Simple to complex with some numerical computations, was completed by mr. Kobayashi pseudometric on hyperkahler manifolds core reader. Most users will download and install just the x64 version. A twistor sphere of generalized kahler potentials on hyperkahler manifolds by malte dyckmanns. Nov 27, 2001 superconformal symmetry and hyperkahler manifolds with torsion. Verbitsky morrisonkawamata cone conjecture for hyperk. 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. As an application we derive a global construction of the fourth order transgression of the chern character forms of hyperholomorphic bundles over compact.
Hence x admits a hyperkahler struc ture by a theorem of yau. Pdf examples of compact holomorphic symplectic manifolds. A hyperkahler manifold is a riemannian manifold with three covariant constant orthogonal automorphisms i, j and k of the. Kahler and symplectic structures on 4manifolds and. Our hyperkahler manifolds have the homotopy type of a union of compact toric varieties intersecting along toric subvarieties. Pdf modifying hyperkahler manifolds with circle symmetry. We will discuss results where this phenomenon re ects interesting geometric aspects, e. Misha verbitsky submitted on 28 aug 2009, last revised 22 may 20 this version, v8 abstract. 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. We also propose an elliptic analogue of these bar involutions.
Lagrangian submanifolds in hyperkahler manifolds, legendre. Download pdf 236 kb abstract a nonlinear generalization of the dirac operator in 4dimensions, obtained by replacing the spinor representation with a hyperkahler manifold admitting certain symmetries, is considered. When the manifold is kahler, a holomorphic hamiltonian vector field is a killing. Pdf toward a classification of compact homogeneous complex manifolds with symplectic structure. Morrisonkawamata cone conjecture for hyperkahler manifolds. 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. 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.
Complex structures on quaternionic manifolds 165 covered by a hyperkahler manifold. Noncommutative automorphism groups of title positive entropy. Mbm loci can be equivalently defined as centers of bimeromorphic contractions. You shall have a manifold which is hyperkaehler globally, but not hyperkaehler in a neighbourhood of any point. 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. 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. 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. 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. 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.
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. Tu 14 june 2 july, 2010 tufts university medford ma usa an introduction to manifolds. Compact hyperkahler manifolds in algebraic geometry. These cycles are modeled on the legendrian varieties studied by freudenthal in his geometric approach to the celebrated titsfreudenthal. Dec 26, 2004 nonnilpotent complex geometry of nilmanifolds and heterotic supersymmetry ugarte, luis and villacampa, raquel, asian journal of mathematics, 2014. A riemannian metric on mis called hermitian if it is compatible with the complex structure jof m, hjx,jyi hx,yi.
The subsequent sections are independent from the introduction. Modifying hyperkahler manifolds with circle symmetry. Furthermore, we shall assume sometimes, implicitly that all hyperkahler. Kahlerricci flow, morse theory, and vacuum structure deformation of n 1 supersymmetry in four dimensions gunara, bobby e. Pdf superconformal symmetry and hyperkahler manifolds. We discuss a category of lagrangian subvarieties and its relationship with the theory of lagrangian intersection. Given a projective hyperkahler manifold with a holomorphic lagrangian fibration, we prove that hyperkahler metrics with volume of the. All elds are assumed of char 0, an all abelian categories over a base eld k. Preliminaries let us recall that a hyper almost complex manifold is a manifold endowed with three almost complex structures, j. Hyperbolic geometry of the ample cone of a hyperkahler manifold. Hyperk ahler manifolds, moduli spaces, derived categories, holomorphic symplectic manifolds. 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. Lagrangian submanifolds in hyperkahler manifolds, legendre transformation by naichung conan leung.
Nonnilpotent complex geometry of nilmanifolds and heterotic supersymmetry ugarte, luis and villacampa, raquel, asian journal of mathematics, 2014. A compact hyperkahler or irreducible holomorphic symplectic manifold is a simply connected compact complex manifold of kahler type x such that h0x. Hyperkahler hk manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Hyperkahler manifolds and birational transformations.
Implosion for hyperkahler manifolds volume 149 issue 9 andrew dancer, frances kirwan, andrew swann. Hyperkahler manifolds from the titsfreudenthal square. Manifold evolves very rapidly, with dozens of new features per week. Finally, we show that if a covariantly constant, generalized spinor satisfies generalized seibergwitten equations, the metric on. Hyperk ahler manifolds and sheaves daniel huybrechts abstract. Superconformal symmetry and hyperkahler manifolds with torsion. Here we give a brief introduction to results of this paper. Pdf superconformal symmetry and hyperkahler manifolds with. In many important cases, such as for calabiyau manifolds cat, teich. Mar 29, 2012 short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. The aim was to describe some background results before describing in detail in subsequent talks the results of arxiv. Product downloads manifold software gis and database tools. Concerning holomorphic properties of the complex manifold 2 it is known that.
Hyperkahler manifolds with torsion hkt manifolds were introduced by p. 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. Summer school and conference on hodge theory and related topics. There is, however, an essential difference between kahler and hyperkahler manifolds. Riemannian geometry of contact and symplectic manifolds. For an excellent survey of these works written from a mathe maticians point of view, the reader is referred to the paper of g. Gbundles on abelian surfaces, hyperkahler manifolds, and. Picard number of the generic fiber of an abelian fibered. 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. A twistor sphere of generalized kahler potentials on hyperkahler manifolds. This satisfactory state of affairs can be used to prove the existence of hyperkahler metrics on many examples of complex manifolds. It was shown that the mbm loci on deformation equivalent hyperkahler manifolds are diffeomorphic.
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. That page provides downloads for x64 64bit and 32bit versions. Our hyperkahler manifolds are constructed as moduli spaces of bridgeland stable objects in derived categories of k3 surfaces. 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. Liu, we have proven the injectivity of the period map on the teichmuller space of polarized and marked calabiyau type manifolds. 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. Tensor category is an abelian category equipped with the. A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies.
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. We determine the mbm loci on a hyperkahler manifold of k3type of low dimension using. Pdf we introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of sun actions. In the case of symmetric products of k3 surfaces our entropy agrees with the black hole entropy of d5d1 system. Papadopoulos hp were much studied in physics literature since then. We shall use the notation m,i or m i to stress that a particu.
If we view hyperk\ahler manifolds as abelian varieties, then those of jacobian type should be viewed as jacobian varieties. Gauging isometries on hyperkahler cones and quaternion kahler 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. Kobayashi pseudometric on hyperkahler manifolds request pdf. The most fundamental is the k3 surface the only nontrivial example in 4 real dimensions. Besides giving a thorough introduction into kahler geometry, my main aims were cohomology of kahler manifolds, formality of kahler manifolds af. 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. In this paper we define the notion of a hyperk\ahler manifold potentially of jacobian type. A kahler metric on a given complex manifold can be modified to another. We study the moduli space mg,a of flat gbundles on an abelian surface a, where g is a compact, simple, simply connected, connected lie group. A hyperkahler manifold naturally possesses a whole 2sphere of complex. Implosion for hyperkahler manifolds compositio mathematica. As for abelian varieties, the interplay between algebraic, arithmetic, and.
Elliptic canonical bases for toric hyperkahler manifolds arxiv. Hyperkahler manifolds and nonabelian hodge theory of irregular curves. 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. We describe how this inequality can be extended to hyperkahler moment maps in some cases, and use morse theory. We then give a few applications for the mordellweil group. Dancer gave a systematic description of hypertoric varieties.
Cohomology of compact hyperkahler manifolds and its applications. 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. Hyperkahler metrics and supersymmetry project euclid. The point of view will be that of an algebraic or complex geometer. This has interest ing implications for the geometry of compact hyperkahler man ifolds and of holomorphic vector bundles over such manifolds. Toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions.
Pdf morrisonkawamata cone conjecture for hyperkahler manifolds. Toric hyperkahler manifolds with quaternionic kahler bases. I certify that this is an original project report resulting from the work completed during this period. A hyperk ahler manifold is a c1riemannian manifold together with three covari antly constant orthogonal endomorphisms i, jand kof. Automorphisms of hyperkahler manifolds dipartimento di. Also, by a small birational contraction, we will mean. Presentation mode open print download current view.
Moduli spaces of hyperk ahler manifolds or of sheaves on them are often nonseparated. 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. For an arbitrary variety v, let v reg denote the nonsingular part of v. Some people always want the latest version, while others prefer to update less frequently. These are notes of lectures on kahler manifolds which i taught at the university of bonn and, in reduced form, at the erwinschr. The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry. K3 surfaces and their higherdimensional analogs, hyperk ahler. Short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. K3 surfaces and their higherdimensional analogs, hyperk ahler manifolds. Toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions by o. Hyperkahler manifolds are special classes of kahler manifolds. Blairriemannian geometry of contact and symplectic manifoldsbirkhauser progress. An mbm locus on a hyperkahler manifold is the union of all deformations of a minimal rational curve with negative selfintersection.
Thus we are able to see relations between certain complex manifolds which are. Download the 64bit access database engine from microsofts access database engine 2016 redistributable page. Hyperkahler manifolds and nonabelian hodge theory of. Sidharth kshatriya under my guidance during the academic year 20062007. Dec 29, 2004 toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions. This is a version of the morrisonkawamata cone conjecture for hyperkahler manifolds. 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.