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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, a hyperkähler manifold is a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
(M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
g and satisfy the quaternionic relations I^2=J^2=K^2=IJK=-1. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were first given this name by
Eugenio Calabi Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equa ...
in 1979.


Early history

Marcel Berger's 1955 paper on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(''n'')·Sp(1). Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan and Kraines who have independently proven that any such manifold admits a parallel 4-form \Omega. Bonan's later results include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms \Omega^\wedge\bigwedge^T^*M=\bigwedge^T^*M.


Equivalent definition in terms of holonomy

Equivalently, a hyperkähler manifold is a Riemannian manifold (M, g) of dimension 4n whose
holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...
is contained in the compact symplectic group . Indeed, if (M, g, I, J, K) is a hyperkähler manifold, then the tangent space is a quaternionic vector space for each point of , i.e. it is isomorphic to \mathbb^n for some integer n, where \mathbb is the algebra of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s. The compact symplectic group can be considered as the group of orthogonal transformations of \mathbb^n which are linear with respect to , and . From this, it follows that the
holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...
of the Riemannian manifold (M, g) is contained in . Conversely, if the holonomy group of a Riemannian manifold (M, g) of dimension 4n is contained in , choose complex structures , and on which make into a quaternionic vector space.
Parallel transport In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
of these complex structures gives the required complex structures I, J, K on making (M, g, I, J, K) into a hyperkähler manifold.


Two-sphere of complex structures

Every hyperkähler manifold (M, g, I, J, K) has a
2-sphere A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center' ...
of complex structures with respect to which the
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
g is Kähler. Indeed, for any real numbers a, b, c such that :a^2 + b^2 + c^2 = 1 \, the linear combination :aI + bJ + cK \, is a complex structures that is Kähler with respect to g. If \omega_I, \omega_J, \omega_K denotes the Kähler forms of (g, I), (g, J), (g, K), respectively, then the Kähler form of aI + bJ + cK is :a \omega_I + b \omega_J + c \omega_K.


Holomorphic symplectic form

A hyperkähler manifold (M, g, I, J, K), considered as a complex manifold (M, I), is holomorphically symplectic (equipped with a holomorphic, non-degenerate, closed 2-form). More precisely, if \omega_I, \omega_J, \omega_K denotes the Kähler forms of (g, I), (g, J), (g, K), respectively, then :\Omega := \omega_J + i\omega_K is holomorphic symplectic with respect to I. Conversely,
Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
's proof of the
Calabi conjecture In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswa ...
implies that a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
, Kähler, holomorphically symplectic manifold (M,I,\Omega) is always equipped with a compatible hyperkähler metric. Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques from
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, sometimes under the name ''holomorphically symplectic manifolds''. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic manifold of complex dimension 2n with H^(M)=1 is exactly ; and if the simply connected Calabi–Yau manifold instead has H^(M)\geq 2, it is just the Riemannian product of lower-dimensional hyperkähler manifolds. This fact immediately follows from the Bochner formula for holomorphic forms on a Kähler manifold, together the Berger classification of holonomy groups; ironically, it is often attributed to Bogomolov, who incorrectly went on to claim in the same paper that compact hyperkähler manifolds actually do not exist!


Examples

For any integer n \ge 1, the space \mathbb^n of n-tuples of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s endowed with the flat Euclidean metric is a hyperkähler manifold. The first non-trivial example discovered is the Eguchi–Hanson metric on the cotangent bundle T^*S^2 of the
two-sphere A sphere (from Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ''center ...
. It was also independently discovered by
Eugenio Calabi Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equa ...
, who showed the more general statement that cotangent bundle T^*\mathbb^n of any
complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
has a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
hyperkähler metric. More generally, Birte Feix and Dmitry Kaledin showed that the cotangent bundle of any
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...
has a hyperkähler structure on a
neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
of its zero section, although it is generally incomplete. Due to Kunihiko Kodaira's classification of complex surfaces, we know that any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
hyperkähler 4-manifold is either a
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with а trivial canonical bundle and irregularity of a surface, irregularity zero. An (algebraic) K3 surface over any field (mathematics), field ...
or a compact
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
T^4. (Every
Calabi–Yau manifold In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...
in 4 (real) dimensions is a hyperkähler manifold, because is isomorphic to .) As was discovered by Beauville,Beauville, A. Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differential Geom. 18 (1983), no. 4, 755–782 (1984). the
Hilbert scheme In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a ...
of points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension . This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties. Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to , where denotes the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s and is a finite
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of , are known as asymptotically locally Euclidean, or ALE, spaces. These spaces, and various generalizations involving different asymptotic behaviors, are studied in
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
under the name gravitational instantons. The Gibbons–Hawking ansatz gives examples invariant under a circle action. Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to certain gauge theory equations which arise from the dimensional reduction of the anti-self dual
Yang–Mills equations In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Eu ...
: instanton moduli spaces, monopole moduli spaces, spaces of solutions to Nigel Hitchin's self-duality equations on
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, space of solutions to Nahm equations. Another class of examples are the Nakajima
quiver A quiver is a container for holding arrows or Crossbow bolt, bolts. It can be carried on an archer's body, the bow, or the ground, depending on the type of shooting and the archer's personal preference. Quivers were traditionally made of leath ...
varieties,Nakajima, H. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365–416. which are of great importance in representation theory.


Cohomology

show that the cohomology of any compact hyperkähler manifold embeds into the cohomology of a torus, in a way that preserves the Hodge structure.


Notes


See also

* Quaternion-Kähler manifold * Hypercomplex manifold * Quaternionic manifold *
Calabi–Yau manifold In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...
* Gravitational instanton * Hyperkähler quotient *
Twistor theory In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should ...


References

* * Kieran G. O’Grady, (2011)
Higher-dimensional analogues of K3 surfaces.MR2931873
* * {{DEFAULTSORT:Hyperkahler manifold Structures on manifolds Complex manifolds Riemannian manifolds Differential geometry Quaternions