Summer school and conference on hodge theory and related topics. 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. Toric hyperkahler manifolds with quaternionic kahler bases. Hyperkahler geometry and invariants of threemanifolds. Lecture notes geometry of manifolds mathematics mit. A hyperkahler manifold is a riemannian manifold with three covariant constant orthogonal automorphisms i, j and k of the. This point of view provides guidance in the search for examples of hyperkahler manifolds, and elucidates the sort of differential equation which. A stable geodesic is a local minima of the length functional among curves in its homotopy class. The point of view will be that of an algebraic or complex geometer. Relative moduli spaces of periodic monopoles provide novel examples of asymptotically locally flat hyperkahler manifolds. A feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3manifolds which is. In conclusion, we want to mention another class of kahler manifolds, namely the socalled hyperkahler manifolds, which are widely discussed in the context of.
Im aware that there are a number of torelli type theorems now proven for compact hyperkahler manifolds. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Let m,i be a compact kahler manifold which admits a holomorphic symplectic form for. 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.
Ricciflat manifolds have been studied for many years and the interest in various aspects of their theory seems still to be growing. Cohomology of compact hyperkahler manifolds and its applications. This is a summary of the work in the authors recent paper with this title written with jerome gauntlett, george papadopoulos and paul townsend, hepth97022012. Quantum invariants of knots and 3 manifolds pdf bulletin new series of the.
Hyperkahler hk manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. We also introduce and study extensively a normalized legendre transformation of lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. In the present work some examples of toric hyperkahler metrics in eight dimensions are constructed. Besides giving a thorough introduction into kahler geometry, my main aims were cohomology of kahler manifolds, formality of kahler manifolds af. Hyperkahler manifolds and nonabelian hodge theory of irregular curves. Of the form m \ d, where m is a del pezzo surface, d. Once you merge pdfs, you can send them directly to your email or download the file to our computer and view. The geometry and topology of toric hyperkahler manifolds. Hyperkahler manifolds and nonabelian hodge theory of. The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry.
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. We discuss a category of lagrangian subvarieties and its relationship with the theory of lagrangian intersection. A general theory of hk manifolds was rst developed thirty years ago by bogomolov, fujiki and beauville. Hyperkahler manifolds 3 iiispaces of representations of a riemann surface group in a complex lie group related to point 1. This volume introduces hyperkahler manifolds to those who have not previously studied them. Find materials for this course in the pages linked along the left. Hence x admits a hyperkahler struc ture by a theorem of yau. Hyperkahler manifolds and birational transformations 2002 cached. Pdf we introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of sun actions.
For an excellent survey of these works written from a mathe maticians point of view, the reader is referred to the paper of g. These manifolds are important in string theory as we can have unbroken supersymmetry if we compactify stringm theory on the corresponding compact manifold. Click add files and select the files you want to include in your pdf. I certify that this is an original project report resulting from the work completed during this period. Sidharth kshatriya under my guidance during the academic year 20062007. K3 surfaces and their higherdimensional analogs, hyperk ahler. We announce the structure theorem for theh 2mgenerated part of cohomology of a compact hyperkahler manifold. It is a natural sequel to my earlier book on topological manifolds lee00. A nonlinear generalization of the dirac operator in 4dimensions. Pdf quaternionic kahler and hyperkahler manifolds with. This satisfactory state of affairs can be used to prove the existence of hyperkahler metrics on many examples of complex manifolds. Question about hyperkahler manifolds which are deformation. Cohomology of hyperkahler manifolds perimeter institute.
First it is described how the calderbankpedersen metrics arise as a consequence of the joyce description of selfdual structures in four dimensions, the jonestod correspondence and a result due to tod and przanowski. Compact hyperkahler manifolds in algebraic geometry. Involution of hyper kahler manifolds and branes 3 henceforth, unless speci ed otherwise, iwould be taken as the complex structure on x. 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. By a theorem of bogomolov, these decompose into a product of complex tori, calabiyau varieties and irreducible symplectic manifolds. Simon donaldson collapsing coassociative fibrations. Hyperkahler manifolds are special classes of kahler manifolds. I will talk about compact hyperkahler manifolds, which generalize the famous k3 surface to the higher dimensions. Alex degtyarev, ilia itenberg, viatcheslav kharlamov abstract. If we view hyperk\ahler manifolds as abelian varieties, then those of jacobian type should be viewed as jacobian varieties. A positive current is a positive p,pform with coefficients in measures. K3 surfaces and their higherdimensional analogs, hyperk ahler manifolds.
This gives a vast class of knot invariants and 3manifold invariants as well as a class of linear. Characteristic numbers of compact hyperkahler manifolds are expressed in graphtheoretical form, considering them as a special case of the curvature invariants introduced by l. Almost hermitian manifolds with closed symplectic form. Pdf examples of compact holomorphic symplectic manifolds. Moduli spaces of hyperk ahler manifolds or of sheaves on them are often nonseparated. By considering the interactions between wellseparated periodic monopoles, we infer the asymptotic behavior of their metrics. Lagrangian submanifolds in hyperkahler manifolds, legendre transformation. Abstract in this paper we define the notion of a hyperk\ahler manifold potentially of jacobian type. Source of description description based on online resource. Hi there, i have some questions about the mirror symmetry of hyperkahler manifold and k3 surface. Once files have been uploaded to our system, change the order of your pdf documents. Certain classes of homogeneous kahler manifolds arising from the classical groups, including explicit kahler potentials, have also been treated in 2. 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.
These are notes of lectures on kahler manifolds which i taught at the university of bonn and, in reduced form, at the erwinschr. Hyperkahler manifolds and birational transformations y. We establish a compact analog of the p w conjecture. M is a smooth anti canonical divisor del pezzo surfaces are complex surfaces classi. Simple to complex with some numerical computations, was completed by mr. 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. In complex dimension four one can find universal bounds for all of these invariants beauville, guan. How to merge pdfs and combine pdf files adobe acrobat dc. For an arbitrary variety v, let v reg denote the nonsingular part of v. 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.
A hyperkahler manifold hk is a simply connected compact kahler manifold whose space of global holomorphic twoforms is spanned by a. Calabiyau theorem, bochner vanishing and hodge theory misha verbitsky cohomology in mathematics and physics euler institute, september 17, 20, st. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to. Part iii provides constructions of modular categories, based on. Irreducible symplectic manifolds and orthogonal modular forms. In this talk we will describe some results about betti, hodge, and characteristic numbers of compact hyperkahler manifolds. Donaldson showed that the hflow is related to the mean curvature flow of surfaces in a 4manifold. Lagrangian submanifolds in hyperkahler manifolds, legendre transformation by naichung conan leung. 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.
The ordered triple j j1,j2,j3 is a hyperkahler structure on m,g. Many interesting moduli spaces in geometry have hyperkahler structure. Existence of the twistor families is a consequence of yaus solution of the calabi conjecture. Simple examples of hyperkahler manifolds are provided by quaternionic vector spaces hk and by the cotangent bundle of complex manifolds. Papadopoulos hp were much studied in physics literature since then. 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. Tu 14 june 2 july, 2010 tufts university medford ma usa an introduction to manifolds. Citeseerx hyperkahler manifolds and multiplyintersecting. A geodesic is stable if its second variation is nonnegative. The most fundamental is the k3 surface the only nontrivial example in 4 real dimensions.
Pdf hyperkahler manifolds with circle actions and the gibbons. Pdf toward a classification of compact homogeneous complex manifolds with. Source of description this bibliographic record is available under the creative. Recall that, with the cupproduct form, h2x b r 3,19. We will discuss results where this phenomenon re ects interesting geometric aspects, e. Pdf merge combine pdf files free tool to merge pdf. Hyper kahler manifolds and branes indranil biswas and graeme wilkin abstract.
Pdf we show that a complete simplyconnected hyperkaehler 4manifold with an isometric triholomorphic circle action is obtained from the. Pdf we present a new construction of hyperkhler metrics that derives from the 3dimensionaln4 nonlinear supermultiplet. A lot of interesting things have been discovered for these manifolds such as mirror symmetry for calabiyau manifold, and homological mirror symmetry involving calibrated sub manifolds. The aim was to describe some background results before describing in detail in subsequent talks the results of arxiv. Hyperkahler manifolds and birational transformations. Lagrangian submanifolds in hyperkahler manifolds, legendre. In the series university of florida digital collections. We study complex lagrangian submanifolds of a compact hyperk ahler manifold and prove two results.
In this paper, the metric on the moduli space of the k 1 sun pe riodic instanton or caloron with arbitrary gauge holonomy at spatial in. Geometry of moduli spaces of meromorphic connections on curves, stokes data, wild nonabelian hodge theory, hyperkahler manifolds, isomonodromic deformations, painleve equations, and relations to. Kobayashi pseudometric on hyperkahler manifolds core reader. Quaternionic maps qmaps between hyperkahler manifolds are. Andersonkronheimerlebrun1 and goto 8 have constructed these hyperkahler manifolds whose metrics arecomplete and.
Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s. The geometry and topology of toric hyperkahler manifolds 731 the inequalities of 2. Also, by a small birational contraction, we will mean. Hyperk ahler manifolds have been studied mainly due to their appearance in the famous bogomolovs decomposition theorem. Hyperkahler manifolds dedicated to the memory of andrey todorov banach center, 1214. Short survey based on talk given at the institut henri poincare january 17th 2012, during program on surface groups. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Z for integer cohomology, we get torsion for small n. You can merge pdfs or a mix of pdf documents and other files.
The most recent addition to the list is the link between hyperkahler geometry and theoretical physics. Hyperk ahler manifolds and sheaves daniel huybrechts abstract. On the other hand hk varieties are very interesting from an algebrogeometric point of. 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. We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperk.
They can be thought of as quaternionic analogues of kahler manifolds. So all holomorphic objects on xare with respect to iunless speci ed otherwise. Hyperkahler manifolds with torsion hkt manifolds were introduced by p. General ihsmhilbert schemes of k3 surfaces algebra generated by h2sn. Do hyperkahler manifolds live in quaternionickahler families. Click, drag, and drop to reorder files or press delete to remove any content you dont want. Instantons, monopoles and toric hyperkahler manifolds. We also describe how these bounds are related to the question. For a holomorphic symplectic variety with a lagrangian. Andre has considered some families of hyperkahler or holomorphic symplectic manifolds in some paper. Tianyau spaces complete nonsingular noncompact hyperkahler space asymptotic to a nilfold bundle over a line.