These manifolds are interesting from several points of view. In differential geometry, these are manifolds with holonomy contained in spn. The ordered triple j j1,j2,j3 is a hyperkahler structure on m,g. In the following section, we describe our constructions, without using supersymmetry, and give examples. By definition, a hyperkahler manifold is a compact simply connected. 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. Andersonkronheimerlebrun1 and goto 8 have constructed these hyperkahler manifolds whose metrics arecomplete and depend on the choice of parameters. A powerful tool for constructing compact hyperkahler manifolds is the famous calabiyau theorem, which provides a ricci. Free riemannian geometry books download ebooks online textbooks. In this article we study the geometry of hyperlagrangian submanifolds in a hyperkahler manifold. Rational curves on hyperkahler manifolds article pdf available in international mathematics research notices 201523 january 2014 with 41 reads how we measure reads. We will discuss results where this phenomenon re ects interesting geometric aspects, e. Besides giving a thorough introduction into kahler geometry, my main aims were cohomology of kahler manifolds, formality of kahler manifolds af. A little more precisely it is a space together with a way of identifying it locally with a euclidean space which is compatible on overlaps.
We prove that the characteristic foliation fon a nonsingular divisor din an irreducible projective hyperk ahler manifold x cannot be algebraic, unless the leaves of fare rational curves or xis a surface. Conversely, any compact k ahler manifold with a holomorphic symplectic form is hyperk ahler. Sidharth kshatriya under my guidance during the academic year 20062007. We prove niteness of the number of smooth blowdowns on fano manifolds and boundedness results for the geometry of non projective fanolike manifolds. As for abelian varieties, the interplay between algebraic, arithmetic, and. Papadopoulos hp were much studied in physics literature since then. In this class and next class, we will focus on 2manifolds, talking about how to. Twistor spaces of hyperkahler manifolds with s1actions. The role of a moment map is to combine the above two situations. This is equivalent to g preserving both the metric g. Donaldson showed that the hflow is related to the mean curvature flow of surfaces in a 4manifold. Hyperkahler quotients and their application to the moduli space of. We say that an action of a group g on m is trisymplectic if it preserves each of the symplectic forms.
Pdf one central problem in real algebraic geometry is to classify the real structures of a given complex manifold. Source of description this bibliographic record is available under the creative. Hyperk ahler manifolds and sheaves daniel huybrechts abstract. Moduli spaces, k3 surfaces, hyperkahler manifolds, mirror symmetry.
Pdf we introduce the notion of a special complex manifold. These strata are hyperkahler manifolds which can be described in terms of open sets in complex symplectic quotients of the cotangent bundle of k c sl n, c by subgroups containing. Compact hyperkahler manifolds in algebraic geometry. A hyperkahler manifold is a riemannian manifold with three covariant constant orthogonal automorphisms i, j and k of the. The main tool of hypercomplex geometry is the socalled obata connection. They can be thought of as quaternionic analogues of 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. Find materials for this course in the pages linked along the left. The example of merge sort that i have on this machine is for a linked list, where you dont run into this problem because the storage for the output list uses the same space as the storage for the input lists. The first construction uses a legendre transform to relate the kahler potentials of certain hyperkahler manifolds to a linear space. Source of description description based on online resource.
Every hyperkahler manifold is complex the complex structure is induced. The point of view will be that of an algebraic or complex geometer. Moduli spaces of hyperkahler manifolds and mirror symmetry ictp. Pdf finiteness of klein actions and real structures on. For an excellent survey of these works written from a mathe maticians point of view, the reader is referred to the paper of g. I certify that this is an original project report resulting from the work completed during this period. Smooth manifolds a manifold, m, is a topological space with a maximal atlas or a maximal smooth structure. Characteristic numbers of compact hyperkahler manifolds are expressed in graphtheoretical form, considering them as a special case of the curvature invariants introduced by l. We can also combine them to form the hyperkahler moment map.
Moduli spaces of hyperk ahler manifolds and mirror symmetry. Hyperkahler manifolds are special classes of kahler manifolds. Verbitsky stable bundles on k ahler manifolds with trivial picard group we would now consider how the space of stable bundles changes when we change the hyperk ahler structure. We express characteristic numbers of compact hyperk\ahler manifolds in graphtheoretical form, considering them as a special case of the curvature invariants introduced by rozansky and witten. Available formats pdf please select a format to send. We establish a characterization, a chern number inequality, topological and geometrical properties of lagrangian submanifolds. Question about hyperkahler manifolds which are deformation. This is an introductory lecture note on the geometry of complex manifolds. Let us now build on hattoris fourdimensional description 15, to combine. Conversely, any compact kahler manifold with a holomorphic symplectic. We develop the foundation of the complex symplectic geometry of lagrangian subvarieties in a hyperkahler manifold. These notes intend to explain the analogue of the product decomposition of the moduli space of metrics on a calabiyau manifold in the hyperk ahler situation and to show how mirror symmetry for k3 surfaces, i. Hyperk ahler manifolds have been studied mainly due to their appearance in the famous bogomolovs decomposition theorem.
Simple to complex with some numerical computations, was completed by mr. Thus we are able to see relations between certain complex manifolds which are. Hyperk ahler manifolds, moduli spaces, derived categories, holomorphic symplectic manifolds. 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. 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. Id need to think hard about that it is a good question. Each point in xc gives rise to a twistor line joining a point in x. K3 surfaces and their higherdimensional analogs, hyperk ahler 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. 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 solution manual is written by guitjan ridderbos. The most fundamental is the k3 surface the only nontrivial example in 4 real dimensions. February 2, 2000 communicated by thomas peternell abstract.
Simons collaboration on special holonomy in geometry. We also introduce and study extensively a normalized legendre transformation of lagrangian subvarieties under a birational transformation of projective hyperkahler manifolds. Kahler manifolds are modelled on complex euclidean space. 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. Simons collaboration on special holonomy in geometry, analysis and physics home page spaces with special holonomy are of intrinsic interest in both mathematics and mathematical physics.
Arnaud beauville the chow ring of hyperkahler manifolds. A stable geodesic is a local minima of the length functional among curves in its homotopy class. Combining this with the fact that cpn admits a kahler metric. The most recent addition to the list is the link between hyperkahler geometry and theoretical physics. Some boundedness results for fanolike moishezon manifolds laurent bonavero and shigeharu takayama received. It is a natural sequel to my earlier book on topological manifolds lee00. 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. Hyperkahler manifolds and nonabelian hodge theory of. 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. In the series university of florida digital collections. We show that the number of equivariant deformation classes of real structures in a given deformation class of compact hyperk. Nonnilpotent complex geometry of nilmanifolds and heterotic supersymmetry ugarte, luis and villacampa, raquel, asian journal of mathematics, 2014. Combining these two maps, we obtain an isomorphism. Simple examples of hyperkahler manifolds are provided by quaternionic vector spaces hk and by the cotangent bundle of complex manifolds. Introduction to differentiable manifolds lecture notes version 2. This satisfactory state of affairs can be used to prove the existence of hyperkahler metrics on many examples of complex manifolds. A few new topics have been added, notably sards theorem and transversality, a proof that infinitesimal lie group actions generate global group actions, a more thorough study of firstorder partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.
Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s. Lagrangian submanifolds in hyperk\ahler manifolds, legendre. Hyperkahler hk manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Calabiyau manifolds and related geometries springerlink. An introduction to 3manifolds 5 in the study of surfaces it is helpful to take a geometric point of view. A hyperk ahler manifold m is a complex manifold with a holomorphic symplectic form. Thus there is more than one natural complex structure on m.
Geometry of moduli spaces of meromorphic connections on curves, stokes data, wild nonabelian hodge theory, hyperkahler manifolds, isomonodromic deformations, painleve equations, and relations to. Some boundedness results for fanolike moishezon manifolds. An instanton e, d on a pseudohyperkahler manifold m is a vector. 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. A hyperkahler manifold is a riemannian 4nmanifold with a family of almost complex. Many interesting moduli spaces in geometry have hyperkahler structure. Lecture notes geometry of manifolds mathematics mit. The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry. Given a hypercomplex manifold m, there exists a unique torsionfree connection on the tangent bundle tm which preserves the quaternion action. The semichiral quotient, hyperkahler manifolds and tduality article pdf available in journal of high energy physics 201210 december 2011 with 25 reads how we measure reads. As an application we derive a global construction of the fourth order transgression of the chern character forms of hyperholomorphic bundles over compact.
Outside of differential geometry, hyperkaehler manifolds are manifolds with quaternionic action on tm. Moduli spaces of hyperk ahler manifolds or of sheaves on them are often nonseparated. Lagrangian submanifolds in hyperkahler manifolds, legendre. We follow the book introduction to smooth manifolds by john m. Then mis called a kahler manifold and the metric on ma kahler metric. Characteristic foliation on nonuniruled smooth divisors on projective hyperkahler manifolds ekaterina amerik, fred eric campana abstract. A geodesic is stable if its second variation is nonnegative. Lets call a coherent sheaf f simple if f has no subsheaves f0. Pdf curvature and characteristic numbers of hyperk\ahler. A hyperkahler manifold is a hypercomplex manifold equipped with a metric g. Hyperkahler manifolds and birational transformations. The latter definition is satisfying, though its tough on people who dont know or like the holonomy. If two compact 1 manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen to preserve the orientation this means that the corresponding cobordism is an oriented 2manifold and its orientation induces on the boundary the given orientation on one of the 1 manifolds and.
All the known hk manifolds are deformations of one of the beauville examples or of one of our two examples. Riemannian holonomy groups and calibrated geometry. Moduli spaces of hyperkaehler manifolds and mirror symmetry. In other words, manifolds are made up by gluing pieces of rn together to make a more complicated whole. A hyperkahler manifold m is a complex manifold with a holomorphic symplec tic form. Manifolds whether you are looking for a custom, hand built, equal length oneoff manifold, or are requiring jig built smallmedium batch work, our wide ranging, multidisciplinary knowledge and expertise can help you with design, development and manufacture of all types of stainless steel exhaust manifolds. Calculus on manifolds a solution manual forspivak1965 jianfei shen school of economics, the university of new south wales sydney, australia 2010. Pdf the semichiral quotient, hyperkahler manifolds and.
This section describes only the manifold part of sagemanifolds. We discuss a category of lagrangian subvarieties and its relationship with the theory of lagrangian intersection. Alex degtyarev, ilia itenberg, viatcheslav kharlamov abstract. The proof relies on the twistor correspondence for hyperkahler manifolds. Hyperkahler manifolds with torsion hkt manifolds were introduced by p. Hyperkahler manifolds and birational transformations y. For example, cit dn c hit and any surface in an oriented four manifold are examples of hyper lagrangian submanifolds.
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