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