifolds and sheaves daniel huybrechts abstract. A hyperkahler manifold naturally possesses a whole 2sphere of complex. Compact hk manifolds are one of the building blocks for algebraic varieties with trivial first chern class and their role in algebraic geometry has grown immensely over the last 20 year. In this paper we define the notion of a hyperk\ahler manifold potentially of jacobian type. Summer school and conference on hodge theory and related topics. We also show that for smooth projective toric varieties x the cotangent bundle of x carries a hyperkahler metric, which is complete only if x is a product of projective spaces. All hyperkahler manifolds are ricciflat and are thus calabiyau manifolds this can be easily seen by noting that sp is a subgroup of the special unitary group su2k.
Some people always want the latest version, while others prefer to update less frequently. Sidharth kshatriya under my guidance during the academic year 20062007. Modifying hyperkahler manifolds with circle symmetry. The most fundamental is the k3 surface the only nontrivial example in 4 real dimensions. A global torelli theorem for hyperkahler manifolds. Hyperbolic geometry of the ample cone of a hyperkahler manifold. Pdf superconformal symmetry and hyperkahler manifolds with. Mbm loci can be equivalently defined as centers of bimeromorphic contractions. A hyperkahler manifold is a riemannian manifold with three covariant constant orthogonal automorphisms i, j and k of the.
Picard number of the generic fiber of an abelian fibered. This is done by proving a vanishing theorem for holomorphic sections of line bundles on the twistor space 2 which is inspired to an analogous result of gauduchon 3. Tu 14 june 2 july, 2010 tufts university medford ma usa an introduction to manifolds. When the manifold is kahler, a holomorphic hamiltonian vector field is a killing. Introduction it has been known for some time 17 that characters of the bps representations of the extended superconformal algebra do not in general have a good modular property. Mar 29, 2012 short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. Sep 07, 2012 pdf we introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of sun actions. Compact hyperkahler manifolds in algebraic geometry. Our hyperkahler manifolds are constructed as moduli spaces of bridgeland stable objects in derived categories of k3 surfaces. This is a version of the morrisonkawamata cone conjecture for hyperkahler manifolds.
All elds are assumed of char 0, an all abelian categories over a base eld k. Misha verbitsky submitted on 28 aug 2009, last revised 22 may 20 this version, v8 abstract. For an excellent survey of these works written from a mathe maticians point of view, the reader is referred to the paper of g. Hyperkahler hk manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. It was shown that the mbm loci on deformation equivalent hyperkahler manifolds are diffeomorphic. The subsequent sections are independent from the introduction. Moduli spaces of hyperk ahler manifolds or of sheaves on them are often nonseparated.
As an application we derive a global construction of the fourth order transgression of the chern character forms of hyperholomorphic bundles over compact. The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry. Furthermore, we shall assume sometimes, implicitly that all hyperkahler. Manifold evolves very rapidly, with dozens of new features per week. Product downloads manifold software gis and database tools. A riemannian metric on mis called hermitian if it is compatible with the complex structure jof m, hjx,jyi hx,yi. Pdf morrisonkawamata cone conjecture for hyperkahler manifolds. A hyperk ahler manifold is a c1riemannian manifold together with three covariantly constant orthogonal endomorphisms i, jand kof the tangent bundle which satisfy the quaternionic relations i 2 j k2 ijk 1. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Morrisonkawamata cone conjecture for hyperkahler manifolds. Implosion for hyperkahler manifolds compositio mathematica.
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