In fact, if u a, b, c e r3 then so that if ii ull 1 we have another covariant constant and hence integrable complex structure iu. Complex geometry studies compact complex manifolds. The aim of these notes is to acquaint the reader with important objects. Hyperkahler manifolds with torsion hkt manifolds were introduced by p. 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. Kobayashi pseudometric on hyperkahler manifolds core reader. A global torelli theorem for hyperkahler manifolds. Compact hyperkahler manifolds in algebraic geometry. Papadopoulos hp were much studied in physics literature since then. Apart the two exceptional examples given by ogrady, the only known classes of hyperkahler manifolds are deformation equivalent to hilbert scheme of point of a k3 and generalized kummer varieties.
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. Short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. Hyperkahler manifolds and nonabelian hodge theory of. Prove that if t is norm preserving, then t is angle preserving. This satisfactory state of affairs can be used to prove the existence of hyperkahler metrics on many examples of complex manifolds. Smooth manifolds a manifold, m, is a topological space with a maximal atlas or a maximal smooth structure. Since the restriction of the riemannian metric of m to n is hermitian and its kahler. Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s. Pdf curvature and characteristic numbers of hyperk. Hyperkahler hk manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Lagrangian submanifolds in hyperkahler manifolds, legendre. Morrisonkawamata cone conjecture for hyperkahler manifolds.
Their rich geometry has been explored with techniques from different branches of mathematics and the interplay of analysis, arithmetic, and geometry makes their theory highly attractive. 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. Construction of automorphisms of hyperkahler manifolds. Free riemannian geometry books download ebooks online. Construction of automorphisms of hyperkahler manifolds volume 153 issue 8. Narasimhan no part of this book may be reproduced in any form by print, micro.
Alex degtyarev, ilia itenberg, viatcheslav kharlamov abstract. The subject is on the crossroad of algebraic and differential geometry. Let n be a complex submanifold of a kahler manifold m. A hyperkahler manifold m,i,j,k, considered as a complex manifold m,i, is holomorphically symplectic equipped with a holomorphic, nondegenerate 2form. The ambient coordinates and the vector space structure ofrnare super. Toric hyperkahler manifolds with quaternionic kahler bases and supergravity solutions. Hyperkahler manifolds 3 iiispaces of representations of a riemann surface group in a complex lie group related to point 1. Kahlerricci flow, morse theory, and vacuum structure deformation of n 1 supersymmetry in four dimensions gunara, bobby e. Except for the latter, the main example is complex projective space endowed with the fubinistudy metric.
The latter definition is satisfying, though its tough on people who dont know or like the holonomy. Cohomology of compact hyperkahler manifolds and its applications. It is a tremendous advantage to be able to work with manifolds as abstract topological spaces, without the excess baggage of such an ambient space. 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. Justin sawon at university of north carolina at chapel hill. Hyperk ahler manifolds have been studied mainly due to their appearance in the famous bogomolovs decomposition theorem.
Get the latest, most powerful release 9 technology in a portable installation. Simple to complex with some numerical computations, was completed by mr. We establish a characterization, a chern number inequality, topological and geometrical properties of lagrangian submanifolds. Pdf hyperkahler manifolds with circle actions and the gibbons. Misha verbitsky submitted on 28 aug 2009, last revised 22 may 20 this version, v8 abstract. Then the inequality holds true in an open neighborhood of xsince f and gare continuous. The file must be placed in the shared folder itself and not within a subfolder of the shared folder. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a pdf plugin installed and enabled in your browser. Tu 14 june 2 july, 2010 tufts university medford ma usa an introduction to manifolds. The most fundamental is the k3 surface the only nontrivial example in 4 real dimensions. Hyperkahler manifolds with torsion hktmanifolds were introduced by p. The linear transformation t is angle preserving if t is 11, and for x. Liu, we have proven the injectivity of the period map on the teichmuller space of. A smooth map taking an open set in the plane onto an open set in a surface is called a smooth parametrisation of that part of the surface.
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. Mar 29, 2012 short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. Liu, we have proven the injectivity of the period map on the teichmuller space of polarized and. 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. Swann, modifying hyperkahler manifolds with circle symmetry, imada preprint pp200511, esi prerpint 1725, eprint arxiv. A general theory of hk manifolds was rst developed thirty years ago by bogomolov, fujiki and beauville. A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. Nonnilpotent complex geometry of nilmanifolds and heterotic supersymmetry ugarte, luis and villacampa, raquel, asian journal of mathematics, 2014. Often, especially in older texts, one xes a covering and coordinate patches and writes any tensor in terms. 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. Hyperk ahler manifolds and sheaves daniel huybrechts abstract. The point of view will be that of an algebraic or complex geometer. Moduli of hyperkahler manifolds and the torelli problem. Presentation mode open print download current view.
Many interesting moduli spaces in geometry have hyperkahler structure. This thesis has studied global torelli problems for projective manifolds. Hyperk ahler manifolds, moduli spaces, derived categories, holomorphic symplectic manifolds. The converse is also true in the case of compact manifolds, due to shingtung yau s proof of the calabi conjecture. I certify that this is an original project report resulting from the work completed during this period. We develop the foundation of the complex symplectic geometry of lagrangian subvarieties in a hyperkahler manifold.
Summer school and conference on hodge theory and related topics. As for abelian varieties, the interplay between algebraic, arithmetic, and. The aim was to describe some background results before describing in detail in subsequent talks the results of arxiv. First and foremost is my desire to write a readable but rigorous introduction that gets the. In fact the hyperkahler manifolds of complex dimen sion two are precisely the k3 surfaces. Prop osition 1 l et m b e an irr e ducible c omp a c t hyp erk. Lecture notes geometry of manifolds mathematics mit. This is a version of the morrisonkawamata cone conjecture for hyperkahler manifolds. Hyperkahler metrics and supersymmetry project euclid. Pdf morrisonkawamata cone conjecture for hyperkahler manifolds. Manifold software free download manifold top 4 download. At the time of the original paper of calabis, it seemed that hyperk. Outside of differential geometry, hyperkaehler manifolds are manifolds with quaternionic action on tm.
Hyperkahler manifolds with circle actions and the gibbonshawking ansatz. A hyperkahler manifold m4n is, by definition, endowed with three complex structures i, j and k. Business software downloads manifold system by cda international and many more programs are available for instant and free download. Di erential geometry is the study of manifolds and uses tensoriol objects which take into account this huge redundancy in the actual way that we may choose to describe a given manifold. 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.
Download fulltext pdf modifying hyperkahler manifolds with circle symmetry article pdf available in asian journal of mathematics 104 november 2005 with 33 reads. Find materials for this course in the pages linked along the left. This theorem states that any manifold with a ricci at metric is, up to a nite cover, a direct product of complex tori, calabiyau manifolds and hyperk ahler manifolds. In particular, we have focused on projective manifolds of calabiyau type, which is a generalization of calabiyau manifolds. Kahler manifolds are modelled on complex euclidean space.
Automorphisms of hyperkahler manifolds dipartimento di. 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. Moduli spaces of hyperk ahler manifolds or of sheaves on them are often nonseparated. In fact, all these manifolds were total spaces of cotangent bundles to k. So r b a f g 2 0since the in tegrand is always nonnegative and is positive on some subinterval of a. We describe two constructions of hyperkahler manifolds, one based on a legendre transform, and one on a symplectic quotient. Bimeromorphic automorphism groups of nonprojective hyperkahler. The exposition begins with the definition of a manifold, explores possible additional structures on manifolds, discusses the classification of surfaces, introduces key foundational results for. We will discuss results where this phenomenon re ects interesting geometric aspects, e. Hyperkahler manifolds and nonabelian hodge theory of irregular curves. Given a compact, kahler, holomorphically symplectic manifold m, i. For an excellent survey of these works written from a mathe maticians point of view, the reader is referred to the paper of g.
Pdf modifying hyperkahler manifolds with circle symmetry. Detecting quaternionic maps between hyperkahler manifolds. Alternatively, you can download the file locally and open with any standalone pdf reader. In mathematics, the name hyperkahler is more common, and hyperkahler is used by uninitiates and rarely. Recent developments in string theory have made it an highly attractive area, both for mathematicians and theoretical physicists. Sidharth kshatriya under my guidance during the academic year 20062007. Lagrangian submanifolds in hyperkahler manifolds, legendre transformation by naichung conan leung. Twistor spaces and compact manifolds admitting both kahler and nonkahler structures kamenova, ljudmila, journal of geometry and symmetry in physics, 2017 on holomorphically projective mappings of equidistant parabolic kahler spaces chuda, hana, mikes, josef, peska, patrik, and shiha, mohsen, 2018. Manifold software free download manifold top 4 download offers free software downloads for windows, mac, ios and android computers and mobile devices. We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperk. This book grew out of a graduate course on 3manifolds and is intended for a mathematically experienced audience that is new to lowdimensional topology. Release 9 downloads manifold software gis and database tools.
Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers. Pdf we show that a complete simplyconnected hyperkaehler 4manifold with an isometric triholomorphic circle action is obtained from the. Our hyperkahler manifolds are constructed as moduli spaces of bridgeland stable objects in derived categories of k3 surfaces. In the second part, we prove that the first chern form of the moduli space of polarized calabiyau manifolds, with the hodge metric or the weilpetersson. This is an introductory lecture note on the geometry of complex manifolds.
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