L2-index theory, the Chern conjecture, and manifolds of...

L2-index theory, the Chern conjecture,and manifolds of special holonomy

DISSERTATION

zur

Erlangung des Doktorgrades (Dr. rer. nat.)

der

Mathematisch-Naturwissenschaftlichen Fakultät

der

Rheinischen Friedrich-Wilhelms-Universität Bonn

vorgelegt von

Michael Hoffmannaus

München

Bonn, Februar 2014

Angefertigt mit Genehmigung der Mathematisch-NaturwissenschaftlichenFakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn

1. Gutachter: Prof. Dr. Werner Ballmann

2. Gutachter: Prof. Dr. Werner Müller

Tag der Promotion: 25.04.2014

Erscheinungsjahr: 2014

Abstract

In this dissertation we study a conjecture attributed to Chern and Hopf, ageneralization of Atiyah’s L2-index theorem used by Gromov to prove saidconjecture for Kähler manifolds, and an extension to Kähler orbifolds and allknown examples of quaternionic Kähler manifolds. The study of a two-formη with values in a bundle G over a quaternionic Kähler manifold motivatesus to show vanishing results for the cohomology of so5(C)-modules. We usethese results to show that in certain degrees all G-valued d∇-closed forms ared∇-exact.

Contents

Introduction 3

1 The Chern conjecture and cohomology vanishing 111.1 (Quaternionic) Hermitian linear algebra and Lefschetz theorems 121.2 L2-vanishing theorems for (quaternionic) Kähler manifolds . . 171.3 Kähler orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 The Chern conjecture for quaternionic Kähler manifolds . . . 21

2 Differential forms and quaternionic geometry 252.1 Forms on quaternionic Hermitian manifolds . . . . . . . . . . 262.2 Parallel forms on quaternionic Kähler manifolds . . . . . . . . 322.3 The map Lη and vanishing theorems . . . . . . . . . . . . . . 35

3 The cohomology of so5(C)-modules 393.1 Lie algebra cohomology . . . . . . . . . . . . . . . . . . . . . . 403.2 The so5(C)-action on

∧∗(Hn)⊗R C . . . . . . . . . . . . . . . 413.3 Cohomology of so5(C)-modules . . . . . . . . . . . . . . . . . 463.4 Applications to quaternionic geometry . . . . . . . . . . . . . 54

4 L2-index theory 574.1 Atiyah’s L2-index theorem for Dirac operators . . . . . . . . . 574.2 G-twisted bundles . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 The algebra of equivariant smoothing operators . . . . . . . . 654.4 The L2-index theorem . . . . . . . . . . . . . . . . . . . . . . 68

1

Introduction

The Chern conjecture, which has long been open, provides a correspondencebetween the sectional curvature K and the Euler characteristic χ(M) of aclosed 2n-dimensional Riemannian manifoldM . It states that K < 0 implies(−1)nχ(M) > 0 while K > 0 implies χ(M) > 0. A variant states thatK ≤ 0 implies (−1)nχ(M) ≥ 0 while K ≥ 0 implies χ(M) ≥ 0. By theGauß–Bonnet theorem this is true for two- and four-dimensional manifolds(see [Che55] for 2n = 4, where Chern attributes the proof to Milnor; Berger[Ber00, page 6/7] attributes the conjecture to Hopf, see [Hop26]).

Consider d + d∗ : Aeven(M) → Aodd(M). By Hodge theory the Eulercharacteristic is ind(d+ d∗).

Let Γ → M → M be a normal Riemannian covering of a closed Rie-mannian manifold and D the pullback of an elliptic operator D on M . ByAtiyah’s L2-index theorem the Γ-index equals the index of D:

indΓ(D) = dimΓ(ker(D))− dimΓ(ker(D∗)) = ind(D)

(see [Ati76, 1.1] or restricted to Dirac operators, Theorem 4.1.3).For complete simply connected 2n-dimensional Riemannian manifolds

N with non-positive sectional curvature Dodziuk (and Singer) conjectured([Dod79, Conjecture 2]) that the space of square integrable harmonic formsHk

2(N) vanishes for k 6= n. Then by Atiyah’s L2-index theorem for theoperator d+ d∗ the conjecture

(−1)nχ(M) = (−1)n ind(d+ d∗) = dimΓ(Hn2 (M)) ≥ 0

holds (with M = N the universal cover ofM). For closed manifolds with neg-ative sectional curvature, if one can show furthermore the non-vanishing ofHn

2 (M), the Chern-conjecture follows. Nevertheless, in [And85, main result]there is a counterexample N (without compact quotient) to the Dodziuk–Singer conjecture.

Gromov showed this results about square integrable harmonic forms forKähler manifolds with negative sectional curvature and compact quotient (as

4

universal cover of a closed orbifold) in [Gro91, Proposition 1.3.A, Theorem2.5.].

To show the vanishing of Hk2(M) for k 6= n, he used the hard Lefschetz

theorem for Kähler manifolds. This was generalized to the Dodziuk–Singerconjecture (i.e., non-positive sectional curvature) for Kähler manifolds in[CX01] and [JZ00]. For quaternionic Kähler manifolds we show the Dodziuk–Singer conjecture except for the degrees n± 1 in Theorem B.

For the non-vanishing of the harmonic square integrable n-forms (see[Gro91, 2.5]) he used, without proof, a twisted variant of Atiyah’s L2-indextheorem for good orbifolds (see [Gro91, 2.3 A and A’]). Ballmann suggesteda more general and systematic notion of G-twisted Dirac bundles. Withoutgiving the proof he states the analog of Atiyah’s L2-index theorem for Diracoperators associated withG-twisted Dirac bundles (see [Bal06, Theorem 8.27]and [Gro91, Remark 2.3 A’] pointing in this direction). Let G be an extensionof Γ by a compact Lie group K, i.e.,

1 −→ K −→ G −→ Γ −→ 1

is a short exact sequence of Lie groups (as above Γ → M → M is a nor-mal Riemannian covering). A G-twisted Dirac bundle E → M is a gradedDirac bundle (in the sense of Gromov–Lawson [GL83]) with a G-action thatis compatible with the Γ-action on M , the Hermitian metric on E, the con-nection ∇E, the Clifford multiplication γ : TM ⊗ E → E, and the grading ofE (cf. Definition 4.2.1)). We do not need to assume that the Γ-action on Mis free. Hence, we get an index theorem for good orbifolds M = M/Γ.

Theorem A (Theorem 4.4.1). Let Γ → M → M be a Riemannian orbifoldcovering, where M is a complete manifold and M = M/Γ a good orientedclosed orbifold. Let E be a G-twisted Dirac bundle over M . Then the G-dimensions of the kernels ker(D+) and ker(D−) are finite and the G-indexis the integral of the index form ωD+ over a compact fundamental domainF ⊂ M :

indG(D+) = dimG(ker(D+))− dimG(ker(D−)) =

∫F

ωD+ .

This generalizes the theorem given by Gromov in [Gro91, Theorem 2.3.A]that is also proven by Marcolli and Mathai [MM99, Theorem 1.1]. The proofof Theorem A uses methods of the heat kernel proof of Atiyah’s L2-indextheorem given in [Roe98, Chapter 13].

First we show that the Dirac bundles we consider have bounded geometry.Then we introduce a trace trG on the algebra AG of G-equivariant smoothing

5

operators. The spaces ker(D±) are closed G-invariant subspaces of L2(E)and the orthogonal projection P± : L2(E) → ker(D±) is in AG. Hence,we can define dimG(ker(D±)) = trG(P±). Furthermore, we prove that theassociated supertrace strG(e−tD

2) is independent of t. Moreover, we establish

the equation

limt→∞

strG(e−tD2

) = strG(Pker(D)) = indG(D+)

by using estimates for bundles of bounded geometry. Finally we can apply thelocal index theorem for Dirac bundles over complete manifolds of boundedgeometry ([Roe88, 2.11]) and get limt→0 strG(e−tD

2) =

∫FωD+ for the index

form ωD+ . In Section 1.3 we discuss the proofs of different generalizations ofthe Chern conjecture to Kähler orbifolds with negative or non-positive sec-tional curvature and different Euler characteristics (part of this was alreadyproven by Gromov).

Furthermore, we discuss the Chern conjecture for other manifolds of spe-cial holonomy. The only other class of manifolds with special holonomy wherethe Chern conjecture is open is the one of the quaternionic Kähler manifoldsthat we introduce now.

Let (M, g) be a Riemannian manifold. A Riemannian metric g is calledHermitian for an almost-complex structure J : TM → TM if it is compatiblein the sense that g(JX, JY ) = g(X, Y ). An (almost-) Hermitian manifold(M,J, g) is Kähler if J is parallel with respect to the Levi–Civita connection.Then the Kähler form ωJ(X, Y ) = g(JX, Y ) is parallel and hence closed.

A manifold M is almost-quaternionic if there is a vector bundle G ⊂End(TM) and a covering (Ui) of M such that G|Ui is spanned by three localalmost-complex structures I, J,K with IJK = − id. An almost-quaternionicRiemannian manifold (M,G, g) is quaternionic Hermitian if g is Hermitianfor all of the local almost-complex structures above. On an open set Ui wecan define ωI(X, Y ) = g(IX, Y ), ωJ(X, Y ) = g(JX, Y ), and ωK(X, Y ) =g(KX,Y ). Then the Kraines four-form Ω and the G-valued fundamentaltwo-form η are given locally by

Ω = ωI ∧ ωI + ωJ ∧ ωJ + ωK ∧ ωK ,

η = ωI ⊗ I + ωJ ⊗ J + ωK ⊗Kand are globally well-defined. A quaternionic Hermitian manifold is quater-nionic Kähler if G is parallel with respect to the Levi–Civita connection ∇.On quaternionic Kähler manifolds the fundamental two-form and the Krainesfour-form and are parallel and hence closed.

The holonomy of Kähler manifolds is contained in U(n), the holonomy ofquaternionic Kähler manifolds in Sp(n) Sp(1). Riemannian manifolds with

6

holonomy in Sp(n) = U(2n)∩Sp(n) Sp(1) are called hyperkähler. We excludethem in the definition of quaternionic Kähler manifolds. Since they are Ricci-flat, the Chern conjecture is not open for them.

Let M be a closed 4n-dimensional quaternionic Kähler manifold. Theodd Betti numbers b2m+1 with 2m + 1 < n or 2m + 1 > 3n vanish (see[SW02, Proposition 5.8]). If, furthermore, the scalar curvature of M is pos-itive, all odd Betti numbers vanish (see [Sal82, Theorem 6.6]). Hence, theChern conjecture holds for quaternionic Kähler manifolds with positive scalarcurvature. We prove that the Chern conjecture holds for locally symmetricquaternionic Kähler manifolds of finite volume using Hirzebruch proportion-ality. There are as of today no other known examples of closed quaternionicKähler manifolds.

We show the following L2-vanishing theorem for quaternionic Kähler man-ifolds. It proves for them the Dodziuk–Singer conjecture with exception ofthe degrees k = 2n± 1.

Theorem B (Theorem 1.2.5). Let (M,G, g) be a simply connected complete4n-dimensional quaternionic Kähler manifold. If the sectional curvature Kis nonpositive, we obtain

Hk2(M,E) = 0 for k /∈ 2n− 1, 2n, 2n+ 1 .

Here Hk2(M,E) is the L2-cohomology with values in a flat Hermitian bundle

E.

To prove this theorem, we use a Lefschetz-type theorem for the Krainesfour-form Ω given in [Bon82, Theorem 1].

We use the map

Lη : Ak(M) −→ Ak+2(M,G), Lη(α) = η ∧ α

and the vanishing theorems for forms on quaternionic Kähler manifolds toprove similar theorems for G-valued forms.

If we extend the wedge product to G-valued forms using the compositionin End(TM), we obtain η ∧ η = −Ω. Moreover, the curvature of G is up toa constant λ the commutator with η:

d∇d∇(α) = λ(η ∧ α− α ∧ η). (1)

Hence, the image of Lη is in the kernel of the curvature of G. The converseis true in many degrees as one can see in the following theorem.

7

Theorem C (Theorem 2.3.1). LetM be a 4n-dimensional quaternionic Käh-ler manifold. Then

0 −→ A`(M)Lη−−→ A`+2(M,G)

d∇d∇−−−→ A`+4(M,G) −→ 0 (2)

is a chain complex. It is exact in A`(M) for ` < 2n. We obtain d∇d∇ = 0if and only if M is hyperkähler. Otherwise, the chain complex is exact inA`+2(M,G) if `+ 2 < 2n or `+ 2 > 3n.

In Theorem 2.1.9 we give a generalization to quaternionic Hermitian man-ifolds. There we use the right-hand side of Equation (1) instead of d∇d∇.

Theorem C allows us to transfer the results about the cohomology ofquaternionic Kähler manifolds to d∇-closed and d∇-exact forms in A∗(M,G).

Theorem D (Theorem 2.3.3). Let (M,G, g) be a complete 4n-dimensionalquaternionic Kähler manifold. If M is closed, the d∇-closed (2m+ 1)-formsin A2m+1(M,G) are d∇-exact for 2m− 1 < n. If furthermore M has positivescalar curvature, this is also true for m < n.

(Theorem 2.3.5) If M is simply connected with nonpositive sectional cur-vature for every d∇-closed (square integrable) m-form α ∈ Am

2 (M,G) withm < 2n, there is an (square integrable) form γ ∈ Am−1

2 (M,G) with d∇(γ) =α.

For all forms α ∈ A∗(M) the equality d∇Lη(α) = Lη(dα) holds. There-fore, all d-closed forms are mapped to d∇-closed forms. Lη is injective onA`(M) for ` < 2n. Consequently, for k < 2n a d∇-closed G-valued k-form βis Lη(α) for a closed k − 2−form α. Therefore, on closed manifolds we canuse the vanishing results of [SW02, Proposition 5.8] and [Sal82, Theorem6]. If M is complete, negatively curved, and simply connected we use ourL2-cohomology vanishing Theorem B.

Theorem C is a corollary of Lemma E about quaternionic Hermitian vec-tor spaces Hn. Define ωI , ωJ , and ωK in

∧2(Hn) as above (we consider Hn in∧∗(Hn) as a real vector space with the extra structure to define these two-forms). Let LI , LJ , and LK :

∧∗(Hn) →∧∗(Hn) denote the wedge product

with ωI , ωJ , and ωK .

Lemma E (Lemma 2.1.10). Let k < 2n or k > 3n, and let αI , αJ , αK ∈∧k(Hn) be three k-forms with

LI(αJ) = LJ(αI), LJ(αK) = LK(αJ), and LI(αK) = LK(αI).

Then there is a β ∈∧k−2(Hn) with

LI(β) = αI , LJ(β) = αJ , and LK(β) = αK .

8

The operators LI , LJ , and LK generate, together with their adjointsΛI ,ΛJ , and ΛK (ΛI = ∗−1 LI ∗), the Lie algebra so5(C) acting on∧∗(Hn)⊗R C. This result of Verbitskiı [Ver90] is an extension of the sl2(C)-action on

∧∗(Cn)⊗R C used to prove the hard Lefschetz theorem on Kählermanifolds. Verbitskiı’s motivation is to describe the cohomology of closedhyperkähler manifolds. Lemma E is applicable to the cohomology of hyper-kähler manifolds. In Chapter 3 we use this action to translate Lemma E intoLie algebra cohomology. It is equivalent to the (1.)-case of Theorem F.

Let n ⊂ so5(C) be the abelian Lie algebra spanned by LI , LJ and LK .We prove Lemma E by showing a vanishing result for the first cohomology of∧∗(Hn)⊗R C as an so5(C)-module using [Kos61, Theorem 5.14]. The coho-mology Hn(n, V ) can be defined as the homology of the Chevalley–Eilenbergcomplex described below (see [CE56, Section XIII.7., XIII.8.]).

Let E(n) be the exterior algebra over n. Then the complex is givenby Hom(E(n), V ). A q-cochain f : E(n) → V is therefore a C-multilinearalternating function of q variables in n with values in V . The coboundary δis defined as

(δf)(x1, . . . , xq+1) =∑

1≤i≤q+1

(−1)i+1xif(x1, . . . , xi, . . . , xq+1)

+∑

1≤i<j≤q+1

(−1)i+jf([xi, xj], . . . , xi, . . . , xj, . . . , xq+1).

For this notion of cohomology and

H = [LI ,ΛI ] = [LJ ,ΛJ ] = [LK ,ΛK ]

we prove the following vanishing theorems:

Theorem F (Theorem 3.3.1). Let n ⊂ so5(C) and H be as above. The coho-mology groups Hq(n,

∧∗(Hn)⊗RC) do not vanish if and only if q ∈ 0, 1, 2, 3.Decompose Hq(n,

∧∗(Hn) ⊗R C) into the eigenspaces of the H-action as⊕j∈ZH

q(n,∧∗(Hn)⊗R C)j with H · vj = jvj for vj ∈ Hq(n,

∧∗(Hn)⊗R C)j.Then

0. The space H0(n,∧∗(Hn) ⊗R C)j does not vanish if n ≤ j ≤ 2n and

vanishes for j > 2n.

1. The space H1(n,∧∗(Hn)⊗R C)j does not vanish if and only if

−2 ≤ j ≤ n− 2.

2. The space H2(n,∧∗(Hn)⊗R C)j does not vanish if and only if

−n− 4 ≤ j ≤ −4.

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3. The space H3(n,∧∗(Hn)⊗RC)j does not vanish if −2n−6 ≤ j ≤ −n−6

and vanishes for j < −2n− 6.

Theorem G (Theorem 3.3.2). Let n ⊂ so5(C) and H be as above. Let Vbe a finite dimensional left so5(C)-module. The cohomology groups Hq(n, V )do not vanish if and only if q ∈ 0, 1, 2, 3. Decompose Hq(n, V ) into theeigenspaces of the H-action as

∑j∈ZH

q(n, V )j.

0. The space H0(n, V )j vanishes for j < 0.

1. The space H1(n, V )j vanishes for j < −2.

2. The space H2(n, V )j vanishes for j > −4.

3. The space H3(n, V )j vanishes for j > −6.

These results are sharp in the sense that for every pair (n, j) not excludedabove there is a V with Hn(n, V )j 6= 0.

The constants −2q in the inequalities j < −2q or j > −2q are explainedby the action on the Eq(n) part:

The action ad(H) = 2 id on n induces ρq(H) = 2q id on the q-formsin Eq(n). The tensor product action of the action of H on V and thedual ρ∗q(H) = −2q id defines an H-action on Hom(E(n)q, V ) and hence onHq(n, V ). If we split V into H-eigenspaces Vi and Hq(n, V )j then

Hq(n, V )j = Hq(n, Vj+q).

For V =∧∗(Hn)⊗R C we find Vj =

∧2n+j(Hn)⊗R C.

We provide a summary of the content and the interdependence of thechapters.

In Chapter 1 we discuss the Chern conjecture and (L2-)cohomology van-ishing results for Kähler and quaternionic Kähler manifolds. The Kähler caseuses Theorem A from Chapter 4.

In Chapter 2 we use the form η to describe the d∇-closed G-valued formson quaternionic Kähler manifolds. The proof combines cohomology vanishingresults from Chapter 1 with Theorem F from Chapter 3.

In Chapter 3 we calculate the cohomology of so5(C)-modules to prove thecentral Lemma C of Chapter 2 as corollary of Theorem F.

In Chapter 4 we prove the L2-Index Theorem A for G-twisted Dirac-bundles over good orbifolds. This theorem was used by Gromov to show theChern conjecture for Kähler orbifolds with negative sectional curvature.

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Acknowledgments

I would like to thank all people who supported me while writing this thesis.First of all I want to express my gratitude to my supervisor Werner Ball-mann for his help in mathematical and personal problems. He proposed thisinteresting topic and encouraged me to find new areas of research. I thankWerner Müller for being my second referee and mentor. I thank all mem-bers of the Arbeitsgruppen Differentialgeometrie and Globale Analysis forthe pleasant atmosphere and the interesting discussions. Special thanks goto Robert Kucharczyk, Antonio Sartori, Rebecca Glover, and Jan Swobodawho read earlier drafts and found errors. I am grateful to Uwe Semmelmann,Misha Verbitskiı, and Rafe Mazzeo who gave valuable advice and encouragedme to work out the proof in Remark 3.4.2. I acknowledge the Max PlanckInstitute for Mathematics for funding this thesis by a scholarship. Last butnot least I want to thank my friends and my family.

CHAPTER1The Chern conjecture and coho-mology vanishing

In this chapter we discuss the Chern conjecture and the vanishing of (L2-)cohomology for Kähler manifolds, Kähler orbifolds, and quaternionic Kählermanifolds.

LetM be a closed 2n-dimensional Riemannian manifold and let χ(M) beits Euler characteristic. Chern conjectured that if M has negative sectionalcurvature then (−1)nχ(M) > 0, and if M has positive sectional curvaturethen χ(M) > 0. This is known to be true for 2- and 4-dimensional manifolds(see [Che55, Theorem 5]).

Gromov showed this for Kähler manifolds (and orbifolds) with negativesectional curvature using results about the L2-cohomology in [Gro91, 2.5.].We explain Gromov’s proof of the Chern conjecture for the case of Kählerorbifolds in the Sections 1.2 and 1.3. It uses the L2-index theory describedin Chapter 4.

Since Kähler manifolds can be defined as Riemannian manifolds withholonomy in U(n), one could hope to prove this conjecture for other classesof Riemannian manifolds with special holonomy. The only other class of man-ifolds with special holonomy (that is not necessarily locally symmetric) forwhich the Chern-conjecture is open is that of quaternionic Kähler manifoldswith holonomy in Sp(n) Sp(1). The 2n-dimensional Calabi–Yau manifoldswith holonomy in SU(n) ⊂ U(n), the 4n-dimensional hyperkähler manifoldswith holonomy in Sp(n) ⊂ SU(2n), the 7-dimensional G2-manifolds, andthe 8-dimensional Spin(7)-manifolds are all Ricci-flat (see [Sal89, page 2]).

11

12 CHAPTER 1. CHERN CONJECTURE AND COHOMOLOGY

Hence, they cannot have positive or negative sectional curvature.For quaternionic Kähler manifolds we have partial results in Sections 1.2

and 1.4.In Section 1.1 we introduce (quaternionic) Kähler manifolds starting with

(quaternionic) Hermitian vector spaces and manifolds. We give the Lefschetztheorems for the Kähler two-form, the Kraines four-form, and the fundamen-tal two-form. We use the L2-versions of these Lefschetz theorems in Section1.2 to get L2-vanishing theorems for simply connected non-positively curved(quaternionic) Kähler manifolds. We discuss in Section 1.3 how this is usedto prove the Chern conjecture for negatively curved Kähler orbifolds.

For quaternionic Kähler manifolds this result is new. But to prove theChern conjecture we would need to understand the L2-cohomology in thethree middle dimensions. Thus, in Section 1.4 we cite Semmelmann–Wein-gart’s and Salamon’s vanishing theorems for closed quaternionic Kähler man-ifolds. This helps us to prove the Chern conjecture for positively curvedquaternionic Kähler manifolds and all known cases of negatively curved ones.The cohomology vanishing theorems for quaternionic Kähler manifolds aretransferred to G-valued forms in Chapter 2.

1.1 (Quaternionic) Hermitian linear algebra andLefschetz theorems

It is natural to ask for the connection between Kähler manifolds, quaternionsand quaternionic Kähler manifolds. Quaternionic Kähler manifolds are nei-ther Kähler nor are they “quaternionic” in the way complex manifolds are“complex” (see [Bes87, Theorem 15.58]). We will define more general quater-nionic and quaternionic Hermitian manifolds (compare [Swa90, Section 1.2]and [Bes87, Section 14.F]). This explains the analogy to the case of Kählermanifolds that can be viewed as (almost-) complex and (almost-) Hermitianmanifolds with parallel almost-complex structure.

Moreover, we will see the following structure of the proof of the hardLefschetz theorem (with Gromov’s L2-version) again.

We start with a (quaternionic) Hermitian vector space. We understand itas a real vector space with extra structure and get a theorem about

∧∗(V ∗)using the exterior forms defined by the extra structure. In the Kähler Lef-schetz case this is the following Lemma 1.1.1. It has well known corollaries forthe (L2-)cohomology of Kähler manifolds. Similarly Theorem 1.1.3 ([Bon82,Theorem 1]), Theorem 3.2.2 ([Ver90, Theorem 1.2]), and Lemma 2.1.10 aboutthe linear algebra of quaternionic Hermitian vector spaces have consequences

1.1. (QUATERNIONIC) HERMITIAN LINEAR ALGEBRA 13

for the closed and exact differential forms on quaternionic Kähler, quater-nionic Hermitian, and hyperkähler manifolds.

Lemma 1.1.1. (see [Huy05, Section 1.2.] and [Bal06, Chapter 5]) Let(V, 〈, 〉, i) be a 2n-dimensional Hermitian vector space, i.e., the complex struc-ture i is compatible with the Euclidean inner product 〈·, ·〉 on V . Defineωi(v, w) = 〈iv, w〉. Then the operator Li :

∧k(V ∗) →∧k+2(V ∗) defined by

Li(α) = α ∧ ωi = ωi ∧ α induces isomorphisms

Lmi :n−m∧

(V ∗) −→n+m∧

(V ∗) for 0 ≤ m ≤ n.

The adjoint of Li is Λi = ∗−1 Li ∗.

Here the complex structure of V determines an orientation. Hence, theinner product defines a volume form vol and a Hodge ∗-operator ∗ by

α ∧ ∗β = 〈α, β〉 · vol .

To prove this kind of lemma one can use elementary but somehow tediouslinear algebra. Here and in the case of Lemma 2.1.10 one can also use moreelegant representation theory. In this case define the counting operator Hby H|∧k(V ∗) = (k − dimC V

∗) id. Then Li,Λi, and H define an sl2(C)-actionon∧∗(V ∗)⊗R C and an sl2(R)-action on

∧∗(V ∗). Now we define the corre-sponding quaternionic notions.

Definition 1.1.2. A quaternionic (right) vector space V is an abelian groupwith a right action of H, where H = R ⊕ Ri ⊕ Rj ⊕ Rk are Hamilton’squaternions,

V ×H→ V

satisfying the vector space axioms

(v + w)α = vα + wα,

v(α + β) = vα + vβ,

v(αβ) = (vα)β,

v1 = v,

for all v, w ∈ V and α, β ∈ H. This amounts to an anti-homomorphismH→ End(V ).

Since H is a division ring every quaternionic vector space has a basis and awell defined dimension (see [Lan02, page 642], the classical proof from linearalgebra in [Lan02, Paragraph III.5] needs no commutativity). Choosing a


basis (v1, . . . , vn) of V gives us an isomorphism to Hn. Identifying the sub-algebra R[i] ⊂ H with C we have (v1, v1j, . . . , vn, vnj) as a basis of V as aC-vector space and (v1, v1i, v1j, v1k, . . . , vn, vni, vnj, vnk) as a basis of V asan R-vector space. The standard Euclidean inner product of V with respectto such an adapted basis is compatible with the H-action in the followingsense:

〈v, w〉 = 〈vi, wi〉 = 〈vj, wj〉 = 〈vk, wk〉 for all v, w ∈ V.

Following [Bon82] we call a quaternionic vector space with a compatiblemetric a quaternionic Hermitian vector space. We will use Hn and “quater-nionic Hermitian vector space V of real dimension 4n” interchangeably. Wealways denote by

∧∗(Hn) or∧∗(V ∗) the exterior algebra over the correspond-

ing real vector space.We define three 2-forms

ωI(v, w) = 〈v(−i), w〉, ωJ(v, w) = 〈v(−j), w〉, and ωK(v, w) = 〈v(−k), w〉.

The sign −1 in above formulas comes from the fact that the tangent spaceof a quaternionic Hermitian manifold is a quaternionic Hermitian left vectorspace. Since αβ = βα we can turn a quaternionic right vector space intoa quaternionic left vector using the scalar multiplication (α, v) 7→ vα (com-pare [Kak95] for a detailed treatment of quaternions and quaternionic Kählermanifolds). Then we obtain ωI(v, w) = 〈Iv, w〉 = 〈vi, w〉 = 〈v(−i), w〉. Wecan define the Kraines four-form

Ω = ωI ∧ ωI + ωJ ∧ ωJ + ωK ∧ ωK ∈4∧

(V ∗)

and get a Lefschetz-type theorem:

Theorem 1.1.3 ([Bon82, Theorem 1]). Let V be a 4n-dimensional quater-nionic Hermitian vector space with the Kraines four-form Ω. The operatorLΩ :

∧k(V ∗) →∧k+4(V ∗) defined by LΩ(α) = α ∧ Ω = Ω ∧ α induces iso-

morphisms

LmΩ :2n−2m∧

(V ∗) −→2n+2m∧

(V ∗) for 0 ≤ m ≤ n.

Similarly, we define the fundamental two-form

η = ωI ⊗ i+ ωJ ⊗ j + ωK ⊗ k ∈2∧

(V ∗)⊗R H

with η ∧ η = −Ω and obtain the following theorem.

1.1. (QUATERNIONIC) HERMITIAN LINEAR ALGEBRA 15

Theorem 1.1.4 ([Bon83, Theorem 4]). Let V be a 4n-dimensional quater-nionic Hermitian vector space with the fundamental two-form η. The opera-tor Lη :

∧k(V ∗) ⊗R H →∧k+2(V ∗) ⊗R H defined by Lη(α) = η ∧ α induces

isomorphisms

Lmη :2n−m∧

(V ∗)⊗R H −→2n+m∧

(V ∗)⊗R H for 0 ≤ m ≤ 2n.

Kähler manifolds can be defined as Riemannian manifolds with holon-omy in U(n) or equivalently as Hermitian manifolds with a parallel (almost-)complex structure J . To define quaternionic Kähler manifolds we first intro-duce the holonomy group Sp(n) Sp(1). Then we give an equivalent definitionas quaternionic Hermitian manifolds with parallel bundle of almost-complexstructures G.

Definition 1.1.5. Let V be a 4n-dimensional quaternionic Hermitian vectorspace. Then we get a right-action of the unit quaternions

H ⊃ S3 = Sp(1) ⊂ SO(4n)

on V . The centralizer of Sp(1) in SO(4n) is the compact symplectic groupSp(n). It can be defined as the group of H-linear orthogonal matrices. Thegroup Sp(n) Sp(1) in SO(n) is isomorphic to Sp(n)×Z/2Z Sp(1) since ± idis the intersection of Sp(1) and Sp(n) (see [Sal89, Chapter 9]). Quater-nionic Kähler manifolds are Riemannian manifolds with holonomy group inSp(n) Sp(1). We exclude the hyperkähler manifolds with holonomy in Sp(n)from this definition.

We prefer the following equivalent approach to quaternionic Kähler man-ifolds.

Definition 1.1.6. An almost-quaternionic manifold is a pair (M,G) withMa 4n-dimensional manifold and G ⊂ End(TM) a subbundle such that for allp ∈ M there is an open neighborhood U 3 p where G|U = 〈IU , JU , KU〉 forthree anticommuting almost complex structures IU , JU , and KU , i.e.,

I2U = J2

U = K2U = IUJUKU = − idU .

A quaternionic Hermitian manifold (M,G, g) is an almost-quaternionic man-ifold equipped with a compatible Riemannian metric g, i.e.,

g(v, w) = g(IUv, IUw) = g(JUv, JUw) = g(KUv,KUw).

We choose 〈A,B〉 = 14n

tr(ABt) as the inner product on G ⊂ End(TM). Sinceg(IUv, w) = g(I2

Uv, IUw) = g(v,−IUw) we obtain I tU = −IU (this is true for


any section of G). For two anticommuting almost complex structures IU andJU we get

〈IU , JU〉 =1

4ntr(IU(−JU)) =

1

4ntr(JUIU) = −〈JU , IU〉 = −〈IU , JU〉 = 0.

Thus, the choice of three anticommuting almost complex structures IU , JU ,and KU corresponds to the choice of an orthonormal frame in G.

A quaternionic manifold is an almost-quaternionic manifold with a com-patible torsion-free connection.

A Quaternionic Kähler manifold is a quaternionic Hermitian manifoldthat is quaternionic since its Levi-Civita connection ∇ is compatible with G

(compare [Bes87, Section 14.F] and [Bes87, Proposition 14.36]). The mani-fold is hyperkähler if the curvature d∇d∇ of G is vanishing.

The tangent space of an almost-quaternionic manifold has the structureof a quaternionic (left) vector space; the tangent space of a quaternionic Her-mitian manifold has the structure of a quaternionic Hermitian (left) vectorspace.

Since the wedge product is defined locally, the transition to (quaternionic)Hermitian manifolds M just replaces

∧∗(V ∗) by A∗(M) (and∧∗(V ∗) ⊗R C

by A∗(M,C)). In the (quaternionic) Kähler case the differential forms areparallel. Thus, the wedge product commutes with the Hodge Laplacian ∆ =(d+d∗)2 (see [Lic76, page 159]) and preserves harmonic forms. Since parallelforms are bounded, the wedge product also preserves L2-forms.

As corollary we get the following Lefschetz type theorem.

Theorem 1.1.7. Let M be a quaternionic Hermitian 4n-dimensional man-ifold. Then

LΩ : Ak(M) −→ Ak+4(M), α 7→ α ∧ Ω

defines isomorphisms

LmΩ : A2n−2m(M) −→ A2n+2m(M) for m < n.

Therefore, for k < n− 1 the map LΩ : Ak(M)→ Ak+4(M) is injective. If Mis a quaternionic Kähler manifold Ω is parallel and ∆ commutes with LΩ.

In the next section we will use the Lefschetz-theorems for Kähler andquaternionic Kähler manifolds to show L2-vanishing results.

1.2. L2-VANISHING THEOREMS 17

1.2 L2-vanishing theorems for (quaternionic) Käh-ler manifolds

By Atiyah’s L2-index theorem (Theorem 4.1.3) for

d+ d∗ : Aeven(M) −→ Aodd(M),

the Chern conjecture for 2n-dimensional negatively curved manifoldsM holdsif the harmonic L2-forms Hk

2(M) on the universal cover M vanish preciselyfor k 6= n.

The argument Gromov used to show that this holds for Kähler manifoldscan be partially transferred to the Kraines four-form Ω of quaternionic Kählermanifolds. We follow [Gro91] and [Bal06, Chapter 8].

Definition 1.2.1. Let E be a flat Hermitian vector bundle over a completeconnected Riemannian manifold M . A differential form α with values in Eis called O(f(r)) for a non-decreasing function f : R+ → R+ if we can choosean origin o ∈M and a constant c > 0 such that for r(p) = d(o, p):

∀p ∈M ‖α(p)‖ ≤ cf(cr(p) + c) + c.

A form α is called d(O(f)) if there is a form β that is O(f(r)) with α = dβ.

Since ∇Ω = 0 = dΩ and ∇ω = 0 = dω, the forms Ω and ω are bounded(i.e., O(1)) and we can apply the following propositions. Note that theirproofs do not use that M is Kähler.

Proposition 1.2.2 ([Bal06, Proposition 8.4]). Let M be a complete and sim-ply connected Riemannian manifold and α a differential form on M of degreek with ‖α‖∞ <∞ and dα = 0.

1. If K ≤ 0 then α = d(O(r)).

2. If K ≤ −c2 < 0 then α = d(O(1)).

The classical application of this proposition is the case of Kähler mani-folds. Gromov defined Kähler-hyperbolic manifoldsM as closed Kähler man-ifolds with a universal cover M where the Kähler form ω is d(O(1)). ClosedKähler manifolds with negative sectional curvature are Kähler hyperbolic(but the notion is much more general, see [Gro91, Examples 0.3.A]). Caoand Xavier extended this terminology to the first case of the proposition:a closed Kähler manifold is called Kähler-parabolic if on its universal coverthe Kähler form is d(O(r)) but not d(O(1)). Closed Kähler manifolds withnon-positive sectional curvature are Kähler-parabolic.


Proposition 1.2.3 ([Bal06, 8.8.]). Suppose α = d(O(r)). Let β be a closedand square integrable differential form with values in a flat Hermitian bundleE. Then d∗(α ∧ β) = 0 implies α ∧ β = 0.

As corollaries we get:

Theorem 1.2.4 ([Bal06, 8.9.],[CX01, Main Theorem],[JZ00]). Let (M,ω) bea simply connected complete Kähler manifold. If the sectional curvature Kis nonpositive (or more generally ω = d(O(r))), then

Hk2(M,E) = 0 for all k 6= dim(M)

2.

Here Hk2(M,E) is L2-cohomology with values in a flat Hermitian bundle E.

We adapt the proof (given in the literature) to the case of quaternionicKähler manifolds.

Theorem 1.2.5 (Theorem B). Let (M,G, g) be a simply connected complete4n-dimensional quaternionic Kähler manifold. If the sectional curvature K isnonpositive (or more generally the Kraines four-form satisfies Ω = d(O(r))),then

Hk2(M,E) = 0 for k /∈ 2n− 1, 2n, 2n+ 1 .

Here Hk2(M,E) is L2-cohomology with values in a flat Hermitian bundle E.

Proof. Assume β is a harmonic k-form with k < 2n− 1 that is in L2(M,E).The Lefschetz Theorem for the operator

LΩ :k∧

(TM∗) −→k+4∧

(TM∗), β 7→ Ω ∧ β

shows that if LΩ(β) = 0 then β = 0 for k < 2n−1 (see [Bon82, Theorem 1]).Since β is harmonic and LΩ preserves harmonic forms ([Lic76, page 159]),d∗(Ω∧β) = 0. This implies β∧Ω = 0 by Propositions 1.2.2 and 1.2.3. Hence,since β = 0 we conclude

Hk2(M,E) = 0 for k < 2n− 1.

By Poincaré duality we obtain

Hk2(M,E) = 0 for k > 2n+ 1.

The Hodge operator ∗ defines Poincaré duality for harmonic forms. See thediscussion on [Bal06, page 9/10] for an extension of the Hodge operator toE-valued forms with the help of the Hermitian inner product of E to extendPoincaré duality to this case.

1.3. KÄHLER ORBIFOLDS 19

The vanishing results of [Gro91, 2.5.], [CX01, Theorem 3], and [JZ00] forKähler hyperbolic and Kähler parabolic manifolds in Theorem 1.2.4 have aclose analog in our Theorem 1.2.5 for quaternionic Kähler manifolds. In try-ing to use Theorem 1.2.5 to settle the Chern conjecture we have the problem

that Ω being a four-form does not tell us anything about Hdim(M)

2±1

2 (M,E).Also the nonvanishing result of [Gro91, 2.5.] needs a 2-form. If one wants toadapt Gromov’s idea, one needs a closed 2-form.

Maybe there is a suitable one; the following ones are not. Wedgingwith the forms ωI , ωJ , and ωK defines local isomorphisms

∧2n−1(TpM∗) →∧2n+1(TpM

∗). However, they are neither closed nor globally well defined ingeneral. The fundamental two-form η locally given by ωI⊗I+ωJ⊗J+ωK⊗Kis globally well defined on a quaternionic Hermitian manifold (see Proposition2.1.1) and parallel on a quaternionic Kähler manifold (see Theorem 2.2.3).Since wedging with ωI , ωJ , and ωK defines local isomorphisms, wedging withη is an injective map

∧2n−1(TpM∗)→

∧2n+1(TpM∗,Gp). This is also a corol-

lary of Theorem 1.1.4. However, for Gromov’s proof η would have to bed(O(1)). The form η is not even d∇-exact. Let α be a G-valued one-formwith d∇(α) = η and hence d∇d∇α = 0. By Theorem C every G-valued k-formin ker(d∇ d∇) for k < 2n or k > 3n can be written as β ∧ η. Thus, thereare no nontrivial G-valued d∇-exact and d∇-closed 2-forms. We will settlethe Chern conjecture for all known examples of closed quaternionic Kählermanifolds in Section 1.4.

1.3 Kähler orbifolds

We can apply Theorem 4.4 to prove that the Chern conjecture holds forclosed Kähler orbifolds. Gromov covers this case in [Gro91, 2.5] for Kähler-hyperbolic manifolds, but Atiyah’s L2-index theorem he uses does not coverit. There are different notions of Euler characteristic for orbifolds. Ourargument applies to the Euler characteristic introduced by Satake in [Sat57,page 481] that we call Euler–Satake characteristic χorb(M) following [FS11,Definition 2.1]. We use the methods of Gromov [Gro91], Cao–Xavier [CX01]and Jost–Zuo [JZ00] as presented in [Bal06, Chapter 8]. The first part of theTheorem is stated in [Gro91].

The Kähler-parabolic case is treated in [CX01] and [JZ00], but only formanifolds.

Theorem 1.3.1. Let M be a 2n-dimensional closed Kähler orbifold withnegative sectional curvature. Then

(−1)nχorb(M) > 0.


If M is of non-positive sectional curvature, we still obtain

(−1)nχorb(M) ≥ 0.

Proof. Since M has non-positive sectional curvature, M is a good orbifold(see [Hae90] and [BH99, 2.16]) with universal cover M . For good orbifoldsit is enough to assume instead of negative (or non-positive) curvature thatM = M/Γ is Kähler-hyperbolic (or -parabolic).

Since the Gauß–Bonnet theorem holds for orbifolds by Satake [Sat57, 2],we have to show (2π)−

n2

∫M

Pf(R) > 0 (or ≥ 0 respectively). But Theorem4.4.1 shows (2π)−

n2

∫M

Pf(R) = indΓ(d+ d∗) for

d+ d∗ : Aeven(M) −→ Aodd(M).

Now indΓ(d + d∗) ≥ 0 holds, since Hk2(M) = 0 for n 6= dimC(M) if M

is complete and Kähler-parabolic (or -hyperbolic) (Theorem 1.2.4 [CX01,2], [JZ00, Theorem 1 & Remark], [Bal06, 8.9]). If M is Kähler-hyperbolic,complete, simply connected, and has a cocompact action by a discrete groupΓ, then also H

p,q2 (M) 6= 0 for p+ q = dimC(M) [Gro91, 2.5]. Thus, indΓ(d+

d∗) > 0 in this case.

We want to indicate how the nonvanishing result [Gro91, 2.5] can beproven in our situation. Here we follow [Bal06, Chapter 8] and considerthe following more general situation. Let E → M be a Γ-equivariant flatHermitian vector bundle. While the theorems in [Gro91],[CX01], and [JZ00]do not cover E-valued harmonic forms, [Bal06, Chapter 8] does not coverorbifolds.

By [Bal06, Theorem 8.9], we obtain Hk2(M, E) = 0 for all k 6= dimC(M)

and every complete simply connected Kähler-parabolic manifold M (here noΓ-action on M is necessary).

In the proof of Hp,q2 (M, E) 6= 0 for p + q = dimC(M) in [Bal06, 8.31]

we need to use Theorems 4.4.1 and 4.2.3 instead of [Bal06, 8.27] and [Bal06,8.29]. The rest is unchanged, since [Bal06, 8.10],[Bal06, 8.17] and [Bal06,8.28] hold independently of the Γ-action. With Theorem 4.4.1, we concludeTheorem 1.3.2 with an analogue of Theorem 1.3.1 for the orbifold Eulercharacteristic χporb(M,E) of the Dolbeault complex Ap,∗(M,E) defined as∫M

td(M) ∧ ch(Ap,0(M,C)⊗ E) as in [Bla96, (3.4)]. For M a manifold, thisis the alternating sum

∑2nq=0(−1)qhp,q(M,E).

Theorem 1.3.2 (See [Gro91, Theorem 0.4.A.] for χp(M) and M a Käh-ler-hyperbolic manifold). Let M be a closed 2n-dimensional Kähler orbifold

1.4. QUATERNIONIC KÄHLER MANIFOLDS 21

with negative sectional curvature. Then M is a good orbifold M = M/Γ. LetE →M be a flat Hermitian vector bundle and E → M its pullback. Then

(−1)n+pχporb(M,E) > 0.

If M is of non-positive sectional curvature K ≤ 0, still

(−1)n+pχporb(M,E) ≥ 0.

Proof. For the twisted Dolbeault operator ∂ + ∂∗ on the Dirac bundle

Ap,∗(M, E) = A0,∗(M,C)⊗ Ap,0(M,C)⊗ E,

the discussion above gives (−1)n+pindΓ(∂+ ∂∗) > 0 in the Kähler-hyperboliccase (or ≥ 0 in the Kähler-parabolic case). The index theorem 4.4.1 gives usindΓ(∂ + ∂∗) =

∫M

td(M) ∧ ch(Ap,0(M,C)⊗E) (compare [Bal06, 8.28]), butthis is by definition χporb(M,E).

Of course the argument above also shows for flat Hermitian bundles E →M that

(−1)nχ(M,E) = dimΓ Hn2 (M, E) > 0,

if M is Kähler-hyperbolic (resp. ≤ 0 if M is Kähler-parabolic). This case is(partially) covered in [Bal06, Theorem 8.22], but not in [Gro91],[CX01], or[JZ00].

The Euler–Satake characteristic χorb(M) is Q-valued. There are otherpossibilities to define Z-valued Euler characteristics for orbifolds discussed in[Bla96, (3.5)] and [Kaw79, page 153] that differ from χorb(M) by local correc-tion terms. For Kähler orbifoldsM and the Euler characteristic χtop(M) ofMas a topological space, there are counterexamples to (−1)nχtop(M) ≥ 0, e.g.topological 2-spheres that are hyperbolic outside of orbifold points. They canbe given by the quotient of the 2-dimensional hyperbolic space by a trianglegroup (see [Kat92, 4.4]).

1.4 The Chern conjecture for quaternionic Käh-ler manifolds

The results [Sal82, Theorem 6.6], [SW02, Proposition 5.8], and Section 1.2allow us to partially settle the Chern conjecture for quaternionic Kählermanifolds.


Theorem 1.4.1 ([Sal82, Theorem 6.6],[SW02, Proposition 5.8]). If M is aclosed 4n-dimensional quaternionic Kähler manifold, the odd Betti numbersb2m+1(M) vanish for 2m+ 1 < n and for 3n < 2m+ 1. If, moreover, M haspositive scalar curvature, all odd Betti numbers vanish.

In the case of positive Ricci curvature this was proven by Salamon usingthe twistor space [Sal82, Theorem 6.6] (recall that since quaternionic Kählermanifolds are Einstein manifolds this is equivalent to positive scalar curva-ture). The twistor space Z ⊂ G is a S2-bundle. Any fiber Zp = π−1(p) con-sists of all almost complex structures in Gp, i.e. Zp = L ∈ Gp | L2 = − idp.In the case of negative Ricci curvature Semmelmann and Weingart showedthe vanishing of the odd Betti numbers b2m+1(M) for 2m + 1 < n and for3n < 2m + 1 [SW02, Proposition 5.8]. For n ≤ 2m + 1 ≤ 3n and negativeRicci curvature it seems to be unknown if b2m+1(M) vanishes.

Proposition 1.4.2 ([Kra66, Theorem 3]). If M is a closed 4n-dimensionalquaternionic Kähler manifold, the Betti numbers b4i(M) are positive for 0 ≤i ≤ n.

Proof. The forms Ωi are closed 4i-forms. For closed manifolds, they are notexact. This is analogous to the corresponding theorem for symplectic orKähler 2-forms. Ωn is a volume form and, by Stokes’ Theorem, not exacton a closed manifold. If Ωm were equal to dα for m < n, we would haved(α ∧ Ωn−m) = Ωn.

Corollary 1.4.3. IfM is a closed 4n-dimensional quaternionic Kähler man-ifold with positive scalar curvature, its Euler characteristic satisfies χ(M) ≥n+ 1.

Proof. We compute

χ(M) =4n∑j=0

(−1)jbj = −2n∑j=0

b2j+1 +n∑i=0

b4i +n−1∑i=0

b4i+2 ≥ n+ 1.

In the general quaternionic Kähler case

χ(M) =4n∑j=0

(−1)jbj ≥ (n+ 1)−∑

n≤2i+1≤3n

b2i+1.

Therefore, to prove the Chern conjecture one would have to show∑n≤2i+1≤3n

b2i+1 < n+ 1.

Here the methods of [SW02] fail.


Definition 1.4.4. A symmetric space is a connected Riemannian manifoldM such that for all p ∈ M there is an isometry sp : M → M, such thatsp(p) = p and dsp(p) = − id. If sp is only locally defined as an isometry of aneighborhood of p, we call M a locally symmetric space.

Example 1.4.5 (see [Bes87, 14.52]). The quaternionic Kähler symmetricspaces of compact type are called Wolf spaces. Among them are the 4n-dimensional Grassmannians

HP n = G1(Hn+1) = Sp(n+ 1)/(Sp(n)Sp(1)),

G2(Cn+2) = SU(n+ 2)/S(U(n)U(2)),

G4(R4+n) = SO(n+ 4)/S(O(n)O(4)).

Furthermore there are five exceptional cases in the dimensions 8, 28, 40, 64,and 112, given by

G2/SO(4),F4/(Sp(3)Sp(1)),E6/(SU(6)Sp(1)),

E7/(Spin(12)Sp(1)), and E8/(E7Sp(1)).

For every symmetric space of compact type, there is a dual space of non-compact type. See for example [Hel62, Ch.V.2] or [Bal06, B.37].

Example 1.4.6. The dual spaces of the Wolf spaces are the 4n-dimensionalexamples

Sp(n, 1)/(Sp(n)Sp(1)), SU(n, 2)/S(U(n)U(2)), SO(n, 4)/S(O(n)O(4)),

and the exceptional cases

G22/SO(4),F−20

4 /(Sp(3)Sp(1)),E26/(SU(6)Sp(1)),

E−57 /(Spin(12)Sp(1)), and E−24

8 /(E7Sp(1)).

Every symmetric space of non-compact type has a compact quotient [Bor63,Theorem A].

For the locally symmetric case we will use the Gauß-Bonnet theorem,Hirzebruch proportionality and Theorem 1.4.3 to prove the Chern conjecture.Since all known examples of closed quaternionic Kähler manifolds (to ourknowledge) are locally symmetric, this may prove the general case. There arealso homogeneous examples of quaternionic Kähler manifolds with negativesectional curvature given by Alekseevskii (see [Bes87, 14.98], [Ale68, 4], andthe corrected classification in [Cor96, Theorem 2.28]). But it is known thatthey have no compact quotients, since every compact quaternionic Kählermanifold with homogeneous universal cover is locally symmetric (see [Ale70,3] and [Bes87, 14.56]).


Theorem 1.4.7. If M is a locally symmetric quaternionic Kähler manifoldof finite volume, then χ(M) > 0.

Proof. The Pfaffian Pf is an ad-invariant homogeneous polynomial of degreem on so2m(R) with Pf(A)2 = detA. By the Gauß–Bonnet theorem, the Eulercharacteristic of a locally symmetric space of finite volume is the integral overthe Euler form Pf

(1

2πR)induced by the Pfaffian ([Har71, Equation (F)])

χ(M) =

∫M

Pf

(1

2πR

).

Since for dual symmetric spaces of dimension 4n = 2m the curvatures areequal up to a sign, their Euler forms Pf

(1

2πR(M−)

)= (−1)2nPf

(1

2πR(M+)

)are equal. Therefore for a locally symmetric quotient of finite volume M−/Γthe following holds (and is called Hirzebruch proportionality, see [Hir58]):

χ(M−/Γ) =

∫M−/Γ

Pf

(1

2πR(M−)

)=

vol(M−/Γ)

vol(M+)

∫M+

Pf

(1

2πR(M+)

)=

vol(M−/Γ)

vol(M+)χ(M+).

Being a quotient of finite volumes, vol(M−/Γ)vol(M+)

is positive, and for a quater-nionic Kähler manifold M+ of positive curvature χ(M+) > 0. Hence, forlocally symmetric quaternionic Kähler manifolds χ(M/Γ) > 0.

CHAPTER2Differential forms and quaternionicgeometry

Recall the analogy between Kähler manifolds (M,J, g) and quaternionic Käh-ler manifolds (M,G, g) described in Section 1.1. The first are Hermitian man-ifolds with parallel almost-complex structure J . The second are quaternionicHermitian manifolds with a parallel rank three bundle of anticommuting al-most complex structures G.

Similarly to the Lefschetz theorems for (quaternionic) Kähler manifoldswe will use Lemma 2.1.10 about the linear algebra of quaternionic Hermitianvector spaces to obtain results about d∇-closed forms on quaternionic Käh-ler manifolds. Also similarly to the Lefschetz case the conceptual proof ofLemma 2.1.10 in the next chapter uses representation theory (and an elemen-tary proof sketched in the end of the next chapter is possible but tedious).

In Section 2.1 we discuss differential forms in the more general case ofquaternionic Hermitian manifolds. We introduce the G-valued two-form

η = ωJ ⊗ J + ωI ⊗ I + ωK ⊗K

and use Lemma 2.1.10 to prove the central Theorem 2.1.9 discussing theexactness of the chain complex

0 −→ A`(M)Lη−−→ A`+2(M,G)

Lκ−−→ A`+4(M,G) −→ 0.

In Section 2.2 we restrict ourselves to quaternionic Kähler manifolds. Asin the Kähler case this is the case where the forms we discussed are parallel.

26 CHAPTER 2. FORMS AND QUATERNIONIC GEOMETRY

Moreover the form κ is up to a constant the curvature of G. Thus, we canuse Theorem 2.1.9 to discuss d∇-closed forms in G.

In Section 2.3 we do this and show that in certain degrees d∇-closedforms are d∇-exact using the vanishing theorems for the (L2-)cohomology ofquaternionic Kähler manifolds given in Section 1.2 and 1.4.

2.1 Forms on quaternionic Hermitian manifolds

Recall Definition 1.1.6 of quaternionic Hermitian manifolds (M,G, g). On aquaternionic Hermitian manifold we can locally define the forms ωI , ωJ , andωK on TpM after choosing an orthonormal basis 〈I, J,K〉 = Gp. In generalwe cannot choose I, J, and K or ωI , ωJ , and ωK globally. They are onlydefined in an open neighborhood of p. The following tensors, however, areinvariant under a change of basis, hence defined globally. We denote by E(E)the smooth sections of E.

Proposition 2.1.1. Let (M,G, g) be a quaternionic Hermitian manifold.Then the following tensors are independent of the choice of the basis I, J,K,of Gp.

J = I ⊗ I + J ⊗ J +K ⊗K ∈ E(G⊗ G),

Ω = ωI ∧ ωI + ωJ ∧ ωJ + ωK ∧ ωK ∈ A4(M),

η = ωI ⊗ I + ωJ ⊗ J + ωK ⊗K ∈ A2(M,G),

κ =

0 ωK −ωJ−ωK 0 ωIωJ −ωI 0

∈ A2(M,End(G)).

Hence, they define global tensors. We call Ω Kraines four-form, η funda-mental two-form and κ curvature two-form. Here the matrix κ is given withrespect to the basis I, J,K.

Proof. It is straightforward to check that these objects behave well under achange of basis of Gp. For the calculation we write ωI , ωJ , ωK , I, J, and Kas ω1, ω2, ω3, J1, J2, and J3 before and as ω′1, ω′2, ω′3, J ′1, J ′2, and J ′3 after thechange of basis. Then there is a matrix A = (Aij) with Ji =

∑3j=1AijJ

′j and

ωi =∑3

j=1Aijω′j. Since

J21 = J2

2 = J23 = J1J2J3 = J ′1

2= J ′2

2= J ′3

2= J ′1J

′2J′3 = − id,

both bases are orthonormal (compare Definition 1.1.6) and A ∈ SO(3). Thus,we get

∑3j=1 AijAkj = δik =

∑3j=1AjiAjk. This shows that the tensors J,Ω, η,

and κ are well defined. The ωi and Ji transfer as components of vectors ω

2.1. FORMS ON QUATERNIONIC HERMITIAN MANIFOLDS 27

and J . The formulas for J,Ω, η, and κ are independent of the basis since theycorrespond to the inner products 〈J, J〉, 〈ω, ω〉, 〈J, ω〉, and the cross productJ 7→ ω × J both invariably defined under the SO(3)-action. We give theexplicit calculation for η; the others are similar (and the other tensors arewell known in the literature):

η =3∑j=1

ωj ⊗ Jj =3∑

i,j,k=1

AjiAjkω′i ⊗ J ′k =

3∑i,k=1

δki ω′i ⊗ J ′k =

3∑i=1

ω′i ⊗ J ′i .

On quaternionic Kähler manifolds one can define many bundles. Thisrichness may lead to some confusion because the same letters are used fordifferent bundles in the literature. Our convention of using G is also usedin [Swa90]. In [Ish74] G is called V , in [Kak95] and [Bes87] E, in [Sal82]and [SW02] Sym2H for the locally defined bundle H corresponding to theSp(1)-action and E is used for the locally defined bundle corresponding tothe Sp(n)-action. The bundle H in the next remark is different from H of[Sal82] and [SW02].

Remark 2.1.2 (The bundles G, R, and H). On an almost-quaternionic man-ifold M , we have the bundle G → M defined locally as 〈I, J,K〉. On anysmooth manifold we have the trivial line bundle R ⊂ End(TM) defined asR id. We define the bundle H ⊂ End(TM) as H = G ⊕ R. Since for anyp ∈M the fiber Hp is, as an R-algebra, isomorphic to the quaternions H withRp = R ⊂ H = Hp we call the projections

< : H −→ R,

= : H −→ G

“real part” and “imaginary part”. We identify A∗(M) with A∗(M,R) byα=α⊗ id.

Remark 2.1.3 (compare [Bal06, 1.1]). For a morphism µ : E1 ⊗ E2 → E3,α1 =

∑φi ⊗ σi ∈ A∗(M,E1), and α2 =

∑ψj ⊗ τj ∈ A∗(M,E2) we define

the form α1 ∧µ α2 ∈ A∗(M,E3) by∑

i,j φi ∧ ψj ⊗ µ(σi ⊗ τj). We will use forµ the following morphisms. Let F be any vector bundle, then there are themorphisms

ε : End(F )⊗ F −→ F , given by evaluation,γ : End(F )⊗ End(F ) −→ End(F ), given by composition,λ : End(F )⊗ End(F ) −→ End(F ), given by the Lie bracket.

Thus, fiberwise ε(σ⊗τ) = σ(τ), γ(σ⊗τ) = στ , and λ(σ⊗τ) = στ−τ σ.


Definition 2.1.4. Since R,G, and H are subbundles of End(TM) we canuse γ to extend Lη : Am(M)→ Am+2(M,G) linearly to

Lη : Am(M,H) −→ Am+2(M,H), α 7→ η ∧γ α.

Similarly Lκ : Am(M,G)→ Am+2(M,G) and Lκ : Am(M,H)→ Am+2(M,H)are defined by Lκ(α) = κ ∧ε α (since G ⊕ R = H we can extend κ ∈A2(End(G)) to A2(End(H)) acting trivially on R). Furthermore, with Ω ∈A4(M)=A4(M,R) ⊂ A4(M,H) we get

LΩ : Am(M,H) −→ Am+4(M,H), α 7→ α ∧γ Ω.

Moreover, we define

L=η : Am(M,H) −→ Am+2(M,H), α 7→ η ∧λ α.

Proposition 2.1.5. Let M be a quaternionic Hermitian manifold (or V aquaternionic Hermitian vector space). Then the operators

Lη, LΩ, Lκ, and L=η : A∗(M,H) −→ A∗(M,H)

(respectively∧∗(V ∗)⊗R H→

∧∗(V ∗)⊗R H with V ∗ as always understood asreal vector space) satisfy the following identities:

Lη Lη = −LΩ,

= LΩ = LΩ =,< LΩ = LΩ <,

0 = < Lη <,1

2L=η = = Lη = = Lκ,

< L=η = 0 = L=η <.

Proof. Since the operators are defined fiberwise we only have to prove theidentities on

∧∗(V ∗)⊗RH. These are the results of the following straightfor-ward calculations:

η ∧γ η = (ωI ⊗ I + ωJ ⊗ J + ωK ⊗K) ∧γ (ωI ⊗ I + ωJ ⊗ J + ωK ⊗K)

= ωI ∧ ωI ⊗ I2 + ωJ ∧ ωJ ⊗ J2 + ωK ∧ ωK ⊗K2

+ ωI ∧ ωJ ⊗ (IJ + JI) + ωJ ∧ ωK ⊗ (JK +KJ)

+ ωK ∧ ωI ⊗ (KI + IK)

= Ω⊗ (−1) = −Ω.

Here [Bon82] differs by a factor 2 in his definition of Ω.


We can decompose any form α ∈∧∗(V ∗)⊗R H as

α = α1 ⊗ 1 + αI ⊗ I + αJ ⊗ J + αK ⊗K. (2.1)

The operators = and < commute with LΩ since

α ∧γ (Ω⊗ 1) = (α1 ∧ Ω)⊗ 1 + (αI ∧ Ω)⊗ I+ (αJ ∧ Ω)⊗ J + (αK ∧ Ω)⊗K.

On the other hand Lη can be decomposed as Lcη + 12L=η where Lcη =

Lη − 12L=η exchanges the real and the imaginary part and 1

2L=η is induced by

the operator Lκ on the imaginary forms.To see this we calculate the Lie bracket on H. It has R as null space,

〈I, J,K〉 as image, and is twice the multiplication for any two out of I, J,K.The operator Lκ does just this. With

1

2[J,K] = JK = I = =I = =J=K,

[I, I] = 0 = =I2 = =I=I,

[I, 1] = 0 = =I=1,

and cyclic permutations we obtain

κ ∧ε α = (ωJ ∧ αK − ωK ∧ αJ)⊗ I+ (ωK ∧ αI − ωI ∧ αK)⊗ J+ (ωI ∧ αJ − ωJ ∧ αI)⊗K

=1

2η ∧λ α = =(η ∧γ =(α)).

With < = = 0 = = < the formula < L=η = 0 = L=η < follows from12L=η = = Lη =.

Remark 2.1.6. Let α ∈ A∗(M,G) locally be given by αI⊗I+αJ⊗J+αK⊗K.Then we obtain the adjoint

L∗η : A∗(M,G)→ A∗(M)

of Lη : A∗(M)→ A∗(M,G) by the following formula

L∗η(α) = ∗−1LωI ∗ (αI) + ∗−1LωJ ∗ (αJ) + ∗−1LωK ∗ (αK).


To check this choose β in A∗(M) and α as above. Since I, J,K is an or-thonormal basis of G we obtain

〈β ∧ η, α〉G = 〈β ∧ ωI , αI〉+ 〈β ∧ ωJ , αJ〉+ 〈β ∧ ωK , αK〉.

Hence, we show with

〈Lη(β), α〉G vol = β ∧ ωI ∧ ∗αI + β ∧ ωJ ∧ ∗αJ + β ∧ ωK ∧ ∗αK= (〈β, ∗−1LωI ∗ (αI)〉+ 〈β, ∗−1LωJ ∗ (αJ)〉

+ 〈β, ∗−1LωK ∗ (αK)〉) vol,

the formula for L∗η.

Bonan gives the following Lefschetz theorem for η:

Theorem 2.1.7 ([Bon83, Theorem 4, Theorem 5]). Let (M,G, g) be a 4n-dimensional quaternionic Hermitian manifold. Then the following maps areisomorphisms for 0 < m ≤ 2n;

Lmη : A2n−m(M,H) −→ A2n+m(M,H).

We call a form φp ∈ Ap(M,H) η-primitive if L2n−p+1η (φp) = 0. Then every

form φp ∈ Ap(M,H) has a unique decomposition

φp =∑`

L`η(µp−2`)

into η-primitive forms µp−2`.

Remark 2.1.8. Note ([Bal06, 1.16]) that for γ, λ (as in Remark 2.1.3) ingeneral

α ∧γ β − (−1)deg(α) deg(β)β ∧γ α = α ∧λ β.In our special case, if we decompose α and β as in Equation (2.1),

(αI ⊗ I) ∧γ (β1 ⊗ 1) = (−1)deg(α) deg(β)(β1 ⊗ 1) ∧γ (αI ⊗ I),

(αI ⊗ I) ∧γ (βI ⊗ I) = (−1)deg(α) deg(β)(βI ⊗ I) ∧γ (αI ⊗ I),

(αI ⊗ I) ∧γ (βJ ⊗ J) = −(−1)deg(α) deg(β)(βJ ⊗ J) ∧γ (αI ⊗ I).

In the definitions of Lω(α) on a Kähler manifold, LΩ(α) on a quaternionicKähler manifold for α ∈ A∗(M) or α ∈ A∗(M,H), respectively, and Lη(α)for α ∈ A∗(M,R), we have ω ∧ α = α ∧ ω, Ω ∧(γ) α = α ∧(γ) Ω, andη ∧γ α = α ∧γ η. The order is a matter of taste since deg(ω), deg(Ω), anddeg(η) are even. By contrast the two linear extensions Lη(α) = η ∧γ α andLη(α) = α ∧γ η of Lη from A∗(M,R) to A∗(M,H) differ by L=η (α).


We now come to the main theorem of this section.

Theorem 2.1.9. Let M be a quaternionic Hermitian 4n-dimensional mani-fold. Then the fundamental two-form η and the curvature two-form κ inducethe chain complex

0 −→ A`(M)Lη−−→ A`+2(M,G)

Lκ−−→ A`+4(M,G) −→ 0. (2.2)

It is exact in A`(M) for ` < 2n and in A`+2(M,G) if `+2 < 2n or `+2 > 3n.

This means that Lκ is injective on A0(M,G) and A1(M,G) and Lη issurjective onto A4n−1(M,G) and A4n(M,G).

Proof. To show that Lκ Lη vanishes on A∗(M) use Proposition 2.1.5 andthe following argument:

Lκ Lη < = = Lη = Lη <= = Lη Lη <= −= LΩ <= −LΩ = <= 0.

Now use an open cover by sets U with G|U = 〈IU , JU , KU〉. To see thatLη : A`(M) → A`+2(M,G) is injective for ` < 2n we can use either theisomorphism theorem for η or the ones for ωI , ωJ , and ωK .

Now we show how to get Ker(Lκ) ⊆ Im(Lη) using Lemma 2.1.10. Assumethat αI ⊗ IU + αJ ⊗ JU + αK ⊗KU is an element of Ker(Lκ)|U . Then

0 = Lκ(αI ⊗ IU + αJ ⊗ JU + αK ⊗KU)

= (αJ ∧ ωK − αK ∧ ωJ)⊗ IU+ (αK ∧ ωI − αI ∧ ωK)⊗ JU+ (αI ∧ ωJ − αJ ∧ ωI)⊗KU .

The pointwise identities

Lk(αJ) = Lj(αK), Li(αK) = Lk(αi), and Lj(αI) = Li(αJ)

allow us to use the following Lemma 2.1.10 to construct a β with

αI = β ∧ ωI , αJ = β ∧ ωJ and αK = β ∧ ωK .

Therefore, we can write the form

αI ⊗ IU + αJ ⊗ JU + αK ⊗KU = β ∧ ωI ⊗ IU + β ∧ ωJ ⊗ JU+ β ∧ ωK ⊗KU

as Lη(β) and this is in Im(Lη)|U .


Let i, j, and k be a triple of anti-commuting complex structures compati-ble with the metric of the quaternionic Hermitian vector space V . We adoptthe notation L1 = Li, L2 = Lj, and L3 = Lk of [Ver90] in the next lemma:

Lemma 2.1.10 (Lemma E). Let V be a 4n-dimensional quaternionic Hermi-tian vector space as above. Let k < 2n or k > 3n, and let α1, α2, α3 ∈

∧k(V ∗)be three k-forms with

L1(α2) = L2(α1), L2(α3) = L3(α2), and L1(α3) = L3(α1).

Then there is a β ∈∧k−2(V ∗) with

L1(β) = α1, L2(β) = α2, and L3(β) = α3.

Proof. This is equivalent to Lemma 3.2.1. By Proposition 1.1.1, β is uniquelydetermined by these properties for k < 2n.

2.2 Parallel forms on quaternionic Kähler man-ifolds

After Section 2.1 where we broadened our focus to discuss quaternionic Her-mitian manifolds we come back to quaternionic Kähler manifolds. Recallthat a Hermitian manifold (M,J, g) is Kähler if and only if its Kähler formωJ is closed ([Bes87, 2.B]). This is the case if and only if the Kähler formωJ or the complex structure J is parallel. Analogous properties single outthe quaternionic Kähler manifolds in the quaternionic Hermitian manifolds(M,G, g). Here the analogs of J and ωJ are the bundle G and the form Ω.

A quaternionic Hermitian manifold (M,G, g) is quaternionic Kähler (orhyperkähler) if and only if one of G,Ω, η, and J is parallel (and hence all ofthem are). Then Ω, η, and J are closed (see [Bal06, 1.18]). The converse fordim(M) ≥ 12 and Ω was proven in [Swa90, 5.2.4] (dΩ = 0⇔ DΩ = 0).

Proposition 2.2.1. A Riemannian manifold admits the structure of a quater-nionic Kähler manifold if and only if there is a parallel subbundle G ⊂End(TM) that is not flat with respect to the Levi–Civita connection suchthat (M,G) is a quaternionic Hermitian manifold. Furthermore (M,G) ishyperkähler if G is parallel and flat.

If we write the local orthonormal frame of G as J1, J2, J3, we get

∇XJ1 = α12(X)J2 − α31(X)J3,

∇XJ2 = −α12(X)J1 + α23(X)J3,

∇XJ3 = α31(X)J1 − α23(X)J2.

(2.3)


with non-vanishing αij(X) for quaternionic Kähler manifolds and vanishingαij(X) for hyperkähler manifolds. Using this formula, one can prove thefollowing theorems.

Theorem 2.2.2 ([Bes87, 14.40], [Ish74, 2.13], [Kak95, 3.8]). Let (M,G) bea 4n-dimensional quaternionic Kähler manifold. Then M is an Einsteinmanifold with Ric = λg for a constant λ 6= 0. The curvature of G is 1

n+2λκ.

Proof. In [Bes87, 14.40] and [Ish74, 2.13] there are proofs; the one in [Bes87,14.40] is corrected by a factor of 2 in [Kak95, 3.8].

Theorem 2.2.3. The tensors η,Ω, and J on a quaternionic Hermitian man-ifold are parallel if and only if M is a quaternionic Kähler manifold or ahyperkähler manifold.

Proof. Compare [Ish74, (1.5),(1.6)], [Bes87, 14.36]: For ωi(Z,W ) := g(JiZ,W )we get

X(ωi(Z,W )) = ∇X(ωi)(Z,W ) + ωi(∇XZ,W ) + ωi(Z,∇XW )

= ∇X(ωi)(Z,W ) + g(Ji(∇XZ),W ) + g(JiZ,∇XW ),

as well as

X(ωi(Z,W )) = (∇Xg)(JiZ,W ) + g(∇X(JiZ),W ) + g(JiZ,∇XW )

= g((∇XJi)Z,W ) + g(Ji(∇XZ),W ) + g(JiZ,∇XW ).

Thus,

∇X(ωi)(Z,W ) = g((∇XJi)Z,W )

= g(αij(X)JjZ,W ) + g(αik(X)JkZ,W )

= αij(X)ωj(Z,W ) + αik(X)ωk(Z,W ),

and hence

∇X(ω1) = α12(X)ω2 − α31(X)ω3,

∇X(ω2) = −α12(X)ω1 + α23(X)ω3,

∇X(ω3) = α31(X)ω1 − α23(X)ω2.


Now the calculation

∇X(ω1 ⊗ J1) = ∇X(ω1)⊗ J1 + ω1 ⊗∇X(J1)

= α12(X)ω2 ⊗ J1 − α31(X)ω3 ⊗ J1

+ α12(X)ω1 ⊗ J2 − α31(X)ω1 ⊗ J3,

∇X(ω2 ⊗ J2) = ∇X(ω2)⊗ J2 + ω2 ⊗∇X(J2)

= −α12(X)ω1 ⊗ J2 + α23(X)ω3 ⊗ J2

− α12(X)ω2 ⊗ J1 + α23(X)ω2 ⊗ J3,

∇X(ω3 ⊗ J3) = ∇X(ω3)⊗ J3 + ω3 ⊗∇X(J3)

= α31(X)ω1 ⊗ J3 − α23(X)ω2 ⊗ J3

+ α31(X)ω3 ⊗ J1 − α23(X)ω3 ⊗ J2

shows us∇X(η) = ∇X(ω1 ⊗ J1 + ω2 ⊗ J2 + ω3 ⊗ J3) = 0.

Therefore, η is parallel. Since

∇X(J1 ⊗ J1) = ∇X(J1)⊗ J1 + J1 ⊗∇X(J1)

= α12(X)J2 ⊗ J1 − α31(X)J3 ⊗ J1,

+ α12(X)J1 ⊗ J2 − α31(X)J1 ⊗ J3

∇X(ω1 ∧ ω1) = ∇X(ω1) ∧ ω1 + ω1 ∧∇X(ω1)

= α12(X)ω2 ∧ ω1 − α31(X)ω3 ∧ ω1

+ α12(X)ω1 ∧ ω2 − α31(X)ω1 ∧ ω3,

similar calculations show that

∇XΩ = ∇X(ω1 ∧ ω1 + ω2 ∧ ω2 + ω3 ∧ ω3) = 0,

∇XJ = ∇X(J1 ⊗ J1 + J2 ⊗ J2 + J3 ⊗ J3) = 0.

Thus, Ω and J are parallel. Assume M is not locally quaternionic Kähleror hyperkähler. Then we write Ei(X) ∈ End(TM) for the part of ∇XJiorthogonal to G and get

∇XJi = Ei(X) + αij(X)Jj − αki(X)J.

We define ωEi(X)(v, w) = g(Ei(X)v, w) and get

∇XΩ = ωE1(X) ∧ ω1 + ω1 ∧ ωE1(X)

+ ωE2(X) ∧ ω2 + ω2 ∧ ωE2(X)

+ ωE3(X) ∧ ω3 + ω3 ∧ ωE3(X)

= 2(ωE1(X) ∧ ω1 + ωE2(X) ∧ ω2 + ωE3(X) ∧ ω3),

2.3. THE MAP Lη AND VANISHING THEOREMS 35

∇Xη = ωE1(X) ⊗ J1 + ω1 ⊗ E1(X)

+ ωE2(X) ⊗ J2 + ω2 ⊗ E2(X)

+ ωE3(X) ⊗ J3 + ω3 ⊗ E3(X),

∇XJ = E1(X)⊗ J1 + J1 ⊗ E1(X)

+ E2(X)⊗ J2 + J2 ⊗ E2(X)

+ E3(X)⊗ J3 + J3 ⊗ E3(X).

If one of these sums is 0, then E1(X) = E2(X) = E3(X) = 0 since thesummands are orthogonal.

2.3 The map Lη and vanishing theorems

Since κ is for quaternionic Kähler manifolds, up to a constant, the curvatureof G, Theorem 2.1.9 can be written as:

Theorem 2.3.1 (Theorem C). Let M be a 4n-dimensional quaternionicKähler manifold. Then

0 −→ A`(M)Lη−−→ A`+2(M,G)

d∇d∇−−−→ A`+4(M,G) −→ 0 (2.4)

is a chain complex. It is exact in A`(M) for ` < 2n and in A`+2(M,G) if`+ 2 < 2n or `+ 2 > 3n.

The exterior derivative d∇ on A∗(M,G) (resp. d on A∗(M)) commuteswith Lη

d∇ Lη = Lη d.

Therefore, d-closed (exact) forms are sent to d∇-closed (exact) forms.With the help of Theorem 2.3.1 we can use Lη to transfer results about

the cohomology of M to the study of d∇-closed and d∇-exact forms withvalues in G.

Corollary 2.3.2. Let m < 2n. A G-valued m-form is in the kernel of d∇d∇if and only if it is in the image of Lη. Moreover, for α ∈ A`(M) with` < 2n− 2 and dα 6= 0 we obtain d∇Lη(α) = Lη(dα) 6= 0.

Hence, Lη is a bijection between d-closed forms in Am−2(M) and d∇-closed forms in Am(M,G).

Since d∇-exact forms are not necessarily in the kernel of d∇ d∇ the mapLη is not necessarily surjective onto the space of d∇-exact forms. For m > 3nthe map Lη is a bijection between the forms α ∈ Am−2(M) with dα ∈ ker(Lη)and the d∇-closed forms in Am(M,G).


Theorem 2.3.3. Let M be a closed 4n-dimensional quaternionic Kählermanifold. Then the d∇-closed (2m + 1)-forms in A2m+1(M,G) are d∇-exactfor 2m − 1 < n. If M has positive scalar curvature, this is also true form < n.

Proof. As we mentioned in Theorem 1.4.1 all odd Betti numbers b2m−1 vanishfor positive scalar curvature [Sal82, Theorem 6.6], and for 2m − 1 < n thisis also true for all closed quaternionic Kähler manifolds [SW02, Proposition5.8]. By Corollary 2.3.2, for 2m + 1 < 2n (⇔ m < n) every d∇-closed(2m + 1)-form α can be written as Lη(β) for a d-closed (2m − 1)-form β.Thus, under the conditions of the theorem for the degree of α there is a γwith dγ = β. Hence,

d∇(Lη(γ)) = Lη(dγ) = Lη(β) = α

and α is d∇-exact.

For forms of even degree this is not true:

Example 2.3.4. On a quaternionic Kähler manifold the forms Lη(Ωm) areparallel and hence d∇-closed (4m+ 2)-forms. For 2m+ 2 < n and M closedthey cannot be d∇-exact. If there were an α with d∇(α) = Lη(Ω

m), we wouldhave d∇d∇(α) = 0. Therefore, α = Lη(γ). Furthermore,

Lη(d(γ)) = d∇(Lη(γ)) = d∇(α) = Lη(Ωm)

gives us dγ = Ωm since Lη is injective. But the form Ωm is not exact, compare1.4.2.

Similarly, we can find a corollary from our vanishing of L2-cohomology onsimply connected negatively curved quaternionic Kähler manifolds. There isan inner product on G ⊂ H ⊂ End(TM) defined by 〈A,B〉 = 1

4ntr(ABt).

This allows us to define L2-norms on A∗(M,G). Denote by A∗2(M,G) thespace of square integrable G-valued forms.

Theorem 2.3.5. Let M be a complete simply connected 4n-dimensionalquaternionic Kähler manifold with nonpositive sectional curvature and m <2n. Then for every d∇-closed square integrable m-form α ∈ Am

2 (M,G) thereis a square integrable form γ ∈ Am−1

2 (M,G) with d∇(γ) = α. Moreover, thesame holds without the square integrable; every d∇-closed m-form α is givenas d∇(γ).

2.3. THE MAP Lη AND VANISHING THEOREMS 37

Proof. For m < 2n every d∇-closed form α ∈ Am(M,G) is given as Lη(β) fora closed form β ∈ Am−2(M). By the theorem of Hadamard-Cartan β = d(δ)and for γ = Lη(δ) we obtain α = d∇γ. This proves the second part of thetheorem. By Theorem 1.2.5 we can choose δ square integrable if β is.

Hence, to prove the theorem we claim that am-form β is square integrableif and only if Lη(β) is square integrable (for m < 2n). Since I, J, and K forman orthonormal basis of Gp we obtain (see Proposition 1.1.1 and Section 3.2for the sl2-representation given by Li, H,Λi):

‖Lη(β)‖2p = ‖ωI ∧ β ⊗ I + ωJ ∧ β ⊗ J + ωK ∧ β ⊗K‖2

p

= ‖ωI ∧ β‖2p + ‖ωJ ∧ β‖2

p + ‖ωK ∧ β‖2p

= 〈LI(β), LI(β)〉p + 〈LJ(β), LJ(β)〉p + 〈LK(β), LK(β)〉p= 〈ΛILI(β), β〉p + 〈ΛJLJ(β), β〉p + 〈ΛKLK(β), β〉p

We can use the computation in [FH91, Claim 11.4] to estimate the eigenvaluesof the three analogous operators ΛILI ,ΛJLJ , and ΛKLK . Decompose

βp =

bm/2c∑`=0

Lì(β`p),

with ΛI(β`p) = 0. Then

ΛILILÌ(β

`p) = (`+ 1)(`+ 2n−m)Lì(β

`p),

lead to the estimates

3(2n−m)‖β‖2p ≤ ‖Lη(β)‖2

p ≤ 3(dm/2e)(2n− dm/2e)‖β‖2p,

for an m-form β. The corresponding estimates for ‖β‖2 and ‖Lη(β)‖2 provethe claim.

CHAPTER3The cohomology of so5(C)-modules

The main ingredient of the results about d∇-closed forms (Theorem D) inthe last chapter is Theorem C discussing the exactness of the chain complex

0 −→ A`(M)Lη−−→ A`+2(M,G)

d∇d∇−−−→ A`+4(M,G) −→ 0

and hence Lemma 2.1.10 about the linear algebra of quaternionic Hermitianvector spaces.

As often in linear algebra, we found an elementary proof of Lemma 2.1.10sketched in Remark 3.4.2 (we omit its tedious details since they are 20 pagesof formulas).

In this chapter we will prove Lemma 2.1.10 with the help of vanishingresults for certain Lie algebra cohomology groups. All Lie algebras and rep-resentations in this chapter are finite dimensional.

In Section 3.1 we state Theorem 3.1.2 ([Kos61, Theorem 5.14]) aboutLie algebra cohomology. In order to do this, we introduce notations in Liealgebra theory and refer to [Kos61], [CE56], [Hum78], and [FH91] for detaileddefinitions.

In Section 3.2 we show, following [Ver90], that the action induced byL1, L2, and L3 on

∧∗(Hn)⊗RC turns the latter into an so5(C)-module. Thenwe specialize all notions introduced in Section 3.1 to the case g = so5(C).

In Section 3.3 we combine the results of the first two sections to describein Theorems 3.3.1 and 3.3.2 the cohomology groups Hj(n,

∧∗(Hn)⊗RC) andHj(n, V ). Here V is any so5(C)-module and n = 〈L1, L2, L3〉C ⊂ so5(C).

Finally, in Section 3.4 we use Theorems 3.3.1 and 3.3.2 to prove Lemma2.1.10 and its generalization Lemma 3.4.1. We also sketch an elementaryproof of Lemma 2.1.10.

40 CHAPTER 3. THE COHOMOLOGY OF so5(C)-MODULES

3.1 Lie algebra cohomology

Given a complex Lie algebra l and an l-module V , we recall the definitionof the Lie algebra cohomology groups Hn(l, V ). Then we introduce certainsubalgebras of a semisimple Lie algebra g to state Theorem 3.1.2. In the nextsection we will specialize this to g = so5(C).

Definition 3.1.1. The Lie algebra cohomology of the l-module V (see [CE56,Section XIII.2.]) is defined as

Hn(l, V ) = ExtnU(l)(C, V ),

where U(l) is the universal enveloping algebra of l.

Let E∗(l) be the exterior algebra over l as a complex vector space (while∧∗(V ) is the exterior algebra over V as a real vector space). In [CE56, SectionXIII.7, XIII.8] it is shown that Hn(l, V ) is the homology of the Chevalley–Eilenberg complex Cq(l, V ) = HomC(Eq(l), V ), which we will now recall. Aq-cochain f : Eq(l)→ V is a C-linear alternating function of q variables in l,with values in V . The coboundary δ is defined as

(δf)(x1, . . . , xq+1) =∑

1≤i≤q+1

(−1)i+1xif(x1, . . . , xi, . . . , xq+1)

+∑

1≤i<j≤q+1

(−1)i+jf([xi, xj], . . . , xi, . . . , xj, . . . , xq+1).

We now introduce the setting of [Kos61].Let g be a semisimple complex Lie algebra. We choose a Borel subalgebra

b ⊂ g (i.e., b is maximal solvable, see [Hum78, Section 16.3]) and a compactreal form gc of g. Let ∗ : g → g be the C-antilinear involution induced byg = gc ⊕ igc and let (v, w) be the Killing form. Then v, w = (v, w∗) is aHermitian inner product and h = b ∩ b∗ is a Cartan subalgebra (see [Kos61,page 349]).

We denote by ∆ ⊂ h′ the set of roots of g and by eφ a choice of rootvectors. Moreover, for subspaces r ⊂ g invariant under the adjoint action ofh we define ∆(r) ⊂ ∆ by

r = (r ∩ h)⊕⊕φ∈∆(r)

〈eφ〉C.

We choose ∆+ = ∆(b) as positive roots and we let ρ be half the sum of thepositive roots.

3.2. THE so5(C)-ACTION ON∧∗(HN)⊗R C 41

A Lie algebra p with b ⊆ p ⊆ g is called a parabolic subalgebra of g (see[Hum78, Section 16.5]). Let p be such a parabolic subalgebra.

We denote by n the maximal nilpotent ideal of p and by g1 the intersectionp∩ p∗. Then g = n∗ ⊕ g1 ⊕ n and p = g1 ⊕ n are orthogonal direct sums (see[Kos61, Proposition 5.3.]).

Assume λ is a dominant integral weight. Let V (λ) be the irreduciblerepresentation of g whose highest weight is λ. Now we define the action βof g1 on Hn(n, V (λ)) (see [Kos61, 5.7]). The exterior algebra E(n) ⊂ E(g)is stable under the adjoint representation of g1 since g1 is in the normalizerof n. Let g′ denote the dual of g with the dual representation. Then E(n)′

is a g1-module. The q-cochain space Cq(n, V ) = Hom(Eq(n), V (λ)) inher-its the tensor representation of g1. This action induces the g1-action β onHn(n, V (λ)).

Let W be the Weyl group of g (see [Hum78, Sections 9 and 10]). Denoteby m the subspace of all nilpotent elements of b and by m1 = m ∩ g1.

We define W1 ⊂ W as the subset of all σ ∈ W with σ−1(∆(m1)) ⊂ ∆+.Furthermore W1(j) shall denote the set of all elements of W1 of length j.For σ ∈W and λ a dominant weight we define ξσ(λ) as σ(ρ+ λ)− ρ.Theorem 3.1.2 ([Kos61, Theorem 5.14]). Let λ be a dominant integralweight. Given a g1-module W , we decompose it into isotropic componentsW =

⊕µW

µ where W µ is a direct sum of g1-modules isomorphic to the sim-ple representation with highest weight µ. With the notation above we obtain

Hj(n, V (λ)) =⊕

σ∈W1(j)

Hj(n, V (λ))ξσ(λ) (3.1)

for all integers j ≥ 0.

3.2 The so5(C)-action on∧∗(Hn)⊗R C

Let H = 〈id, J1, J2, J3〉R be the division ring of Hamilton’s quaternions andendow Hn with the standard Euclidean product 〈·, ·〉. We denote by

∧∗(Hn)the exterior algebra over the real vector space Hn and by ωi the two-form〈Ji·, ·〉. We can decompose a form α ∈

∧∗(Hn) ⊗R C into Re(α) + i Im(α)with Re(α) and Im(α) in

∧∗(Hn). Denote by

Li :∗∧

(Hn)⊗R C→∗∧

(Hn)⊗R C

the wedge product with ωi. Since ωi is a real two-form Li :∧∗(Hn)→

∧∗(Hn)is well defined. Moreover, Li commutes with Im and Re. Thus, Lemma 2.1.10is equivalent to the following complex version.


Lemma 3.2.1. Let k < 2n or k > 3n, and let α1, α2, α3 ∈∧k(Hn)⊗R C be

three k-forms such that

L1(α2) = L2(α1), L2(α3) = L3(α2), and L1(α3) = L3(α1).

Then there is a β ∈∧k−2(Hn)⊗R C with

L1(β) = α1, L2(β) = α2, and L3(β) = α3.

Verbitskiı showed in [Ver90] that there is a natural so5(C)-action on∧∗(Hn) ⊗R C and a natural so4,1(R)-action on∧∗(Hn). This leads to a

hyperkähler analog of the Lefschetz Theorem for Kähler manifolds based onProposition 1.1.1. We introduce the operators that define the so5(C)-action.Then we present an explicit proof of Verbitskiı’s result since there is a signerror in the concise [Ver90].

DefineH on∧∗(Hn)⊗RC as counting operator, H|∧k(Hn)⊗RC = (k−2n) id,

and Λi as the adjoint ∗−1Li∗ of Li. By [Huy05, Section 1.2] (see Proposition1.1.1) the operators L1,Λ1, L2,Λ2, L3,Λ3, and H induce three sl2(C)-actionson∧∗(Hn)⊗R C and three sl2(R)-actions on

∧∗(Hn), i.e.

[Lj,Λj] = H, [H,Lj] = 2Lj, [H,Λj] = −2Λj. (3.2)

It is easy to see that the Lie algebras n := 〈L1, L2, L3〉C and n∗ := 〈Λ1,Λ2,Λ3〉Care commutative. However, the commutators Kjk := [Λj, Lk] for j 6= k donot vanish in general.

Theorem 3.2.2 (See also [Ver90, Theorem 1.2]). The operators L1, L2, L3,Λ1,Λ2,Λ3, K12, K23, K31, and H define an so5(C)-action on

∧∗(Hn)⊗R C .

Proof. We first present the commutator relations of these operators in Lemma3.2.3. Then we give a change of basis to the standard one given in [FH91,page 270].

Lemma 3.2.3. The operators defined above satisfy the relations

Kjk = [Λj, Lk] = [Lj,Λk] = −[Λk, Lj] = −Kkj (3.3)

[Kjk, Kkl] = 2Kjl, [Kkj, Lj] = 2Lk, [Kkj,Λj] = 2Λk, (3.4)

0 = [Kjk, Ll] = [Kjk,Λl] = [Kjk, H], (3.5)

for j, k, l pairwise different and Equation (3.2).


Proof. For any complex structure J compatible with 〈·, ·〉 we obtain Lefschetzoperators LJ(α) = α ∧ ωJ (with ωJ(v, w) = 〈Jv, w〉) and ΛJ = ∗−1 LJ ∗satisfying Equations (3.2) by Proposition 1.1.1. Since 1√

2(Jj + Jk) is of this

type for j 6= k ∈ 1, 2, 3 we get

H =1

2[Lj + Lk,Λj + Λk]

=1

2([Lj,Λj] + [Lk,Λk] + [Lj,Λk] + [Lk,Λj])

= H +1

2([Lj,Λk] + [Lk,Λj]),

which proves Kjk = −Kkj (Equation (3.3)). Using the Jacobi identities

[[Λj, Lk], Li] + [[Lk, Li],Λj] + [[Li,Λj], Lk] = 0, (3.6)

[[Lj,Λk],Λi] + [[Λk,Λi], Lj] + [[Λi, Lj],Λk] = 0, (3.7)

for i, j, k pairwise different we obtain (by permutations and using [Lk, Li] =[Λi,Λj] = 0 and Kjk = −Kkj)

[Kjk, Li] = [Kik, Lj] = [Kij, Lk] = [Kkj, Li] = −[Kjk, Li],

[Kjk,Λi] = [Kik,Λj] = [Kij,Λk] = [Kkj,Λi] = −[Kjk,Λi],

[Kij, Lk] = 0 = [Kij,Λk].

Moreover, for i = j 6= k the Equations (3.6) and (3.7) give us

[Kjk, Lj] = −[[Lj,Λj], Lk] = −[H,Lk] = −2Lk,

[Kjk,Λj] = [[Lj,Λj],Λk] = [H,Λk] = −2Λk.

Finally, the Jacobi identities

[[Λj, Lk], H] + [[Lk, H],Λj] + [[H,Λj], Lk] = 0,

[[Λj, Lk], Kki] + [[Lk, Kki],Λj] + [[Kki,Λj], Lk] = 0,

give us for i, j, k pairwise different

[Kjk, H] = 2[Lk,Λj] + 2[Λj, Lk] = 0 and [Kjk, Kki] = 2Kji.

Define so5(C) (see [FH91, page 270]) as the space of complex 5×5 matricesA such that

Q(Ax, y) +Q(x,Ay) = 0


for the symmetric bilinear form

Q(x, y) = xt ·

0 0 1 0 00 0 0 1 01 0 0 0 00 1 0 0 00 0 0 0 1

· y.We give an isomorphism of the Lie algebra g spanned by the operators abovewith the relations (3.2)-(3.5) and so5(C).

We choose the Cartan subalgebra h to be spanned by1

2H = H1 7→ E1,1 − E3,3 and

1

2iK12 = H2 7→ E2,2 − E4,4.

Let ε1, ε2 be the dual basis with 〈Hi, εj〉 = δij of h′. Then the corre-sponding roots are

∆ = ±ε1,±ε2,±ε1 ± ε2.The roots ε1 and ε2 have root vectors

− 1√2L3 = U1 7→ E1,5 − E5,3 and

1

2√

2(K31 + iK23) = U2 7→ E2,5 − E5,4.

The roots −ε1 and −ε2 have root vectors1√2

Λ3 = V1 7→ E3,5 − E5,1 and1

2√

2(K31 − iK23) = V2 7→ E4,5 − E5,2.

With this choice we obtain [V1, U1] = H1 and [V2, U2] = H2. The commuta-tors of Ui and Vi complete the basis of so5(C);

[U2, U1] = Y, [V2, V1] = Z, [V2, U1] = X1,2, and [V1, U2] = X2,1.

We obtain the following; The roots ε1 + ε2 and −ε1 − ε2 have root vectors1

2(L1 − iL2) = Y 7→ E1,4 − E2,3 and − 1

2(Λ1 + iΛ2) = Z 7→ E3,2 − E4,1.

The roots ε1 − ε2 and ε2 − ε1 have root vectors1

2(L1 + iL2) = X1,2 7→ E1,2 − E4,3 and

1

2(Λ1 − iΛ2) = X2,1 7→ E2,1 − E3,4.

It is not hard to check that the remaining (non-vanishing) commutator rela-tions are the following:

[Z, Y ] = H1 +H2, [X1,2, X2,1] = H1 −H2,

[Y, V1] = −U2, [Y, V2] = U1, [Z,U1] = −V2, [Z,U2] = V1

[X1,2, V1] = −V2, [X1,2, U2] = U1, [X2,1, U1] = U2, and [X2,1, V2] = −V1.

This proves that the mapping is a morphism of Lie algebras.


Remark 3.2.4. The subalgebra g0 = 〈H1, H2, U1, V1, U2, V2, X1,2, X2,1, Y, Z〉Ris a split real form of so5(C). We need a compact real form gc of so5(C)and the corresponding conjugate linear involution ∗ : so5(C) → so5(C) (see[Kos61, 3.3.]). We use the instruction on [FH91, page 436] to construct theconjugate linear involution ∗ corresponding to gc using our basis of g0. Wedefine H∗i = −Hi, U∗i = Vi, V ∗i = Ui, X∗i,j = Xj,i, Z∗ = Y , and Y ∗ = Z.Hence, ∗ is given in Verbitskiı’s basis by H∗ = −H, K∗ij = Kij, L∗i = −Λi,and Λ∗i = −Li. As a compact form we find the Lie algebra gc ∼= so5(R) fixedby ∗ and equal to

〈K12, K23, K31, iH, L1−Λ1, L2−Λ2, L3−Λ3, iL1 + iΛ1, iL2 + iΛ2, iL3 + iΛ3〉R.

Given the compact and split forms in terms of Verbitskiı’s basis it is nothard to see that 〈L1, L2, L3,Λ1,Λ2,Λ3, K12, K23, K31, H〉R defines an so4,1(R)-action on

∧∗(Hn) (see [Ver90, Theorem 1.2.]).

We choose

b = 〈H,K12, L1, L2, L3, K31 + iK23〉C,

consistent with our choice of h = 〈H, iK12〉C = b∩b∗, such that the nilpotentpart is

m = 〈L1, L2, L3, K31 + iK23〉C.

The definition ∆(b) = ∆+ gives us α = ε1 − ε2 and β = ε2 as simple rootsα, β = Π ⊂ ∆. The positive roots are

∆+ = α, β, α + β, α + 2β = ε2, ε1 + ε2, ε1, ε1 − ε2.

Let ρ = 12(3ε1 + ε2) be half the sum of the positive roots. We obtain the

following root diagram (compare [FH91, page 277])


−ε1 + ε2 ε2 = β ε1 + ε2 = α + 2β

ε1 = α + β−ε1

ρ

−ε1 − ε2 −ε2 ε1 − ε2 = α

Let us fixp = 〈H,L1, L2, L3, K12, K23, K31〉C.

We obtaing1 = 〈H,K12, K23, K31〉C = p ∩ p∗,

n = 〈L1, L2, L3〉C as maximal nilpotent ideal of p, and n∗ = 〈Λ1,Λ2,Λ3〉C(which agrees with the notation in the beginning of the section).

Moreover, we get

m1 = m ∩ g1 = 〈K31 + iK23〉C,

so that ∆(m1) = β.

3.3 Cohomology of so5(C)-modules

After the preliminaries in the last two sections we can use Theorem 3.1.2to prove Theorems 3.3.1 and 3.3.2. In the next section we will use theirrespective (1.)-parts to prove Lemmas 3.2.1 and 3.4.1.

3.3. COHOMOLOGY OF so5(C)-MODULES 47

Theorem 3.3.1 (Theorem F). Let n = 〈L1, L2, L3〉C ⊂ so5(C) and H beas above. The cohomology groups Hq(n,

∧∗(Hn) ⊗R C) vanish if and onlyif q /∈ 0, 1, 2, 3. Decompose Hq(n,

∧∗(Hn) ⊗R C) into the eigenspaces ofthe H-action as

⊕j∈ZH

q(n,∧∗(Hn) ⊗R C)j with H · vj = jvj for vj ∈

Hq(n,∧∗(Hn)⊗R C)j. Then

(0.) The space H0(n,∧∗(Hn) ⊗R C)j does not vanish if n ≤ j ≤ 2n and

vanishes for j > 2n.

(1.) The space H1(n,∧∗(Hn)⊗R C)j does not vanish if and only if

−2 ≤ j ≤ n− 2.

(2.) The space H2(n,∧∗(Hn)⊗R C)j does not vanish if and only if

−n− 4 ≤ j ≤ −4.

(3.) The space H3(n,∧∗(Hn)⊗RC)j does not vanish if −2n−6 ≤ j ≤ −n−6

and vanishes for j < −2n− 6.

We give the proof together with the one for the case of general so5(C)-modules treated in the following theorem.

Theorem 3.3.2 (Theorem G). Let n = 〈L1, L2, L3〉C ⊂ so5(C) and H beas above. Let V be a finite dimensional left so5(C)-module. The cohomologygroups Hq(n, V ) vanish if and only if q /∈ 0, 1, 2, 3. Decompose Hq(n, V )into the eigenspaces of the H-action as

⊕j∈ZH

q(n, V )j with H · vj = jvj forvj ∈ Hq(n, V )j. Then

(0.) The space H0(n, V )j vanishes for j < 0.

(1.) The space H1(n, V )j vanishes for j < −2.

(2.) The space H2(n, V )j vanishes for j > −4.

(3.) The space H3(n, V )j vanishes for j > −6.

These results are sharp in the sense that for every pair (n, j) not excludedabove there is a V with Hn(n, V )j 6= 0.

Proof. After some preliminary remarks we will go through the cases (0.)−(3.)using Theorem 3.1.2.

Recall that g1 is the direct sum of the center 〈H〉C and 〈K12, K23, K31〉C ∼=so3(C) ∼= sl2(C). The Weyl group of so5(C) is generated by the simple


reflections sα and sβ. Thus, the subset W1 is given by id, sα, sαsβ, sαsβsαwith lengths ranging from 0 to 3.

With our choice of simple roots α, β, the dominant weights λ are in thefirst octant D (colored in blue and green in the following picture).

β

α

(0, 0) (2n, 2n)

(n, 2n)

Theorem 3.1.2 deals with irreducible representations V (λ) with λ ∈ Dan integral dominant weight, e.g. one of the points above. It is no problemto apply it to any finite dimensional so5(C)-module V and

∧∗(Hn)⊗R C bydecomposing V as sum of irreducible representations V =

⊕λ∈D V (λ)⊕kλ .

Now we specialize to the action on

V =∗∧

(Hn)⊗R C.

We can decompose∧∗(Hn) ⊗R C as

⊕λ∈D V (λ)⊕kλ for some nonnegative

integers kλ and find results for which λ we obtain kλ ≥ 1. Corresponding toour choice of 〈H, iK12〉C as Cartan subalgebra we take J3 as complex structureof Hn. Thus, we can split

∧∗(Hn) into (p, q)-forms. On∧p,q(Hn) ⊗R C the

action of H is given by p+q−2n and the action of iK12 by q−p (see [Ver90]).Hence, for λ = (x, y) for x + y > 2n the kλ = 0 in the decomposition of∧∗(Hn)⊗RC (We choose the basis in a way that the x-coordinate correspond


to theH-action and the y-coordinate to the iK12 action). In the above picturethis means that the highest weights can only be in the blue triangle of thefirst picture and not in the green part. For all integral dominant weights λwith x + y = 2n we will obtain kλ > 0. For the x + y < 2n part we do notknow if kλ = 0. This is the reason why we get no “if and only if“ in the parts(0.) and (3.).

The space of (p, q)-forms is(

2np

)(2nq

)-dimensional, and the space of (p, 2n)-

forms is(

2np

)-dimensional. Therefore, the number k(p,2n−p) of irreducible sub-

representations of∧∗(Hn) ⊗R C with highest weight (p, 2n − p), p ≤ n, is(

2np

)−(

2np−1

)> 0. These weights are represented by the colored dots in the

first octant.

We will describe in (0.) − (3.) the possible ξσ(λ). Since the g1-actioncommutes with H, the eigenvalues of the H-actions are the x-coordinates ofξσ(λ).

(0.) Zeroth, W1(0) = id and ξid(λ) = λ. The action of g1 lives in the firstand eight octants. Thus, the action of H on H0(n, V λ) has no negativeeigenvalues. Since

∧∗(Hn) ⊗R C has subrepresentations with highestweight (2n−q, q), q ≤ n the space H0(n,

∧∗(Hn)⊗RC)j does not vanish


for all n ≤ j ≤ 2n.

(2n, 2n)

(n, 2n)

(1.) First, we obtain W1(1) = sα. This is the main case for H1(n, V )jthat we will need to prove Lemmas 3.2.1 and 3.4.1 in the next section.Hence, we will explain the proof in detail. For the analogous cases (2.)and (3.) we will be concise. The two points in the first octant are thepossible dominant integral weights λ. To find ξsα(λ) = sα((3, 1) + λ)−(3, 1) we reflect λ at the diagonal through −ρ. Since we are interestedin H = 2H1 (and iK12) it is convenient to choose coordinates of h′ suchthat ε1 = (2, 0) and ε2 = (0, 2). Of course all constructions are definedin a coordinate-free manner in terms of α, β and ρ.


(0, 0)

(−3,−1)

The black line y = 0 is sent to the red line x = −2. Hence, the highestweights of the so5(C)-action V (λ) in the first octant are sent to thehighest weights of the g1-action on H1(n, V (λ)) in the second octantstarting in (−2, 2) = sα((3, 1) + (0, 0))− (3, 1). These octants are filledblue and green in the next picture. The weights of the action of g1 onH1(n, V (λ)) live in the first and fourth quadrants around (−2, 0) sincethe 〈K12, K31, K23〉C gives us a vertical sl2(C)-action commuting withthe H-action. Hence H1(n, V (λ)) is an eigenspace of the H-action witheigenvalue the x-coordinate of ξsα(λ) = sα((3, 1) + λ)− (3, 1).

Hence, since ξsα(λ) is always right of the red line x = −2 the action ofH on H1(n, V (λ)) has no eigenvalues less than −2. Since the integralweights on the diagonal x = y are dominant weights (the gray dots inthe picture above) every integral H-eigenvalue with x ≥ 2 appears.

Now we specialize to the action on∧∗(Hn)⊗RC. As we see in the next

picture, this gives us a further restriction of the possible eigenvalues


for H acting on H1(n,∧∗(Hn) ⊗R C) to −2,−1, . . . , n − 2 (H acts

as multiplication with p + q − 2n on (p, q)-forms). Furthermore thereis nonzero cohomology in H1(n,

∧∗(Hn) ⊗R C) with every integral H-eigenvalue between −2 and n− 2.

(n, n) (2n, 2n)

(n, 2n)

p+ q = p+ q =

2n− 2 3n− 2

(2.) Second, W1(2) = sαsβ and ξsαsβ(λ) = sαsβ((3, 1)+λ)−(3, 1). There-


fore the action of H on H2(n, V λ) has no eigenvalues greater than −4.If we specialize to H2(n,

∧∗(Hn) ⊗R C), we have eigenvalues exactlybetween −n− 4 and −4.

(n, n) (2n, 2n)

(n, 2n)

p+ q =p+ q =

2n− 4n− 4

(3.) Third, W1(3) = sαsβsα and ξsαsβsα(λ) = sαsβsα((3, 1) + λ) − (3, 1).Thus, the action of H on H3(n, V λ) has no eigenvalues greater than−6. If we specialize to H2(n,

∧∗(Hn) ⊗R C), there are no eigenvaluesbetween −2n−6 and −n−6 and no eigenvalues less than −2n−6. Theentire cohomology H3(n, V ) lives in the fourth and fifth octant startingin (−6, 0).


(n, n) (2n, 2n)

(n, 2n)

p+ q =p+ q =

2n− 6−6

(4.) Finally, W1(j) = ∅ for j ≥ 4. Thus, Hj(n, V ) vanishes for j ≥ 4.

3.4 Applications to quaternionic geometry

In this section we prove Lemma 3.2.1 as a corollary of Theorem 3.3.1. As inthe last section we combine the proof with the one of the following general-ization of its k < 2n part to any so5(C)-module as a corollary of Theorem3.3.2.

Lemma 3.4.1. Let V be a finite dimensional so5(C)-module. Decompose Vinto H-eigenspaces Vj. Given m > 0 and three elements α1, α2, α3 ∈ V−m.Then the equations

L1(α2) = L2(α1), L2(α3) = L3(α2), and L1(α3) = L3(α1) (3.8)

hold if and only if there is a β ∈ V−m−2 with

L1(β) = α1, L2(β) = α2, and L3(β) = α3. (3.9)

Proof. We will explain why this is just the case (1.) of Theorem 3.3.2. Simi-larly Lemma 3.2.1 is the case (1.) of Theorem 3.3.2.

3.4. APPLICATIONS TO QUATERNIONIC GEOMETRY 55

First, we explain how these propositions are statements about the firstcohomology H1(n, V ). Define a crossed homomorphism f : n → V as a C-homomorphism with

x(f(y))− y(f(x)) = f([x, y]) for all x, y ∈ n,

and a principal crossed homomorphism f : n→ V as one of the form

f(x) = xb

for some fixed b ∈ V . It is not hard to see that the group H1(n, V ) isthe quotient of the crossed homomorphisms Z1(n, V ) modulo the principalcrossed homomorphisms B1(n, V ) in H1(n, V ) (see [CE56, XIII.2.]).

In our setting a crossed homomorphism f ∈ Z1(n, V ) is determined by atriple f(L1) = α1, f(L2) = α2, and f(L3) = α3 satisfying Equation (3.8). Aprincipal crossed homomorphism in f ∈ B1(n, V ) is given by a β ∈ V suchthat Equation (3.9) holds.

Now we have to understand how H acts on H1(n, V ). For all Li ∈ n weobtain ad(H)(Li) = [H,Li] = 2Li. Hence, H acts as 2q id on L ∈ Eq(n)(recall H ∈ g1 ⊂ so5(C)). On the space

Cq(n, V ) = HomC(Eq(n), V ) = Eq(n)′ ⊗C V.

we obtain the tensor representation

−2q id(L′)⊗ α + L′ ⊗ ρ(H)(α).

Lemmas 3.4.1 and 3.2.1 translate to H1(n, V )m = 0 for m < −2 andH1(n,

∧∗(Hn)⊗R C)m = 0 if and only if m < −2 or m > n− 2.

Remark 3.4.2. We found an elementary but lengthy proof for Lemma 3.4.1about V−m except for m = 2. We sketch the idea. For details contact theauthor.

First assume that the crossed homomorphism f : n→ V given by f(Li) =Ai satisfies Λ1(A1) = Λ2(A1) = Λ3(A1) = 0. Assume that H(Ai) = −mAi.

Then we obtain with KijAi = mAj the equations

K212(A1) = −m2A1 and K2

31(A1) = −m2A1. (3.10)

Recall that 〈K12, K23, K31〉C is isomorphic to so3(C) and sl2(C) withK12 7→ −i(x + y), K23 7→ (x − y), and K31 7→ −ih. Equation (3.10)leads to h2(A1) = m2A1 and (x + y)2(A1) = m2A1. In the decompositionof A1 into h-eigenspaces we get A1 = Am1 + A−m1 . If m > 2 or m = 1,


we obtain (x + y)2(Am1 ) = 0 as well as (x + y)2(A−m1 ) = 0 and henceAm1 = 0 = A−m1 = A1.

One can always assume Λ1(A1) = 0 (by subtracting a principal crossedhomomorphism). Since V is a finite dimensional representation there is anN with Λn

2 Λ`3(A1) = 0 for all n + ` > N . For crossed homomorphisms we

get equations similar to Equation (3.10) for the Λn2 Λ`

3(A1) with n + ` = N .Therefore, we can do an induction by showing that Λn

2 Λ`3(A1) = 0 already

for n+ ` = N .It is quite tedious to check this argument, since we have to calculate many

commutators to obtain from Li(Aj) = Lj(Ai) via Λn2 Λ`

3Li(Aj) = Λn2 Λ`

3Lj(Ai)equations of the form An,`,i,jΛ

n2 Λ`

3(A1) = 0.

CHAPTER4L2-index theory

In this chapter we prove a generalization of Atiyah’s L2-index theorem to so-called G-twisted Dirac bundles. In Section 4.1 we introduce Dirac bundles,the Γ-dimension, and Atiyah’s L2-index theorem. Then we discuss in Section4.2 G-twisted bundles and show that they cover the cases used by Gromovand are examples of bundles with bounded geometry. In Section 4.3 weintroduce the notion of equivariant smoothing operators. Finally we stateand prove the L2-index theorem for G-twisted bundles in Section 4.4.

4.1 Atiyah’s L2-index theorem for Dirac oper-ators

We recall the notion of graded Dirac bundles.

Definition 4.1.1. A graded Dirac bundle E (in the sense of Gromov–Lawson,see [Bal06, C.1] and [GL83]; compare also the definition of a Clifford bundlein [Roe88, 2.3] and a Clifford module in [BGV92, 3.32]) over a Riemannianmanifold M is a Hermitian vector bundle with a Hermitian connection ∇E,a Clifford multiplication γ : TM ⊗ E → E, and a Z/2Z-grading (a parallelunitary involution µ : E → E with eigenbundles E = E+ ⊕ E−) that arecompatible in the sense that the following equations hold:

(γ(X)σ, τ) = −(σ, γ(X)τ),

γ(X)γ(Y )σ + γ(Y )γ(X)σ = −2〈X, Y 〉σ,

58 CHAPTER 4. L2-INDEX THEORY

γ(X)(E±) ⊂ E∓,

∇EY (γ(X)σ) = γ(∇M

Y (X))σ + γ(X)∇EY (σ),

for all sections σ, τ of E and all vector fields X and Y on M . The associatedDirac operator D is the composition:

E(E)∇E−→ E(E ⊗ TM∗)

]−→ E(E ⊗ TM)γ−→ E(E),

where the maps are defined by the connection, the metric and the Cliffordmultiplication. For a local orthonormal frame Xj we can write

Dσ =∑

γ(Xj)∇EXjσ.

Since the Clifford multiplication exchanges E+ and E− the Dirac operatordecomposes as

D =

(0 D−

D+ 0

)under the grading E = E+ ⊕ E− (i.e. for D+ : E(E+) → E(E−) andD− : E(E−)→ E(E+) we obtain D = D+ +D−).

Remark 4.1.2. We use [BGV92, 4.2] to describe the index form ωD+. Onclosed manifolds the Atiyah–Singer index theorem for Dirac operators gives

ind(D+) =

∫M

ωD+ .

Since the Z/2Z-grading on E is parallel the connection ∇E is the sum of∇E+

: E+ → E+ and ∇E− : E− → E−. Hence, d∇E : A(M,E) → A(M,E)is a superconnection (compare [BGV92, 1.4]). Denote by

Ω(X, Y ) = ∇EX∇E

Y −∇EY∇E

X −∇E[X,Y ]

the curvature two-form of ∇E. Since d∇Ed∇Eα = Ω ∧ α the form Ω is alsothe curvature of the superconnection d∇

E . Let RE be the action of the Rie-mannian curvature on the Dirac bundle

RE(X, Y ) =1

4

∑k,l

(R(X, Y )Xk, Xl)γ(Xk)γ(Xl).

Define the twisting curvature of E as FE/S = Ω−RE, the chirality operatorΓ as in [BGV92, 3.28, 3.17], and the relative supertrace as

StrE/S(a) = 2−n/2 StrE(Γa).

4.2. G-TWISTED BUNDLES 59

Then we can define the relative Chern character of the bundle E as

ch(E/S) = StrE/S(exp(−FE/S)).

The notation generalizes the classical case E = S ⊗ W with S the spinorbundle and W the twisting bundle. In this case the twisting curvature is thecurvature of the twisting bundle and the relative Chern character the Cherncharacter of the twisting bundle.

The A-genus form A(M) on a Riemannian manifold M of dimension n

is defined as Pf( R/2sinh(R/2)

) (here Pf denotes the Pfaffian).Finally, the index form ωD+ is the n-form component of the product of

ch(E/S) and (2πi)−n2 A(M).

We will generalize Atiyah’s L2-index theorem [Ati76, 1.1] for normal cov-erings in the case of Dirac operators. Let M →M be a Riemannian normalcovering over a closed manifold M . Let E → M be a graded Dirac bundle,E → M its pullback under the covering, and let D, D be the associated Diracoperators.

For a closed subspace H ⊂ L2(E) and an orthonormal basis Φn of H,the function

f(p) : M −→ R+ ∪ ∞, p 7→∞∑n=1

|Φn(p)|2(

= tr(kPH (p, p)

))is independent of the choice of Φn. If H is invariant with respect to theaction of the group of covering transformations Γ on L2(E), f is also invari-ant. In this case we can define the Γ-dimension 0 ≤ dimΓ(H) ≤ ∞ of H asthe integral

∫Ff vol for a fundamental domain F of the Γ-action on M . If E

is the pullback to M of a graded Dirac bundle E on M , the kernels ker(D+)and ker(D−) of D+ and D− = D+∗ are Γ-invariant subspaces of L2(E). IfdimΓ(ker(D+)) and dimΓ(ker(D−)) are finite, the Γ-index is well-defined as

indΓ(D+) = dimΓ(ker(D+))− dimΓ(ker(D−)).

Theorem 4.1.3 (Atiyah’s L2-index theorem [Ati76, 1.1]). Under these as-sumptions

indΓ(D+) = ind(D+) =

∫F

ωD+ .

4.2 G-twisted bundles

For his work on Kähler hyperbolic manifolds Gromov uses a generalized ver-sion of Atiyah’s L2-index theorem [Gro91, 2.3]. He considers twists by trivial


Hermitian line bundles with Hermitian connections with Γ-invariant curva-ture (but not Γ-invariant connection). This case is covered by the followingsetting introduced by Ballmann [Bal06, 8.27,8.29], compare [Gro91, Remark2.3 A’]:

Definition 4.2.1. Let G be an extension of Γ by a compact Lie group K,i.e.,

1 −→ K −→ Gρ−→ Γ −→ 1

is a short exact sequence of Lie groups. Then a G-twisted Dirac bundle Eover a manifold M with a Riemannian Γ-action is a graded Dirac bundlewith a Hermitian G-action that is compatible with the Γ-action and withall structures of the Dirac bundle. Thus, the G-action has to satisfy thefollowing properties:

1. It is compatible with the action of Γ on M , i.e.,

π g = ρ(g) π for all g ∈ G.

Hence, the action on sections σ ∈ E(M, E) is

(gσ)(p) = g(σ(ρ(g)−1(p))).

2. It is Hermitian, i.e., (gv, gw) = (v, w).

3. It leaves the connection ∇E of E invariant, i.e.,

∇Edρ(g)(X)(gσ) = g∇E

X(σ).

4. It leaves the Clifford multiplication of E invariant, i.e.,

γ(dρ(g)(X))gv = gγ(X)v.

5. It leaves the splitting E = E+ ⊕ E− invariant, i.e.,

(gv)± = g(v±).

To show that our setting generalizes the one considered by Gromov andMarcolli–Mathai, we generalize [Bal06, Lemma 8.29] to general Γ-actions.

Lemma 4.2.2. Let M be a simply connected smooth manifold with a Rie-mannian Γ-action (or let γ : M → M be an isometry). Assume F → M is atrivial Hermitian line bundle with a Hermitian connection ∇F such that thecurvature form Ω of F is Γ-invariant. Moreover, let p ∈ M, γ ∈ Γ and letu : Fp → Fγp be a unitary map.

Then there is a unique lift of γ to an isomorphism g of F covering γ withgp = u such that g preserves the Hermitian metric and the connection of F .


Proof. We define gp : Fp → Fγp by u and want to construct a map gq : Fq →Fγq for every q 6= p.

In order to do this, choose a smooth path α from q to p. Then γα connectsγq with γp. Define gq : Fq → Fγq by composing the parallel translationPα : Fq → Fp with u = gp : Fp → Fq and the parallel translation Pγ(α−1) :

Fγ(p) → Fγ(q).By assumption g preserves the connection and hence the parallel trans-

port. Moreover g preserves the Hermitian metric since u and the paralleltransport does this. So we only have to show that the above construction iswell defined and uniqueness follows.

Since M is simply connected α is unique up to homotopy. Moreover Fhas rank 1 and U(1) is abelian.

Let αt be a (piecewise) smooth homotopy with α = α0. By the γ-invariance of the curvature and the dependence of the parallel translationon the curvature we get

Pγ(α1) Pγ(α−10 ) u = u Pα1 Pα−1

0.

Thus, we obtain

Pγ(α−10 ) u Pα0 = Pγ(α−1

1 ) u Pα1 ,

and g is well defined.

Lemma 4.2.3. Let F be as in Lemma 4.2.2. If ∇F is not flat, there isan extension G of Γ by Hol0p(F ) = U(1) and a Hermitian G-action on F

compatible with the connection ∇F and the Γ-action on M .

Proof. Define Λp as

(α, γ)|γ ∈ Γ, α : I → M, α(0) = p, α(1) = γ(p),

the set of piecewise smooth curves α : I → M starting in p and ending insome γ(p) in the orbit Γ(p). Since the Γ-action is not free in general, wedefined elements of Λp as pairs (α, γ). Hence, for γ(p) = γ′(p) = α(1) andγ 6= γ′ we obtain (α, γ) 6= (α, γ′). Define an equivalence relation ∼ on Λp by(α, γ) ∼ (α′, γ′) if and only if γ = γ′ and Pα = Pα′ . Then set G = Λp/∼.

The product of two elements [(α1, γ1)] and [(α2, γ2)] of G is defined as[(α1(γ1 α2), γ1γ2)]. Then the inverse of [(α, γ)] is [(γ−1 α−1, γ−1)].

For g = [(α, γ)] ∈ G we get a unitary map gp given by the paralleltranslation Pα : Fp → Fγ(p). By Lemma 4.2.2 we may extend it to a G-actionon F .


The map ρ : G→ Γ given by [(α, γ)] 7→ γ is a surjective homomorphism.We have g : Fq → ˜Fρ(g)(q). The kernel of ρ is the group Hol0p(F ) of all paralleltranslations by piecewise smooth closed curves starting and ending in p (recallthat M is simply connected). Since ∇F is Hermitian Hol0p(F ) is a connectedLie subgroup of U(1). Therefore, it is either U(1) or 1. The bundle F isflat if and only if Hol0p(F ) 6= U(1).

Since G-twisted Dirac bundles are closed under tensoring with Hermitianline bundles F with G′-action compatible with the Hermitian connection ∇F ,the bundles considered by Gromov over manifolds and by Mathai–Marcolliover orbifolds are G-twisted in our sense. Let E be a G1-twisted Diracbundle and let F have a Hermitian G2-action compatible with ∇F . Let G1

be a K1-extension of Γ and G2 a K2 extension of Γ. Then G3 = G1×ΓG2 is a(K1×K2)-extension of Γ. The action of G3 on E⊗F given by (g1, g2)(e⊗f) =g1e⊗ g2f is compatible with the induced Dirac bundle structure on E ⊗ F .We prove some properties of G-twisted Dirac bundles that allow us to statean L2-index theorem. First we check how the invariance of the structures ofthe Dirac bundle leads to the invariance of all associated objects.

Lemma 4.2.4. The associated Dirac operator D± : L2(E) → L2(E) is G-equivariant, i.e., gD± = D±g.

Proof. In Definition 4.1.1 we defined D as the composition of

E(E)∇E→ E(E ⊗ TM∗)

]→ E(E ⊗ TM)γ→ E(E)

Since all these maps are equivariant by assumption, the composition is alsoequivariant. We get D =

∑γ(Xj)∇E

Xjσ for a local orthonormal frame Xj.

Since the covering transformation ρ(g) is an isometry, dρ(g)(Xj) is a localorthonormal frame and (here (3.) and (4.) refer to Definition 4.1.1)

D(gσ) =∑

γ(Xj)∇EXjgσ

(3.)=∑

γ(Xj)g∇Edρ(g)(Xj)

σ

(4.)=∑

gγ(dρ(g)(Xj))∇dρ(g)(Xi)σ = gDσ.

Definition 4.2.5. A Riemannian manifold has bounded geometry if thereare global bounds for the injectivity radius, the curvature and all covariantderivatives of the curvature.

Note that closed manifolds have bounded geometry. Complete Rieman-nian manifolds with an isometric cocompact group action have bounded ge-ometry since bounds restricted to a compact fundamental domain lead toglobal bounds.


To use the heat kernel estimates of [Roe88, 2.10] for G-twisted Diracbundles over coverings of closed manifolds or orbifolds, we need to checkthat the Dirac bundle also has bounded geometry.

Definition 4.2.6. A Dirac bundle has bounded geometry if there are globalbounds for the curvature and all covariant derivatives of the curvature.

Lemma 4.2.7. Let E be a G-twisted Dirac bundle. Then E has boundedgeometry if there is a fundamental domain F ⊂ M of the Γ-action suchthat there are bounds for the curvature and all covariant derivatives of thecurvature of E|F .

Thus, G-twisted bundles over coverings of closed manifolds or closed orb-ifolds have bounded geometry because they have compact fundamental do-mains.

Proof. Since ∇Edρ(g)(X) = g∇E

Xg−1, the curvature form

Ω(X, Y ) = ∇EX∇E

Y −∇EY∇E

X −∇E[X,Y ]

transforms under the G-action as follows:

Ω(dρ(g)(X), dρ(g)(Y )) = ∇Edρ(g)(X)∇E

dρ(g)(Y ) −∇Edρ(g)(Y )∇E

dρ(g)(X) −∇Edρ(g)[X,Y ]

= g∇EX∇E

Y g−1 − g∇E

Y∇EXg−1 − g∇E

[X,Y ]g−1

= gΩ(X, Y )g−1.

Now assume there are bounds for bounds for the curvature and all co-variant derivatives of the curvature of E|F . Let p ∈ M . Then there exists ag with ρ(g)p ∈ F . For X, Y, Z1, . . . , Zn ∈ TpM we have

Ω(X, Y ) = g−1Ω(dρ(g)(X), dρ(g)(Y ))g,

∇EZ1· · · ∇E

ZnΩ(X, Y ) = g−1∇Edρ(g)(Z1)g · · · g−1∇E

dρ(g)(Zn)g

g−1Ω(dρ(g)(X), dρ(g)(Y ))g

= g−1∇Edρ(g)(Z1) · · · ∇E

dρ(g)(Zn)Ω(dρ(g)(X), dρ(g)(Y ))g.

Hence, since g is Hermitian there are global bounds for the curvature and allcovariant derivatives of the curvature of E on M .

The group G acts on the curvature Ω of the bundle E, the Riemanniancurvature two-form RE and the twisting curvature F E/S = Ω − RE by con-jugation. Therefore, we obtain the following Lemma.


Lemma 4.2.8. Let E be a G-twisted Dirac bundle, then the index form

ωD+ = (2πi)−n2 A(M) ch(E/S)

is Γ-invariant.

Proof. For the curvature of the connection ∇E we have already seen in thelast lemma that

gΩ(X, Y )g−1 = Ω(dρ(g)X, dρ(g)Y ).

Similarly, the Riemannian curvature two-form RE ∈ A2(End(E)) satisfies

gRE(X, Y )g−1 = RE(dρ(g)X, dρ(g)Y )

since

RE(dρ(g)X, dρ(g)Y ) =∑k,l

R(dρ(g)X, dρ(g)Y, dρ(g)Xk, dρ(g)Xl)

γ(dρ(g)Xk)γ(dρ(g)Xl)

=∑k,l

R(X, Y,Xk, Xl)gγ(Xk)g−1gγ(Xl)g

−1

= gRE(X, Y )g−1.

Now by the definition of the twisting curvature as

F E/S = Ω−RE

the first two cases give us directly:

gF E/S(X, Y )g−1 = F E/S(dρ(g)X, dρ(g)Y ).

The index form ωD+ = (2πi)−n2 A(M) ch(E/S) (compare [BGV92, 4.2])

is Γ-invariant since the A-genus form A(M) = Pf( R/2sinh(R/2)

) is invariant un-der the Riemannian action of Γ on M , and the relative Chern character isinvariant under conjugation of F E/S :

ch(E/S) = strE/S(exp(−F E/S)) = strE/S(exp(−g−1F E/Sg)).

4.3. EQUIVARIANT SMOOTHING OPERATORS 65

4.3 The algebra of equivariant smoothing op-erators

Since D± is G-equivariant the kernels ker(D±) are G-invariant subspaces ofL2(E±). For a closed G-invariant subspace H ⊂ L2(E) and the orthogonalprojection PH : L2(E)→ H we can define as in Section 4.1:

dimG(H) =

∫F

f vol =

∫F

tr(kP (p, p)

)vol = trG(PH),

where f(p) =∑∞

n=1 |φn(p)|2 = tr(kPH (p, p)) for an orthonormal basis φn ofH (see [Ati76, Paragraph 2]). For the projections P : L2(E) → ker(D) andP± : L2(E±) → ker(D±) elliptic regularity shows that kP±(p, q) is smooth[Ati76, 2.4], hence bounded on a compact fundamental domain F . Therefore,the G-dimensions of ker(D±) are finite and

indG(D+) = trG(P+)− trG(P−) = trG(µP ) = strG(P )

is well-defined, where µ is the grading operator.Since M is complete, one can argue with the help of an idea of Chernoff

([Che73, (2.)], compare [Roe98, 13.3]) that D is essentially self-adjoint be-cause of the finite propagation speed of the solution of the wave equation.We use D in the following for a self-adjoint extension of the Dirac operator.Hence, we can use functional calculus to define the heat operator e−tD2 .

We will see that we can define strG(e−tD2) by showing that e−tD2 is in the

following algebra defined analogously to [Roe98, 13.4]:

Definition 4.3.1. The algebra AG of G-equivariant smoothing operators isdefined as the set of bounded operators A : L2(E)→ L2(E) that satisfy:

1. A is G-equivariant, i.e., Ag = gA for all g ∈ G;

2. A is represented by a smoothing kernel kA(p1, p2) (as in [Roe98, 13.4]kA(p1, ·) and kA(·, p2) are smooth) such that

Aσ(p1) =

∫M

kA(p1, p2)σ(p2) vol;

3. A and A∗ map L2(E) continuously to UC(E), the space of boundedcontinuous sections of E.


It is not hard to see that AG is an algebra (compare [Roe98, Lemma15.2]). If A1, A2 ∈ AG, we obtain A1A2g = A1gA2 = gA1A2 for all g ∈ G.Furthermore, A1A2 is represented by the smoothing kernel

(p1, p2) 7→∫M

kA1(p1, q)kA2(q, p2) vol .

Since A1, A2, A∗1, and A∗2 map L2(E) continuously to UC(E) and L2(E), so

do A1A2 and A∗2A∗1.

Lemma 4.3.2. Let f be in R(R), the Fréchet space of rapidly decaying func-tions on R, i.e., f is continuous and for every k there is a Ck such that

|f(x)| ≤ Ck(1 + |x|)−k.

Then f(D) ∈ AG.

Proof. Since D is G-equivariant, so is f(D). Roe shows that as

Dkf(D) : L2(E)→ L2(E)

is bounded for all k ∈ N, the bounded geometry of E gives a Sobolev em-bedding theorem [Roe88, 2.8] such that f(D) : L2(E) → UC∞(E) is stillcontinuous and f(D) is represented by a uniformly bounded smoothing ker-nel [Roe88, 2.10] (compare also [Bun91, 3]).

Since A ∈ AG is G-equivariant its kernel is equivariant:

kA(ρ(g)p, ρ(g)q

)= gkA

(p, q)g−1.

Thus, the trace of kA(p, p) is Γ-invariant

tr(kA(ρ(g)p, ρ(g)p

))= tr

(gkA(p, p)g−1

)= tr

(kA(p, p)

),

and the following definition is independent of the fundamental domain F .

Definition 4.3.3. The G-trace of an operator A ∈ AG is defined as

trG(A) =

∫F

tr(kA(p, p)

)vol .

Lemma 4.3.4. If A,B ∈ AG, then trG(AB) = trG(BA).

4.3. EQUIVARIANT SMOOTHING OPERATORS 67

Proof. As in [Roe98, 13.10],

trG(AB −BA) =

∫F×M

tr(kA(p1, p2)kB(p2, p1)− kB(p1, p2)kA(p2, p1)

)vol,

and using 3. in Definition 4.3.1 we see that this integral converges absolutely(compare [Roe98, 13.5]). Consequently, we obtain the decomposition

trG(AB −BA) =∑γ∈Γ

∫F×F

tr(kA(p1, γp2)kB(γp2, p1)

− kB(p1, γp2)kA(γp2, p1))

vol.

Since we have no Γ-action on E, we have to choose for each γ ∈ Γ a g ∈ Gwith ρ(g) = γ, i.e., a set-theoretic section of ρ : G→ Γ. Then

tr(kA(p1, γp2)kB(γp2, p1)− kB(p1, γp2)kA(γp2, p1)

)= tr

(kA(p1, ρ(g)p2)kB(ρ(g)p2, p1)− kB(p1, ρ(g)p2)kA(ρ(g)p2, p1)

)= tr

(gkA(ρ(g−1)p1, p2)g−1gkB(p2, ρ(g−1)p1)g−1−

− gkB(ρ(g−1)p1, p2)g−1gkA(p2, ρ(g−1)p1)g−1)

= tr(kA(γ−1p1, p2)kB(p2, γ

−1p1)− kB(γ−1p1, p2)kA(p2, γ−1p1)

)= −tr

(kA(p2, γ

−1p1)kB(γ−1p1, p2)− kB(p2, γ−1p1)kA(γ−1p1, p2)

).

But since trG(AB −BA) is also equal to∑γ−1∈Γ

∫F×F

tr(kA(p2, γ

−1p1)kB(γ−1p1, p2)− kB(p2, γ−1p1)kA(γ−1p1, p2)

)vol,

we may conclude

trG(AB −BA) = −trG(AB −BA) = 0.

Parallel to the classical case, it follows that strG(e−tD2) is independent of

t byd

dtstrG(e−tD

2

) = − strG(D2e−tD2

)

= −1

2strG(DDe−tD

2

+ De−tD2

D) = “ − 1

2strG([D, De−tD

2

]s)” = 0


since the supertrace vanishes on supercommutators. Here we have to dealwith the fact that D is not a smoothing operator as in the classical case (see[Roe98, 13.14,8.6]). Since Dne−sD

2 is a smoothing operator for n ∈ N, s ∈R+, we obtain −1

2strG(DDe−tD

2+ De−tD

2D) = 0 by

strG(DDe−tD2

) = trG(µDDe−tD2

) = trG(µD2e−t2D2

e−t2D2

)

= trG(e−t2D2

µDDe−t2D2

) = trG(De−tD2

µD)

= − trG(µDe−tD2

D) = − strG(De−tD2

D).

4.4 The L2-index theorem

Now we can state the main theorem of this chapter.

Theorem 4.4.1. (Theorem A) Let Γ→ M → M be a Riemannian orbifoldcovering, where M is a complete manifold and M = M/Γ a good orientedclosed orbifold. Let E be a G-twisted Dirac bundle over M . Then the G-dimensions of the kernels ker(D+) and ker(D−) are finite and the G-indexis the integral of the index form ωD+ over a compact fundamental domainF ⊂ M :

indG(D+) =

∫F

ωD+ .

Proof. We adapt the heat kernel proof (based on the McKean-Singer formulaindG(D+) = strG(e−tD

2)) to our situation.

We will verify that for the projection P : L2(E)→ ker(D) one has

indG(D+) = strG(P ) = strG(e−tD2

) =

∫F

ωD+ ,

where ωD+ = (2πi)−n2 A(M) ch(E/S) is the index form. First, we consider

the limit as t→∞. Since we can choose the fundamental domain F compact,to see limt→∞ strG(e−tD

2) = strG(P ) it suffices to show that

limt→∞

ke−tD2 (p, q) = kP (p, q)

uniformly on compact subsets of M × M.To see this we use that M and E have bounded geometry since M is the

covering of a compact orbifold (compare 4.2.7). Hence, we can use [Roe88,2.10] to see that for t → ∞ the kernels ke−tD2 (p, q) form a bounded, hencerelatively compact, subset of the Fréchet space C∞(E E). Thus, for any

4.4. THE L2-INDEX THEOREM 69

sequence tj → ∞ there is a subsequence ti with ke−tiD2 (p, q) uniformlyconvergent over compact subspaces. From Lemma 4.4.2 below follows thatlimi→∞ e

−tiD2= P holds in the strong operator topology. Hence

limi→∞

ke−tiD

2 (p, q) = kP (p, q)

uniformly on compact subsets. Then by an elementary argument (compare[Roe98, 13.14]) limt→∞ ke−tD2 (p, q) = kP (p, q).

The local index theory of Atiyah–Bott–Patodi [ABP73, (7.)], Getzler[Get86, theorem], and Gilkey [Gil84] for closed manifolds also works in ourcontext of Dirac bundles of bounded geometry by [Roe88, Proposition 2.11].Hence, we obtain

indG(D+) = strG(P ) = limt→∞

strG(e−tD2

) = limt→0

strG(e−tD2

) =

∫F

ωD+ .

Since the proof of [Roe88, Proposition 2.11] is a bit concise, we refer to[Bun91, Chapter 4] for a detailed treatment ([Bun91] is in general a goodreference for a detailed treatment of Dirac bundles and bounded geometrywhile [Roe88] is concise):

The main point is that by the finiteness of the propagation speed of thesolution of the wave equation the short term asymptotic of the heat kernelon a complete manifold M near the diagonal depends only on the local data.Let E → M be a graded Dirac bundle on a complete Riemannian manifoldand U ⊂ M an open precompact set. Let E → M be a graded Dirac bundleon a closed Riemannian manifold with U ⊂ M isometric to U ⊂ M and E|Uisomorphic to E|U . Then by [Bun91, Lemma 4.12] for every ε > 0, T > 0,and δ > 0 there is a C < ∞ such that for 0 < t < T ,p, q with B2ε(q) ⊂ Uand B2ε(p) ⊂ U we get the following estimate:

‖ke−tD2 (p, q)− ke−tD2 (p, q)‖ ≤ Ce−ε2

(4+δ)t .

Lemma 4.4.2. Let E → M be a Dirac bundle over a complete Riemannianmanifold. Then limt→∞ e

−tD2= P holds in the strong operator topology,

where D is a self-adjoint extension of the Dirac operator and P the projectiononto its kernel; i.e., for all σ ∈ L2(E) we have limt→∞ e

−tD2σ = Pσ.

Proof. Since D is self-adjoint, the orthogonal complement of ker(D) is theclosure of the range of D. Therefore, we can show limt→∞ e

−tD2σ = Pσ by

checking the cases σ ∈ ker(D), σ in the range of D and σ in the closure ofthe range of D.


If σ is in the kernel of D, then e−tD2σ = σ = Pσ for every t > 0.

Consequently,limt→∞

e−tD2

σ = σ = Pσ.

If σ is in the range of D, we can write σ = Dτ . Then we use that

limt→∞‖e−tD2

σ‖ = limt→∞‖e−tD2

Dτ‖ = limt→∞‖De−tD2

τ‖ = 0.

This holds since supλ ‖λe−tλ2‖ = 1√

2et−

12 . Hence, we obtain ‖e−tD2

σ‖ ≤1√2et−

12‖τ‖. To see this we calculate

∂

∂λλe−tλ

2

= (1− 2tλ2)e−tλ2

and 1 − 2tλ2 = 0 for λ = (2t)−12 (and check that for λ = 0 and λ → ∞ we

get λe−tλ2 → 0 for t > 0). We obtain λe−tλ2= (2t)−

12 e−t(2t)

−1= 1√

2et−

12 .

If σ is in the closure of the range of D, the idea is to use that ‖e−tD2‖ ≤ 1and that the pointwise limit of uniformly Lipschitz continuous functions iscontinuous:

For every t > 0 the operator e−tD2 is bounded with norm

‖e−tD2‖ ≤ supλ∈R e−tλ2

= 1.

We have already shown that for all Dτ in the range of D,

limt→∞

e−tD2

Dτ = PDτ = 0.

Thus, we can use a classical ε2-argument to show for σ in the closure of the

range of D thatlimt→∞

e−tD2

σ = Pσ = 0.

For every ε > 0 we choose a Dτ with ‖Dτ − σ‖ < ε2and a t0 such that

‖e−tD2Dτ‖ < ε

2for all t > t0. Then for all t > t0,

‖e−tD2

σ‖ ≤ ‖e−tD2‖‖σ − Dτ‖+ ‖e−tD2

Dτ‖ < ε

2+ε

2= ε.

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Zusammenfassung

Die Chern-Vermutung besagt, dass 2n-dimensionale, geschlossene, riemann-sche MannigfaltigkeitenM positiver Schnittkrümmung positive Eulercharak-teristik χ(M) haben, und solche negativer Schnittkrümmung Eulercharakter-istik mit Vorzeichen (−1)n haben.

Nach Hodge-Theorie ist χ(M) der Index des Operators

d+ d∗ : Agerade(M)→ Aungerade(M).

Sei Γ → M → M eine normale Überlagerung mit Fundamentalbereich F ,E → M ein Dirac-Bündel (im Sinne von Gromov-Lawson), E → M seinRückzug und dimΓ die Γ-Dimension. Dann besagt Atiyahs L2-Indexsatz fürden E zugeordneten Dirac-Operator D, dass

indΓ(D) = dimΓ(Ker(D))− dimΓ(Ker(D∗)) = indΓ(D) =

∫F

ωD.

Hier ist ωD die Indexform von D, welche der Atiyah-Singer-Indexsatz liefert.Sei M eine 2n-dimensionale einfach zusammenhängende vollständige rie-

mannsche Mannigfaltigkeit negativer Schnittkrümmung. Die Dodziuk-Singer-Vermutung besagt, dass die harmonischen L2-Formen Hk

2(M) genau dannverschwinden, wenn k 6= n. Dann würde mit

ind(d+ d∗) = indΓ(d+ d∗) = (−1)n dimΓ(Hn2 (M))

die Chern-Vermutung für negative Schnittkrümmung folgen.Gromov zeigte diese Vermutung für Kählermannigfaltigkeiten (M, g, J).

Sein Beweis für Hn2 (M) 6= 0 benutzt ohne Beweis eine Verallgemeinerung

von Atiyahs L2-Indexsatz für Dirac-Operatoren. Diese ist ein Spezialfall desfolgenden Satzes, welchen ich mithilfe von beschränkter Geometrie bewiesenhabe.

Sei K eine kompakte Liegruppe, 1 → K → G → Γ → 1 eine exakte Se-quenz und E → M ein Dirac-Bündel mit G-Wirkung, sodass alle Strukturenvon E äquivariant unter der G-Wirkung ist. Dann gilt indΓ(D) =

∫FωD.

77

78 CHAPTER 4. ZUSAMMENFASSUNG

Hierbei ist E → M nicht notwendig das Pullback eines Bündels E →M undM = M/Γ darf auch ein Orbifold sein.

Um zu zeigen, dass Hk2(M) = 0, benutzt Gromov den Lefschetz Satz

für die Kähler-Zweiform ωJ(v, w) = g(Jv, w). Quaternionische Kählerman-nigfaltigkeiten (M, g,G) lassen sich durch ein paralleles Vektorbündel G ⊂End(TM), welches lokal durch I, J und K mit

I2 = J2 = K2 = IJK = − id

aufgespannt wird, definieren. Für 4m-dimensionale negativ gekrümmte ein-fach zusammenhängende vollständige quaternionische Kählermannigfaltig-keiten konnte ich mit der Kraines-Vierform Ω = ωI ∧ωI +ωJ ∧ωJ +ωK ∧ωKbeweisen, dass Hk

2(M) = 0 für k /∈ 2m− 1, 2m, 2m+ 1.Für quaternionische Kählermannigfaltigkeiten positiver Skalarkrümmung

ist es bekannt, dass die ungeraden Bettizahlen verschwinden, woraus folgt,dass die Eulercharakteristik positiv ist. Für lokal symmetrische Räume neg-ativer Schnittkrümmung folgt die Vermutung aus dem positiven Fall. Dennder Proportionalitätssatz von Hirzebruch besagt, dass die Eulercharakter-istik eines negativ gekrümmten lokal symmetrischen Raumes, bis auf denQuotienten der Volumina, die des positiv gekrümmten dualen Raumes (alsopositiv) ist. Die Frage, ob alle geschlossenen quaternionischen Kählerman-nigfaltigkeiten lokal symmetrisch sind, ist bislang offen.

Die G-wertige Zweiform η = ωI ⊗ I + ωJ ⊗ J + ωK ⊗K ist wohldefiniertund parallel. Wir definieren das Dachprodukt G-wertiger Formen durch dieKomposition. Dann ist −Ω = η ∧ η und die Krümmung d∇d∇(α) von G bisauf eine Konstante α ∧ η − η ∧ α. Für Lη(α) = α ∧ η ist

0 −→ A`(M)Lη−−→ A`+2(M,G)

d∇d∇−−−→ A`+4(M,G) −→ 0

ein Kettenkoplex. Er ist in A`(M) für ` < 2n exakt. In A`+2(M,G) ist erfür ` + 2 < 2n und ` + 2 > 3n exakt. Um die Exaktheit in A`+2(M,G) zuzeigen, haben wir einen Verschwindungssatz für die Kohomologie von so5(C)-Modulen bewiesen. Durch die Exaktheit des Kettenkomplexes kann man mitden Verschwindungssätzen für die (L2-)Kohomologie von quaternionischenKählermannigfaltigkeiten beweisen, dass d∇-geschlossene G-wertige Formenin vielen Graden d∇-exakt sein müssen.

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