quaternion-kähler manifolds and where to find them · theory modern and recent developments wolf...

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History and MotivationExamples

TheoryModern and recent developments

Quaternion-Kahler Manifolds and Where to FindThem

Henrik Winther

Masaryk University, Brno, Czech Republic

2019, September 16

Henrik Winther Quaternion-Kahler Manifolds

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Beginnings

Special Holonomy

The subject of quaternionic geometry starts with the classification ofpossible non-symmetric holonomy algebras g of torsion-free affineconnections by M.Berger in 1955. His list contains the algebrasp(1)⊕ sp(n) amongst a few other possibilities. This algebra correspondsto a compact Lie group, and so is a reduction of Riemannian geometry.

This special holonomy property means the structure can be entirelyreconstructed from its metric alone.

The first examples were constructed in the 60’s and 70’s, and thisrepresents the start of several mathematical disciplines:

Quaternion-Kahler geometry, a special holonomy classQuaternionic Geometry, a bundle geometry (and example ofparabolic geometry)Quaternion-Hermitian geometry, a tensor geometry

Remark

Note that in older papers “quaternionic” means QK as it was not yetgeneralized.


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Beginnings

Riemannian properties and HK-QK split

The Berger list also contains a proper subalgebra sp(n) ⊂ sp(1)⊕ sp(n).The holonomy algebra is very important to Riemannian geometers, andso they consider it a different class to QK. Thus metrics g with holonomysp(n) are called hyperKahler. Any metric g of special holonomy asubalgebra of sp(1)⊕ sp(n) is Einstein, satisfying

Ric(g) = λg

for an Einstein constant λ. Quaternion-Kahler metrics have vanishingscalar curvature if and only if they are actually hyperKahler. Thus wehave three classes:

1 Quaternion-Kahler metrics with strictly positive scalar curvature2 HyperKahler metrics, zero scalar curvature3 Quaternion-Kahler metrics with strictly negative scalar curvature

Definition

From now on we will use the Riemannian convention of takingQuaternion Kahler to mean special holonomy a subgroup of sp(1)⊕ sp(n)with non-zero scalar curvature, explicitly excluding the hyperKahler case.


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Wolf SpacesAlekseevskian Spaces

Homogeneous Spaces

Definition

A homogeneous space is a smooth manifold M equipped with a smoothLie group action G ×M → M which is transitive

Pointed homogeneous spaces are equivalent to coset spaces G/H fora closed subgroup H.

Geometric structures (bundles, tensors, differential equations, etc.)on the manifold M which are invariant with respect to the G-actionare called homogeneous geometric structures

Remark

Lie theory provides a formalism allowing efficient and effectivecomputation in the setting of homogeneous structures. See for exampleNomizu’s and Wang’s theorems about homogeneous connections.

This formalism works better for geometry than for analysis, in somesense. Eg. it can only represent constant functions on M. . .


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Riemannian Symmetric Spaces and Holonomy

A (simply connected) Riemannian symmetric space is equivalently

A Riemannian manifold (M, g) with parallel curvature: ∇R = 0.

A Riemannian manifold (M, g) with isometric geodesic involutions .

A homogeneous Riemannian manifold M = G/K with K thecompact stabilizer of an arbitrary point, admitting a decompositiong = k⊕m such that [m,m] ⊂ k

Symmetric spaces are distinguished by being the exceptions to Berger’sholonomy classification. For a Symmetric space (G/K , g), we have

hol(g) = k = Lie(K )

Remark

Sometimes people say that the holonomy should be precisely Sp(1)Sp(n)to be quaternion-Kahler. But that would rule out most QK symmetricspaces. What they actually want is non-trivial projection to Sp(1).


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Quaternion-Kahler Symmetric Spaces

It was show by J.Wolf in 1965 that

Associated to the highest root in a compact simple Lie algebra g,there is a sp(1) subalgebra.

The normalizer k = Ng(sp(1)) realizes (g, k) as a symmetric pair, i.e.[m,m] ⊂ k for a reductive complement m to k.

To each such a pair there is associated a unique irreducible simplyconnected homogeneous space M = G/K up to conjugacy etc.

This is space has holonomy a subalgebra of sp(1)⊕ sp(n) foraforementioned reasons, and

This construction exhausts the simply connected quaternion-Kahlersymmetric spaces of positive scalar curvature.

Remark

These are the only known complete QK manifolds of positive scalarcurvature.


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M = G/K dimH MSp(n + 1)/Sp(1)Sp(n) nSU(n + 2)/S(U(2)U(n)) nSO(n + 4)/SO(4)SO(n) nG2/SO(4) 2F4/Sp(1)Sp(3) 7E6/SU(2)SU(6) 10E7/Sp(1)Spin(12) 16E8/Sp(1)E7 28

Table: Wolf spaces

Remark

Unlike many other classifications, you may not take products of Wolfspaces to get further examples. This list is complete (for positive scalarcurvature).


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Alekseevskian Spaces

While the Wolf spaces arise from simple algebras, D.Alekseevsky classifiedthe opposite end of the homogeneous spactrum in 1975. He classified thehomogeneous QK manifolds with solvable symmetry group. 3 families:

T -spaces, tp = Rp ⊗ V ⊕ Λ20V ⊕ Rω0 where Rp is euclidean and V

is symplectic.

W-spaces, wp = Rp ⊗W ⊕ Rp ⊗W ∗ ⊕ Λ2W ⊕ Rh where Rp iseuclidean and W is euclidean and 4-dimensional.

V-spaces, vk,l = S ⊗ Rl ⊕ U ⊕ Rh where S ,U are the spinor andvector representations of so(3, 3 + k), the map Λ2S → U is given interms of Clifford multiplication, and R l is euclidean.

In each case the final term acts as a derivation in the Lie algebra, and themetric Lie algebra structure is otherwise naturally defined. Of course,these QK manifolds are strictly negative definite.


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In terms of AQH-structuresTwistor theory

Preliminaries: Connections

Affine connections ∇ are some of the key structures in differentialgeometry.

Problem

We cannot immediately compare the values geometric objects fromnearby points on a smooth manifold. Therefore, we also cannotinvariantly differentiate them.


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We can solve this problem by introducing a connection, which isequivalently possible to define using

1 A system of parallel transport, where given a curve γ : I → M wemay transport tangent vectors, forms and tensors from Tγ(0)M toTγ(1)M.

2 A differential operator ∇ : Γ(TM)× Γ(TM)→ Γ(TM), which isC∞-linear in the first argument and satisfies the Leibniz rule

∇X (fY ) = df (X )Y + f∇X y

for the second.

3 A horizontal sub-bundle H in T (TM) defined by requiring that thetangents of parallel transports are horizontal.

We naturally extend ∇ to covariant tensors by(∇X θ)(Y ) = ∇X (θ(Y ))− θ(∇X Y ), and to general tensor fields usingthe Leibniz rule.


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Properties of Connections

We can derive some invariant objects from connections

The torsion tensor T∇ is given byT∇(X ,Y ) = ∇X Y −∇Y X − [X ,Y ]

The curvature tensor R∇ is given by

R∇(X ,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X ,Y ]Z

The holonomy algebra hol(∇) is the Lie algebra of infinitesimalparallellogram transport. It is given byhol(∇) = Span(Im(R∇)) ⊂ End(Tx M) by the famousAmbrose-Singer theorem.

Also of importance to us is that any pseudo-Riemannian metric gprovides a unique torsion-free metric connection ∇, called theLevi-Civita-connection. The curvature and holonomy of this connectionare precisely the Riemannian curvature and holonomy.


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An almost quaternion-Hermitian structure on a manifold M is aRiemannian metric g together with a compatible almost quaternionicstructure Q, i.e. a rank 3 vector sub-bundle Q ⊂ End(TM) which admitsa local frame around any point consisting of local almost complexstructures I , J,K such that

I 2 = J2 = K 2 = IJK = −1,

and g is Hermitian with respect to each of I , J,K . In particulardim M = 4n.


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Let n > 1. The main tensorial invariant of an almostquaternion-Hermitian structure is its fundamental four-form,

ωI ∧ ωI + ωJ ∧ ωJ + ωK ∧ ωK = Ω ∈ Ω4(M),

where ωI , etc are Kahler forms for an arbitrary local frame I , J,K (but Ωis globally defined).


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Equivalent formulations

Theorem

The following are equivalent (for n ≥ 3):

1 The AQH-structure (g ,Q) is either quaternion-Kahler orhyperKahler.

2 The covariant derivative of the fundamental form Ω vanishes:∇Ω = 0

3 The fundamental form Ω is closed: ∂Ω = 0

4 The Levi-Civita connection ∇ is a quaternionic connection:∇ : Γ(Q)→ Γ(Q)

Remark

A manifold M equipped with an algebraic fundamental form Ω ∈ Ω4(M)is AQH. Therefore, it seems that QK geometry can be formulated entirelywithin the DeRham complex.


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Riemannian geometry with torsion

Next we will consider some conditions which are not equivalent toQK/HK. Let’s discuss the general Riemannian situation first.

We have a Riemannian metric g , together with some geometricstructures (usually differential forms, in our case Ω).This is equivalent to a structure group reduction to a propersubgroup G ⊂ SO(n) (in our case G = Sp(1)Sp(n)).The structure is not assumed parallel. Thus the Levi-Civitaconnection ∇ is not a g-connection.

In this situation, we must replace ∇ with an adapted g-connection

∇ = ∇+ A

where in general A ∈ Γ(T ∗ ⊗ so(n)) depends on the covariant derivativeof the extra structure.

Definition

This yields a minimal adapted connection ∇ which has non-zero torsionT (X ,Y ) = AX Y − AY X .


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First order classes

Let ∇ be the Levi-Civita connection of g . Then we may consider thedecomposition into simple modules over K of the space of formalcovariant derivatives of 4-forms,

T ∗x M ⊗ Λ4T ∗x M =∑α

Vα

and the associated equivariant projections πα. Then the equations

πα(∇Ω) = 0

are natural differential equations for almost quaternion-Hermitianstructures. The first order classes, analogous to Gray-Hervella classesfrom Hermitian geometry, are then given by assuming that a structuresatisfies some subset of these equations.


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For n ≥ 3, the intrinsic torsion is contained in 6 simple modules. Thereare 4 for n = 2. Thus we get 64 and 16 classes in general. In theEH-formalism of Salamon, Swann, Cabrera, we writeTx M = E ⊗C H = EH, where E is C2n as an Sp(n)-module and H is Has an Sp(1)-module. Then∑

α

Vα = (Λ30E ⊕ K ⊕ E )⊗ (H ⊕ S3H)

Three classes have particular importance. These are denoted EH, calledLocally conformally Quaternion Kahler, (K + E )H, calledquaternion-Kahler with torsion (QKT) introduced by Ivanov, and(Λ3

0E + K + E )H, called quaternion-Hermitian.


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Locally Conformally QK

The intrinsic torsion of Riemannian reductions can be decomposed into 3modules with respect to so(n):

T ∗ ⊗ Λ2TM = TM ⊕ C (TM)⊕ Λ30TM

These are called connections with torsion exclusively supported in one ofthe three modules are called connections with Vectorial, Cartan andSkew-Symmetric Torsion, respectively.

When the minimal adapted connection has vectorial torsion, thestructure is called locally conformally . . .

In our case . . . is QK. When the torsion form θ is exact, we may globallyrescale the metric to a QK/HK manifold. But generally this is not thecase.


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Twistor spaces of Quaternionic Manifolds

Given an almost quaternionic manifold (M,Q), one may define thesphere-bundle Z → M with fibre

Zx = S2(Qx ) = J ∈ Qx |J2 = −1.

Definition

The total space Z is called the twistor space of the almost quaternionicmanifold M.

Let ∇ be a minimal quaternionic connection (torsion-free if possible).Then the bundle Q is parallel under ∇, so ∇ restricts to a connection onthe total spaces Q and Z . This yields a decomposition into horizontaland vertical subspaces

Ty Z∇= H ⊕ V ' Tx M ⊕ TJX

S2(Qx ),

and since points in Z are of the form y = (x , Jx ), Jx ∈ Qx , we may usethis isomorphism to left multiply by Jx , turning Z into an almost-complexmanifold of complex dimension 2n + 1.


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Integrability of Twistor Spaces

Remark

The decomposition H ⊕ V allows us to make a choice of sign on eachcomponent when defining J. We will always choose the same sign onboth components. The other possible choice has no good properties.

Theorem

Let (M,Q) be an almost quaternionic manifold with twistor space (Z , J).Then J is integrable, i.e. Z is a complex manifold, if and only if Q isquaternionic, i.e. the bundle Q is invariant with respect to sometorsion-free connection.


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Abstract Twistor spaces

Proposition

The twistor space of a quaternionic manifold comes equipped with:

1 A family of projective lines (CP1) with normal bundles C2n ⊗O(1),called twistor lines.

2 A real structure τ without fixpoints, but which leaves some twistorlines invariant. These are called real twistor lines.

Definition

A Twistor Space is a complex manifold of dimension 2n + 1 equippedwith the above structures.

Theorem

Twistor spaces and quaternionic manifolds are in bijectivecorrespondance. The manifold can be recovered as the set of real twistorlines.


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Structures on Twistor space

Many geometric structures on quaternionic manifolds are equivalent tostructures on twistor spaces. Most important to us, connections yielddistributions, and in particular

Theorem

Let Z be the twistor space of a quaternion pseudo-Kahler manifold. ThenZ is a holomorphic contact manifold, i.e. there is a holomorphic contact1-form θ with values in some line bundle, such that

θ ∧ (dθ)n 6= 0

The distribution ker θ is τ -invariant.

Theorem

Let Z be a holomorphic contact twistor space where the contactdistribution is transversal to real twistor lines. Then Z is the twistorspace of a quaternion pseudo-Kahler manifold.


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Positive Sc., finiteness and Conjecture

Even more striking results are possible by leveraging twistor spaces.

Definition

A Riemannian manifold is positive if it is complete and has positive scalarcurvature. In particular such manifolds are compact.

Theorem

For any quaternionic dimension n, There are only finitely many positivequaternion-Kahler manifolds up to isometry and homothethy.

Conjecture

If a quaternion-Kahler manifold is positive, then it is a Wolf space.

This conjecture is true up to at least n = 2.


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Functors and constructionsReference listing

The HK/QK correspondance

The following recent results belong to Alekseevsky, Cortes, Mohaupt andgeneralizes earlier work by Haydys.

Theorem

Let (M, g , I , J,K ) be pseudo-hyperKahler admitting a non-lightlikeKilling vector field Z stabilizing I and rotating the J,K -plane and the

form ωI (Z , ·) = −df is exact, and both f and f − g(Z ,Z)2 are nonzero.

Then one may construct a pseudo-quaternion-Kahler manifold (M ′, g ′)with dim M = dim M ′.

The signature can be controlled by the data to produce QK manifolds.


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HK/QK-formula

In another recent paper, additionally coauthored by Dyckmanns, theyalso produced an explicit formula,

g ′ =1

2|f |gP |M′

gP = gP −2

f

3∑a=0

(θPa )2

with P → M is a certain S1-principal bundle and M ′ is a submanifold inthe total space, and gP = 2

f− g(Z,Z)2

η2 + g , θP0 = 1

2 df , θP1 = η + 1

2 gZ ,

θP2 = 1

2ωK Z and θP3 = −1

2 ωJ Z


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Previously Feix and Kaledin constructed hyperKahler structures on thecotangent spaces of Kahler manifolds. This was generalized byD.Calderbank and A.Borowka to produce twistor spaces from C-projectivestructures. They call this the qFK construction. This locally classifiesquaternic manifolds admitting a maximal totally complex submanifoldfixed invariant under some symmetry. This year Borowka released a paperconstructing quaternion-Kahler manifolds:

Theorem

Let (S , ω, L) be a real-analytic Kahler manifold with holomorphic linebundle L with connection ∇L such that the induced connection onL⊗O(1) is unitary with curvature c · ω. Then we may construct aholomorphic contact twistor space by qFK corresponding to a QKmanifold M. The scalar curvature is 2c and the manifold admits anisometric S1-action with fixpoint-manifold S.


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There is also a converse to Borowka’s theorem in that all QK-manifoldswith such submanifolds S arise in this way.

Remark

The construction is compatible with the HK/QK correspondance.

Remark

Borowka and Winther wrote a paper showing that the qFK constructionis an “infinitesimal functor”. The jury is still out on whether it can bemade into an actual functor.


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Early history, Berger’s list

M. Berger, Sur les groupes d’holonomie homogenes de varietes a connexionaffine et des varietes riemanniennes Bulletin de la Societe Mathematique deFrance, 83, 279-330 (1955).

K. Nomizu, On infinitesimal holonomy and isotropy groups, Nagoya Math. J. 11,111-114 (1957).

K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J.Math., 76, 33–65 (1954)

Beginnings of Quaternion-Kahler geometry

S. Salamon, Quaternionic Kahler Manifolds, Invent. Math. 67, 143–171 (1982).

J. A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetricspaces, J. Math. Mech. 14, 1033-1047 (1965).

D. V. Alekseevskii, Classification of quaternionic spaces with a transitive solvablegroup of motions, Math. USSR-Izv. 9, 297–339 (1975).


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Riemannian structures with torsion and almost Quaternion-Hermitiangeometry

F.Cabrera, A. Swann, The intrinsic torsion of almost quaternion-Hermitianmanifolds, Ann. Inst. Fourier (Grenoble), 58 (5), 1455-1497 (2008)

F. M. Cabrera, Almost quaternion-Hermitian manifolds, Ann. Glob. Anal. Geom.25, 277–301 (2004).

S. Ivanov, I. Minchev, Quaternionic Kahler and hyper-Kahler manifolds withtorsion and twistor spaces, Journal fur die Reine und Angewandte Math. (Crelle’sJournal) 567, 215–233 (2004).

Twistor spaces

S. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. EcoleNorm. Sup. 19, 31–55 (1986)

H. Pedersen, Y. S. Poon, Twistorial construction of quaternionic manifolds,Proceedings of the Sixth International Colloquium on Differential Geometry,Santiago de Compostela, 207–218 (1988)

LeBrun, C., Quaternionic Kahler Manifolds and Conformal Geometry,Math.Ann.,284, 353–376 (1989)


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More Twistor spaces

C. LeBrun, S. Salamon, Strong rigidity of positive quaternion-Kahlermanifolds, Invent. Math. 118, 109–132 (1994)

S. Ivanov, I. Minchev, S. Zamkovoy, Twistors of AlmostQuaternionic Manifolds, arxiv.org/math.DG/0511525

Modern and recent developments

D.V. Alekseevsky, V. Cortes, T. Mohaupt, Conification of Kahler andHyper-Kahler Manifolds, Commun. Math. Phys. 324, 637–655 (2013)

D.V. Alekseevsky, V. Cortes, M. Dyckmanns, T. Mohaupt, Quaternionic Kahlermetrics associated with special Kahler manifolds, J. Geom. Phys. 92, 271–287,(2015)

A. Borowka, D. Calderbank, Projective geometry and the quaternionicFeix-Kaledin construction, Trans. Amer. Math. Soc., Electronically January 4,(2019)

A. Borowka, Quaternion-Kahler manifolds near maximal fixed points sets ofS1-symmetries, arxiv.org/abs/1904.08474


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More modern and recent developments

A. Borowka, H. Winther, C-projective symmetries of submanifolds inquaternionic geometry, Ann Glob Anal Geom 55, 395-416 (2019)


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