Infinitesimal automorphisms of conical hyper-K¨ ahler manifolds

quater-nionic K¨ahler metric on M_±⁰ is given by (compare Eq. (2.15))

g⁰_±= ˘g±|_M±^{0 (3.26)}= 1

r²g±−4σ r⁴

3

X

a=0

(ˆθa)²

M_±⁰ (3.42)

= Pn

µ=1(dφ^µdφ¯^µ+dψ_µdψ¯_µ) 1±(kφk²+kψk²)

∓

Pn

µ=1( ¯φ^µdφ^µ+ ¯ψ_µdψ_µ)

2+

Pn

µ=1(φ^µdψ_µ−ψ_µdφ^µ)

2

1±(kφk²+kψk²)2 .

Since J1, J2, J3 preserve T M_±⁰ ⊂ T M±, the quaternionic structure Q on M_±⁰ is spanned by the standard complex structures J₁⁰, J₂⁰, J₃⁰ on M_±⁰ ⊂ Hⁿ (see Ex.

3.1.7).

Note that while (M₊⁰, g⁰₊) is isometric to (HPⁿ)^o := {q⁰ 6= 0} ⊂ HPⁿ and thus incomplete, (M₋⁰, g⁰₋) is isometric to the symmetric spaceHHⁿand thus complete (see Examples 2.1.14 and 2.1.15). The normalization of (M_±⁰ , g_±⁰ ) is again such that the reduced scalar curvature is ν=±4.

3.3 Infinitesimal automorphisms of conical

Proof: Since ˆX preserves ˆg, ˆJ_αandξ, we haveLXˆθˆ_α = 0. From this, we obtain dµ^X_α^ˆ =−σ d(ι_Xˆθˆ_α) = −σ(LXˆθˆ_α−ι_Xˆdθˆ_α)^(3.14)= ˆω_α( ˆX,·).

Since Lξgˆ^(3.28)= 2ˆg and Lξ( ˆJ_αξ)^(3.18)= 0, we have Lξθˆ_α = 2ˆθ_α and hence ξ(µ^X_α^ˆ) =−σLξ(ˆθ_α( ˆX)) =−2σθˆ_α( ˆX) = 2µ^X_α^ˆ.

Remark 3.3.2 We call the map µ^Xˆ given by the above proposition the ho-mogeneous hyper-K¨ahler moment mapassociated with ˆX.

From now on, we assume that ˆX is space-like or time-like.

We consider the level set

P :={µ^Xˆ = (−σ,0,0)} ⊂Mb (3.46) of the hyper-K¨ahler moment map and define the following data on P:

gP := ˆg|P ∈Γ(Sym²T^∗P), θ_α^P :=σθˆ_α|_P = 1

2g( ˆˆ J_αξ,·)

P ∈Ω¹(P) (α= 1,2,3), f :=σr²

2

P = g(ξ,ξ)ˆ 2

P ∈C^∞(P), θ₀^P := 1

2df = 1

2ˆg(ξ,·) P = σ

2rdr

P ∈Ω¹(P), X_P :=σX|ˆ _P ∈X(P),

η:=σ ˆg( ˆX,·) ˆ

g( ˆX,X)ˆ P

∈Ω¹(P), f1 := 2

ˆ

g( ˆX,X)ˆ P

∈C^∞(P)

Z₁^P := ˆZ|_P = ˆJ₁ξ|_P ∈X(P). (3.47) The fact that ˆZ = ˆJ₁ξ is tangent to P follows from

dµ^X_α^ˆ( ˆZ) =LZˆµ^X_α^ˆ =−2δ_2αµ^X₃^ˆ + 2δ_3αµ^X₂^ˆ

since µ₂

P =µ₃

P = 0. The last equation used LJˆαξµ^X_β^{ˆ (3.28)}=

(3.18) −2µ^X_γ^ˆ. (3.48)

Remark 3.3.3 Note that, if non-empty, every level set of the hyper-K¨ahler moment map µ^Xˆ is a smooth submanifold of codimension 3 in Mb, due to Eq.

(3.45).

Proposition 3.3.4 Assume that P is non-empty and let M⁰ ⊂ P be a codi-mension one submanifold that is transversal to Z₁^P. Then

g⁰ = 1 2|f|

g_P − 2 f

3

X

a=0

(θ^P_a)² M⁰

(3.49) is a quaternionic (pseudo-)K¨ahler metric on M⁰.

Proof: Sinceµ^X₁^ˆ

P 6= 0,ξ(µ^X_α^ˆ) = 2µ^X_α^ˆ and (3.48) imply thatP ⊂Mb is transver-sal toξ,Jˆ₂ξ,Jˆ₃ξ. Hence,M⁰ ⊂Mb is transversal to D^v = span_R{ξ,Jˆ₁ξ,Jˆ₂ξ,Jˆ₃ξ}.

According to Theorem 3.2.6, g⁰ = ˘g

M⁰ =1 r²gˆ− σ

r²dr²− 4σ r⁴

3

X

α=1

(ˆθ_α)² M⁰⊂P

= 1

2|f|g_P − σ

f²(θ₀^P)²− σ f²

3

X

α=1

(θ_α^P)² M⁰

is a quaternionic (pseudo-)K¨ahler metric on M⁰.

Remark 3.3.5 Note that if (M,b g,ˆ Jˆα, ξ) is the Swann bundle (see Section 3.6) over a quaternionic (pseudo-)K¨ahler manifold ( ¯M,¯g, Q), ˆπ : Mb → M¯, then the groupR^>0 × SO(3) generated byξ,Jˆ1ξ,Jˆ2ξ,Jˆ3ξacts as the standard conformal linear groupCO(3) on the three-dimensional vector space spanned by the func-tions µ^X_α^ˆ. ThenZ₁^P induces a free S¹-action on P and P/S_(Z¹ P

1) is diffeomorphic to ¯M^o := ¯M\ˆπ({µ^Xˆ = 0}).

Let ¯µ^X be the quaternionic K¨ahler moment map associated with the Killing vec-tor fieldX ∈X( ¯M) induced by ˆX. On ¯M^o,J := (kµ¯^Xk⁻¹µ¯^X)|M¯^o defines an inte-grable complex structure (see, e.g., [Ba, Prop. 3.3.]). The quaternionic (pseudo-) K¨ahler stuctures defined on codimension one submanifolds in P transversal to

Z₁^P via Theorem 3.2.6 patch together to the quaternionic (pseudo-)K¨ahler struc-ture (^|ν|₄ ¯g|M¯^o,RJ|M¯^o ⊕V), where V = J^⊥ ⊂ Q|M¯^o ⊂ EndTM¯^o is a rank two vector bundle whose unit sphere bundle is isomorphic to theS¹-principal bundle P →M¯^o.

Note that in general, ¯M^o can be equal to ¯M. In the case where ( ¯M,g) is positive¯ definite, complete and of positive scalar curvature however, ¯M^o must be a proper subset of ¯M and, hence ( ¯M^o,^|ν|₄ ¯g|M¯^o) is incomplete. The latter is due to the fact that on a compact quaternionic K¨ahler manifold of positive scalar curvature, there exists not even a compatible almost complex structure [AMP, Th. 3.8.].

In the following example, we again treat quaternionic vector space with posi-tive and quaternionic Lorentzian signature simultaneously (see Example 3.2.10).

The choice of appropriate codimension four submanifoldsM_±⁰ again leads to (lo-cal) realizations of quaternionic projective and quaternionic hyperbolic space.

This time, we do not choose the canonical (local) sections M_±^0can. := {q⁰ = 1}

in the H^∗-bundles Hⁿ⁺¹>0 → HPⁿ, respectively H^n,<0¹ → HHⁿ. Instead, we choose submanifoldsM_±⁰ that are contained in the respective level setsP± of the hyper-K¨ahler moment map associated with the diagonal S¹-action on quaternionic vector space. This allows us to establish the HK/QK correspondence between, e.g., a certain subset in T^∗(CHⁿ) on the hyper-K¨ahler side and HHⁿ on the quaternionic K¨ahler side in Section 4.4.

Example 3.3.6 Let

Mb₊ =Hⁿ⁺¹>0 =Hⁿ⁺¹\{0}, Mb−=H^n,1<0 ={hq, qi₍₋₎ <0} ⊂H^n,1

with the conical (pseudo-)hyper-K¨ahler structure (ˆg±,Jˆ₁,Jˆ₂,Jˆ₃, ξ) defined by Eqs. (3.37)–(3.40) as in Example 3.2.10. We consider the vector field ˆX induced by the action q =z+jw 7→e^itq of e^it ∈ S¹ on Mb± at t = 0, scaled by a factor of two for convenience:

Xˆ = 2i

n

X

I=0

z^I ∂

∂z^I −w_I ∂

∂w_I −z¯^I ∂

∂z¯^I + ¯w_I ∂

∂w¯_I

. (3.50)

The components µ^X_α^ˆ =−¹₂ˆg( ˆJ_αξ,X) of the homogeneous hyper-K¨ˆ ahler moment

map associated with ˆX are given by µ^X₁^ˆ =−(hz, zi_(±)

| {z }

±|z⁰|²+P

|z^µ|²

− hw, wi_(±)),

µ^X₊^ˆ =µ^X₂^ˆ +iµ^X₃^ˆ = 2ihz,wi¯ = 2i ±z⁰w₀+

n

X

µ=1

z^µw_µ

. (3.51)

Let

λ:=hz, zi_(±) − hw, wi_(±), χ:=hz,wi¯_(±) =±z⁰w₀+

n

X

µ=1

z^µw_µ. (3.52)

To introduce appropriate coordinates, we restrict ourselves to the chart{z⁰ 6= 0}

in

{λ >0} ⊂ {z 6= 0} ⊂Hⁿ⁺¹>0 =Mb₊ and to

{λ <0} ⊂ {hz, zi₍₋₎ <0} ⊂ {z⁰ 6= 0} ⊂H^n,1<0 =Mb−, i.e. we consider

Mb^o₊ :={q =z+jw∈Hⁿ⁺¹ | hz, zi₍₊₎ >hw, wi₍₊₎, z⁰ 6= 0},

Mb^o₋ :={q =z+jw∈H^n,1 | hz, zi₍₋₎ <hw, wi₍₋₎, hq, qi₍₋₎<0}. (3.53) We endow Mb^o_± with coordinates (λ, φ, χ, ζ^µ, η_µ)µ=1, ..., n, where λ, χ are given by Eq. (3.52) and

φ:= argz⁰, ζ^µ:= (z⁰)⁻¹z^µ, η_µ:=z⁰w_µ (µ= 1, . . . , n). (3.54) The coordinates are chosen such that the level sets of φ are transversal to the S¹-action ( ˆX(φ) = 2), whileλ, χ, ζ^µ, η_µareS¹-invariant, so in these coordinates, Xˆ

Mb^o_± = 2_∂φ^∂ . When we set λ = ±1 and χ = 0, this induces coordinates (φ, ζ^µ, η_µ)µ=1, ..., n on the level set

P±={µ^X₁^ˆ =−σ =∓1, µ^X₂^ˆ =µ^X₃^ˆ = 0}={λ=±1, χ= 0} ⊂Mb^o_± (3.55) of the homogeneous hyper-K¨ahler moment map associated with ˆX.

Note that in the current example, ξ,Jˆ₁ξ,Jˆ₂ξ,Jˆ₃ξ generate a free H^∗-action on Mb±. Submanifolds M_±⁰ ⊂Mb± that intersect each H^∗-orbit at most once can be identified via the projection map with subsets ofHPⁿ, respectivelyHHⁿ. Under

this identification, the induced metric g⁰ is independent of the choice of section according to Proposition 3.2.5.

With the choice M_±⁰ = {φ = 0} ⊂ P± of (Z₁^P± = ˆJ₁ξ

P±)-transversal submani-fold, we recover the chart {q⁰ 6= 0} in

(HPⁿ)^o := (3.56)

HPⁿ\{[q =z+jw]_H^∗

right |(z, w)∈C²ⁿ⁺²\{0}, kzk² =kwk², z·w= 0}

for the case M₊⁰, while for M₋⁰ , we recover the whole symmetric space HHⁿ (see Remark 3.3.5). One way of seeing this is by Remark 3.3.5 and the fact that Mb±/Z² is the Swann bundle over HPⁿ, respectively HHⁿ. (HPⁿ)^o is the complement of the zero level set of the quaternionic K¨ahler moment map asso-ciated with the Killing vector field X on HPⁿ that is induced by ˆX. It is an open and everywhere dense submanifold of HPⁿ. Mb− has empty intersection with the zero level set of the homogeneous hyper-K¨ahler moment: The reverse Cauchy-Schwarz inequality (RCS) for complex Lorentzian vector spaces gives the following implication for q=z+jw ∈Mb−∩ {λ= 0}:

hq, qi<0, hz, zi=hw, wi

⇒ hz, zi=hw, wi<0

⇒ |χ|² =|hz,wi|¯ ^{2 RCS}≥ hz, zihw, wi>0.

This shows thatMb−∩ {µ^Xˆ = 0}=∅.

Note that while all three almost complex structures induced on the canonical choice of submanifold M_±^0can. = {q⁰ = 1} ⊂ Mb± (see Example 3.2.10) are inte-grable, the almost complex structuresJ₂⁰, J₃⁰ induced on M_±⁰ are non-integrable.

J₁⁰ is proportional to the quaternionic K¨ahler moment map associated with X and hence integrable (see Remark 3.3.5 and Proposition 4.1.9).

In the following remark, we specify the image Nb± of the coordinate function (λ, φ, χ, ζ^µ, ηµ)µ=1, ..., n : Mb^o_± → R² ×C²ⁿ⁺¹ defined in the above example and give the inverse map from Nb± to Mb^o_±.

Remark 3.3.7 In the above example, the coordinates (λ, φ, χ, ζ, η) on Mb^o_±

take their values in

Nb+ :=R^>0×R×C×Cⁿ×Cⁿ, Nb− :=

{(λ, φ, χ, ζ, η)∈R^<0×R×C×Cⁿ×Cⁿ| kζk² <1, λ²+ 4hζ,ˆ ζihˆˆ η,ηiˆ >0},

respectively. Here, hζ,ˆ ζiˆ :=±|ζ⁰|²+

n

X

µ=1

|ζ^µ|²

| {z }

=:kζk²

and hˆη,ηiˆ :=±|η₀|²+kηk², where

ζ⁰ := 1, η₀ :=±(χ−

n

X

µ=1

ζ^µη_µ). (3.57)

The inverse map from Nb± toMb^o_± is given by

z^I =ρ_λe^iφζ^I, w_I = (ρ_λ)⁻¹e^−iφη_I (I = 0, . . . , n), (3.58) where

ρ_λ := 1 q

±2hζ,ˆ ζiˆ r

±λ+ q

λ² + 4hζ,ˆ ζihˆˆ η,ηiˆ . (3.59) For future reference, we determine the differentials of z^I and wI in terms of the coordinates (λ, φ, χ, ζ, η):

dz^I =z^I ρ⁻¹_λ dρ_λ+idφ

+z⁰dζ^I, dwI =−wI ρ⁻¹_λ dρλ+idφ

+ (z⁰)⁻¹dηI (3.60) with dζ⁰ = 0, dη₀ =±(dχ−P

(ζ^µdη_µ+η_µdζ^µ)) and ρ⁻¹_λ dρ_λ =± 1

2 q

λ²+ 4hζ,ˆ ζihˆˆ η,ηiˆ

dλ−ρ_λ²dhζ,ˆ ζiˆ +ρ⁻²_λ dhˆη,ηiˆ

. (3.61)

For use in Section 4.4, we explicitly determine the geometric data defined in Eq. (3.47) for Example 3.3.6:

Remark 3.3.8 We want to express the geometric data on P± ={λ =±1, χ = 0} ⊂ Mb^o_± defined in Eq. (3.47) for Example 3.3.6 in terms of the coordinates (φ, ζ, η) defined above. Using θ_a^P = σθˆ_α

_P, a = 0, . . . ,3, we

obtain the following:

θ₂^P +iθ₃^{P (3.41)}= 1 2

±(z⁰dw₀−w₀dz⁰) +

n

X

µ=1

(z^µdw_µ−w_µdz^µ) _P

±

(3.60)

=

−χ(ρ⁻¹_λ dρ_λ+idφ) + 1 2dχ−

n

X

µ=1

η_µdζ^µ P±

=−

n

X

µ=1

η_µdζ^µ, (3.62)

θ₀^P +iθ₁^{P (3.41)}= 1 2

±(¯z⁰dz⁰+ ¯w₀dw₀) +

n

X

µ=1

(¯z^µdz^µ+ ¯w_µdw_µ) P±

(3.60)

= 1 2

λ(ρ⁻¹_λ dρ_λ+idφ) +

n

X

µ=1

(ρ_λ²ζ¯^µdζ^µ+ρ⁻²_λ η¯_µdη_µ)±ρ⁻²_λ η¯₀dη₀ P±

=±1

2ρ⁻¹_± dρ±± i

2dφ+1

2 ρ±2 ∂Jˆ1hζ,ˆ ζiˆ

P±+ρ⁻²_± ∂Jˆ1hˆη,ηiˆ P±

(3.61)

= ± 1

2√

1±r˜² hη,ˆ ηiˆ dhζ,ˆ ζiˆ +hζ,ˆ ζiˆ dhη,ˆ ηiˆ

± i

2dφ+ i

4 ρ±2d^chζ,ˆ ζiˆ +ρ⁻²_± d^chη,ˆ ηiˆ

(3.63)

=±1 4d √

1±r˜² +i

±1

2dφ+ 1

4d^c ±√

1±r˜²∓2 logρ±

, where³

ρ±:=ρ_λ|_P± = 1 q

±2hζ,ˆ ζiˆ q

1 +√

1±r˜² ∈C^∞(P±)

˜

r² :=±4hζ,ˆ ζihˆ η,ˆ ηi ∈ˆ C^∞(P_±), r˜² ≥0. (3.64) In Eq. (3.63), d^c = i( ¯∂ −∂) acts on the holomorphic functions (ζ^µ, η_µ)µ=1, ..., n. It will later turn out to be the d^c-operator associated with the first complex structure of the hyper-K¨ahler quotient Mb±///_P±S_(diag.)¹ ≈ M_±⁰ = {φ = 0} ⊂ P±

(see Example 3.4.7).

The other geometric data defined in (3.47) can be calculated as well:

f =±

√1±r˜²

2 , f₁ =± 1 2√

1±r˜²,

3By abuse of notation, hζ,ˆ ζiˆ := hζ,ˆ ζi|ˆ P± = ±1 + kζk² ∈ C^∞(P_±) and hˆη,ηiˆ :=hˆη,ηi|ˆ P± =±|Pn

µ=1ζ^µη_µ|²+kηk²∈C^∞(P_±), compare Eq. (3.57) and above.

X_P =±2 ∂

∂φ, Z₁^P = ∂

∂φ + 2i

n

X

µ=1

η_µ ∂

∂η_µ −η¯_µ ∂

∂η¯_µ , η=±1

2dφ+ 1 4√

1±r˜² ρ±2d^chζ,ˆ ζi −ˆ ρ⁻²_± d^chˆη,ηiˆ

=±1

2 dφ−ρ⁻¹_± d^cρ±

, g_P =±√

1±r˜² (dφ−ρ⁻¹_± d^cρ±)²+

n

X

µ=1

ρ±2dζ^µdζ¯^µ+ρ⁻²_± dη_µdη¯_µ

±ρ⁻²_± d

n

X

µ=1

ζ^µη_µ

2∓4√

1±˜r² |ρ⁻¹_± ∂ρ±|². (3.65)

From the above example, we get the following corollary. It gives the realiza-tion of quaternionic hyperbolic space that we will later obtain when we apply the HK/QK correspondence to a certain subset inT^∗(CHⁿ) in Section 4.4.

Corollary 3.3.9

N₋⁰ :={(ζ, η)∈Cⁿ×Cⁿ | kζk² <1,r˜² <1}, (3.66) together with⁴

g⁰₋= (ρ⁻¹₋ dρ−)² (3.67)

+ 1

√1−r˜² Xⁿ

µ=1

ρ−2dζ^µdζ¯^µ+ρ⁻²₋ dη_µd¯η_µ

−ρ⁻²₋ d

n

X

µ=1

ζ^µη_µ

2

+ 1

1−r˜²

4

n

X

µ=1

η_µdζ^µ

2+1 4 d√

1−˜r²2

+ 1

4 d^c(−√

1−r˜²+ 2 logρ−)2 is isometric to HHⁿ.

Proof: From the argument at the end of Example 3.3.6, we know that M₋⁰ ⊂ Mb− = H^n,<0¹ defines a global section of the H^∗-bundle H^n,<0¹ → HHⁿ. By Proposition 3.2.5, (M₋⁰, g⁰₋) with the metric g⁰₋ defined by Eq. (3.49) is isometric to (M₋^0can., g₋^0can.). Here g^0can.₋ is obtained from the canonical section M₋^0can. = {q⁰ = 1} ⊂ Mb− = H^n,<0¹ via Theorem 3.2.6 (see Eq. (3.42) in Ex-ample 3.2.10). The latter is isometric to HHⁿ (see also Example 2.1.15 and

4Recall that on N₋⁰ , r˜² = 4(1 − kζk²)(−|Pn

µ=1ζ^µηµ|² + kηk²) and ρ−=√ ¹

2(1−kζk²)

p1 +√ 1−r˜².

Remark 2.1.16). The coordinates (ζ^µ, η_µ)µ=1, ..., n defined in Example 3.3.6 give a diffeomorphism from

M₋⁰ ={q =z+jw ∈H^n,1 | hw, wi= 1+hz, zi, argz⁰ = 0,hz,wi¯ = 0,hq, qi<1}

to N₋⁰ ≈ {−1} × {0} × {0} ×N₋⁰ ⊂ Nb−. In these coordinates and with the geometric data calculated in Remark 3.3.8, the metric g⁰ defined by Eq. (3.49) reads as in Eq. (3.67).

Remark 3.3.10 Similarly, N₊⁰ :={(ζ, η)∈Cⁿ×Cⁿ} endowed with the metric

g₊⁰ =−(ρ⁻¹₊ dρ₊)² (3.68)

+ 1

√1 + ˜r² Xⁿ

µ=1

ρ₊²dζ^µdζ¯^µ+ρ⁻²₊ dη_µd¯η_µ

+ρ⁻²₊ d

n

X

µ=1

ζ^µη_µ

2

− 1 1 + ˜r²

4

n

X

µ=1

η_µdζ^µ

2+1 4 d√

1 + ˜r²2

+ 1

4 d^c(+√

1 + ˜r²−2 logρ₊)2

is isometric to {q⁰ 6= 0} ⊂(HPⁿ)^o, where (HPⁿ)^o is the complement of the zero level set of the quaternionic K¨ahler moment map with respect to the diagonal S¹-action that was defined in Example 3.3.6 (see Eq. (3.56)).

Im Dokument The hyper-Kähler/quaternionic Kähler correspondence and the geometry of the c-map (Seite 45-54)