On the rigidity of ℂℙ2n × ℂℙ1

Stuart James Hall

doi:10.1515/coma-2024-0008

Article Open Access

On the rigidity of ℂℙ²ⁿ × ℂℙ¹

Stuart James Hall

Published/Copyright: February 20, 2025

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From the journal Complex Manifolds Volume 12 Issue 1

Abstract

We revisit Koiso’s original examples of rigid infinitesimally deformable Einstein metrics. We show how to compute Koiso’s obstruction to the integrability of the infinitesimal deformations on CP 2 n × CP 1 using completely elementary complex differential geometry.

Keywords: Einstein manifolds; deformations

MSC 2010: 53C25; 53C55; 32Q20

1 Introduction

In the early 1980s, Koiso developed the structure theory for the moduli space of Einstein metrics around a given Einstein metric g on a closed manifold ℳ (see [6], [7], and [8]). The given metric g is said to be infinitesimally deformable if one can find tensors that look like the tangents to putative one-parameter families of geometrically distinct Einstein metrics passing through g (see the next section for more details). Koiso [7] found an obstruction to being able to integrate an infinitesimal deformation to form a genuine one-parameter family of Einstein metrics. He considered the case of symmetric spaces and proved the following result (which we paraphrase).

Theorem

(Koiso [7], Theorem 6.12) The product Kähler-Einstein metric on CP 2 n × CP 1 admits infinitesimal Einstein deformations (EIDs), none of which is integrable to second order.

This gave the first examples of Einstein metrics that admitted infinitesimal deformations but were nevertheless isolated in the moduli space; metrics that are isolated are referred to as being rigid.

Koiso [6] demonstrated that the Kähler-Einstein metric on a complex Grassmannian of m -dimensional subspaces of an ( n + m ) -dimensional vector space admits EIDs when 1 < m , n (i.e. when the Grassmannian is not CP n ). Recent work of the author with Schwahn and Semmelmann [5] has proved that, in the case when n + m is odd, none of the deformations is integrable to second order and so the metric is rigid^[1] (the m = 2 case of this result was proved by Nagy and Semmelmann [10]). The product metric on CP n × CP 1 is Kähler-Einstein. In this article, we give another proof of Koiso’s theorem using the “local” complex-geometric approach that was applied to the Grassmannians in [5].

1.1 Differences from Koiso’s original approach

The infinitesimal deformations of the product CP n × CP 1 were described explicitly by Koiso (note we do not require the complex dimension of the first factor to be even here); the space of deformations is isomorphic to the space of eigenfunctions for the smallest non-zero eigenvalue of the ordinary Laplacian on CP n . If we write the product Einstein metric, with Einstein constant λ , as g 1 ⊕ g 0 , then the isomorphism is given by

f → Hess ( f ) + λ f g 1 + ( 1 − n ) λ f g 0

(Proposition 2.3). The obstruction to integrability to second order found by Koiso is the non-vanishing of an integral of various “cubic” quantities involving the infinitesimal deformation and its second derivatives (Lemma 2.2). Given the form of the deformations, the kinds of quantities Koiso calculated are cubic expressions in f and its covariant derivatives, for example,

⟨ ∣ Hess ( f ) ∣ 2 , f ⟩ L 2 , ⟨ ∇ i ∇ k f ⋅ ∇ k ∇ j f , ∇ i ∇ j f ⟩ L 2 , and ⟨ ∇ i ∇ j ∇ k ∇ l f , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 .

Koiso directly manipulated such expressions in an impressive feat of computation, turning on the bread-and-butter techniques one learns as a student of Riemannian geometry (integration by parts, Ricci identity for tensor fields, etc.).

We use a different method, which is actually implicit in Koiso’s paper [7], and has been used recently to resolve similar rigidity questions. We give a brief outline of the principle here but refer the reader to [1] and [5] for further details. The group G = S U n + 1 acts isometrically on CP n × CP 1 by its standard action on the first factor of the product. The space of EIDs is equivariantly isomorphic to g = su n + 1 and the terms in the obstruction can be thought of as G -invariant homogeneous cubic polynomials in g . The key to the calculations is that the space of such polynomials is one-dimensional and thus all integrals, such as those given previously, can be realised as a multiple of a fixed generator, which we take to be ⟨ f 2 , f ⟩ L 2 . We refer to this principle as the standard argument regarding cubic quantities in the obstruction.

To find the multiple in each case, we work in the standard holomorphic coordinates on an open dense subset of CP n . We choose a particular eigenfunction, which is very simple in these coordinates, and make the calculations using elementary complex differential geometry. The integrals over C n which arise can be easily worked out explicitly (Lemma 4.1) and so each quantity in the obstruction can be directly compared with ⟨ f 2 , f ⟩ L 2 . In this manner, we recover easily many of the identities used by Koiso in the proof of his theorem and that were proved using the standard techniques of Riemannian geometry.

2 Einstein deformations

2.1 General theory

A good general exposition of the theory is given in [2]. For brevity, we simply state the equations that are satisfied by an EID h ∈ Γ ( s 2 ( T ∗ ℳ ) ) .

Definition 2.1

(EID) Let ( ℳ n , g ) be an Einstein manifold such that Ric ( g ) = λ g with λ > 0 . An EIDs is a section h ∈ Γ ( s 2 ( T ∗ ℳ ) ) satisfying the following conditions:

(2.1) tr g ( h ) = 0 ,

(2.2) div ( h ) = 0 ,

(2.3) Δ h + 2 Rm ( h ) = 0 ,

where Δ is the connection Laplacian given by Δ h = tr 12 ( ∇ 2 h ) and Rm is the curvature operator acting on symmetric 2-tensors. We denote the space of EIDs by ε ( g ) .

If the manifold ( ℳ n , g ) is Kähler-Einstein and the EID h is also invariant with respect to the complex structure J , then the equations definining an EID can be conveniently expressed as conditions involving the two-form associated with h , σ ∈ Ω ( 1 , 1 ) ( ℳ ) :

(2.4) Λ ( σ ) = 0 ,

(2.5) ∂ ¯ ∗ σ = 0 ,

(2.6) Δ ∂ ¯ σ = λ σ ,

where Λ is the adjoint of the Lefschetz operator, ∂ ¯ ∗ is the usual adjoint of the Dolbeault operator ∂ ¯ , and Δ ∂ ¯ = ∂ ¯ ∂ ¯ ∗ + ∂ ¯ ∗ ∂ ¯ .

If h ∈ ε ( g ) , the path g ( t ) = g + t h solves the Einstein equation to first order at t = 0 (i.e. h solves the linearised Einstein equation). For a given EID, Koiso developed an obstruction to being able to produce a curve of Einstein metrics solving the Einstein equations to second order; again, for brevity, we only state the obstruction.

Lemma 2.2

(Koiso, Lemma 4.3 in [7]) Let ( ℳ , g ) be an Einstein metric with Einstein constant λ > 0 and let h ∈ ε ( g ) . Then, an obstruction to the integrability of h to order two is given by the nonvanishing of the quantity

(2.7) ℐ ( h ) ≔ 2 λ ⟨ h i k h k j , h i j ⟩ L 2 + 3 ⟨ ∇ i ∇ j h k l , h i j h k l ⟩ L 2 − 6 ⟨ ∇ i ∇ l h k j , h i j h k l ⟩ L 2 ,

where each of the brackets denotes the L 2 -inner product induced by the metric g on the appropriate bundle.

Remark

Nagy and Semmelmann [10] have reformulated the obstruction (2.7) in a coordinate free manner using the Frölicher-Nijenhuis bracket; one could also use this approach to compute ℐ ( h ) and reprove Koiso’s theorem.

Koiso’s obstruction ℐ ( h ) can really be thought of as trying to detect the vanishing of the L 2 -projection to ε ( g ) of a symmetric map E : ε ( g ) × ε ( g ) → s 2 ( T ∗ ℳ ) , i.e.

ℐ ( h ) = ⟨ E ( h , h ) , h ⟩ L 2 .

The deformation is unobstructed at second order if and only if the projection vanishes. For a symmetric space G ⁄ K , the projection can be thought of as an element of Hom G ( s 2 ( g ) , g ) ; in the case that G = S U n , this space is one-dimensional. In the case that G = S U 2 n + 1 , one can write down an explicit generator and check that it has no non-trivial zeroes (see [1] for more details). Hence, if there exists an h ∈ ε ( g ) such that ℐ ( h ) ≠ 0 , then all EIDs of CP 2 n × CP 1 are obstructed to second order.

In the CP 2 n − 1 × CP 1 ( G = S U 2 n ) case, the zeros of the generator of Hom G ( s 2 ( g ) , g ) are easy to characterise (see [1], [5], and [10]). Thus, the set of EIDs that are unobstructed at second order is explicitly known, and it might be possible to prove that they are in fact obstructed at third order (see e.g. [9]), which would prove the rigidity of CP n × CP 1 in general.

2.2 Infinitesimal deformations for CP n × CP 1

Koiso completely characterised the infinitesimal deformations of symmetric spaces. In particular, he proved that for the product metric g 1 ⊕ g 0 on CP n × CP 1 ,

ε ( g 1 ⊕ g 0 ) ≅ su n + 1 .

Of course su n + 1 ≅ iso ( g 1 ) and, as g 1 is a Kähler-Einstein metric, we also have by Matsushima’s theorem iso ( g 1 ) ≅ E 1 , where E 1 is the eigenspace of the smallest non-zero eigenvalue of the ordinary Laplacian; here, the smallest eigenvalue will be 2 λ , where λ is the Einstein constant. We give an explicit description of the EIDs in terms of the eigenfunctions and include a brief proof that the tensors satisfy equations (2.4)–(2.6). However, we will omit the proof that these are all the EIDs.

Proposition 2.3

(cf. Koiso [7], Theorem 5.7) Let g = g 1 ⊕ g 0 be the product Kähler-Einstein metric on CP n × CP 1 with Einstein constant λ > 0 and let f be an eigenfunction of the ordinary Laplacian on ( CP n , g 1 ) with eigenvalue 2 λ . Then

h = Hess ( f ) + λ f g 1 + ( 1 − n ) λ f g 0

is an EID.

Proof

To check equation (2.1) (or (2.4)), we take the trace of h

tr g ( h ) = − 2 λ f + 2 n λ f + 2 ( 1 − n ) λ f = 0 .

For (2.5), let σ be the ( 1 , 1 ) -form associated with h ; if the Kähler forms of g 1 and g 0 are ω 1 and ω 0 , then

σ = i ∂ ∂ ¯ f + λ f ω 1 + ( 1 − n ) f ω 0 .

Taking the codifferential yields

∂ ¯ ∗ σ = i ∂ ¯ ∗ ∂ ∂ ¯ f + λ ∂ ¯ ∗ ( f ω 1 ) + ( 1 − n ) λ ∂ ¯ ∗ ( f ω 0 ) .

If we denote the differential and codifferential operators associated with the metrics g j by a subscript j , we have

∂ ¯ ∗ σ = i ∂ ¯ 1 ∗ ∂ 1 ∂ ¯ 1 f + λ ∂ ¯ 1 ∗ ( f ω 1 ) + ( 1 − n ) λ f ∂ ¯ 0 ∗ ( ω 0 ) .

Let L j ( − ) = − ∧ ω j denote the usual Lefschetz operator associated with the metric ω j . Using the Kähler identities ∂ ¯ ∗ ∂ = − ∂ ∂ ¯ ∗ and [ ∂ ¯ ∗ , L ] = i ∂ , we compute

∂ ¯ ∗ σ = − i ∂ 1 ∂ ¯ 1 ∗ ∂ ¯ 1 f + i λ ∂ 1 f + 0 = − i λ ∂ 1 f + i λ ∂ 1 f + 0 = 0 ,

where we have used ∂ ¯ 1 ∗ ∂ ¯ 1 f = 1 2 Δ 1 f = λ f .

For equation (2.6), we can compute

Δ ∂ ¯ σ = Δ ∂ ¯ 1 ( i ∂ 1 ∂ ¯ 1 f ) + λ Δ ∂ ¯ 1 ( L 1 ( f ) ) + ( 1 − n ) λ Δ ∂ ¯ ( f ω 0 ) .

Applying the Kähler identities (the Laplacian commutes with every operator) along with the fact that, for any smooth function on the first factor F : CP n → R ,

Δ ∂ ¯ ( F ω 0 ) = ( Δ ∂ ¯ 1 F ) ω 0 ,

yields the result.□

3 Objects in local coordinates

We consider CP n with the standard dense open set U 0 , described in homogeneous coordinates by

U 0 = { [ z 0 : z 1 : … : z n ] ∈ CP n : z 0 ≠ 0 } .

Let π : U 0 → C n be the coordinate chart given by

π ( [ z 0 : z 1 : … : z n ] ) = z 1 z 0 , z 2 z 0 , … , z n z 0

and denote the corresponding holomorphic coordinates ( w 1 , w 2 , … , w n ) ∈ C n , where w i = z i ⁄ z 0 . We will use the notation

‖ w ‖ 2 = w 1 2 + w 2 2 + … + w n 2 .

We can write the metric and some associated geometric objects in the w coordinates. This material is completely standard and so we omit the proof.

Lemma 3.1

(Metric objects in coordinates) In the w coordinates, the Fubini-Study metric g is given by

(3.1) g k l ¯ = 1 1 + ‖ w ‖ 2 δ k l − w ¯ k w l 1 + ‖ w ‖ 2 ;

the inverse metric g − 1 (in the sense that g k l ¯ g j l ¯ = δ k j ) is given by

(3.2) g k l ¯ = ( 1 + ‖ w ‖ 2 ) ( δ k l + w k w ¯ l ) ;

the volume form d V g is given by

(3.3) d V g = 1 ( 1 + ‖ w ‖ 2 ) n + 1 − 1 2 n d w 1 ∧ d w ¯ 1 ∧ d w 2 ∧ d w ¯ 2 ∧ … ∧ d w n ∧ d w ¯ n ;

the Christoffel symbols are given by

(3.4) Γ i j k = − 1 1 + ‖ w ‖ 2 ( δ j k w ¯ i + δ i k w ¯ j ) ;

the curvature tensor is given by

(3.5) Rm i k j l = ( δ i j δ k l + δ i l δ k j ) ;

the Ricci tensor is given by

(3.6) Ric ( g ) k l ¯ = ( n + 1 ) g k l ¯ .

If we denote E 1 the space of eigenfunctions for the ordinary Laplacian with eigenvalue 2 ( n + 1 ) , there is an isomorphism F : su n + 1 → E 1 given in coordinates by

F ( η ) ( w ) = W ¯ η W t 1 + ‖ w ‖ 2 ,

where W = ( 1 , w 1 , w 2 , … , w n ) . Where relevant, for η ∈ g , we will denote the eigenfunction F ( η ) by f η . Once and for all, we fix γ ∈ su n + 1 to be

(3.7) γ = Diag ( − n , 1 , 1 , … , 1 ︸ n terms ) .

We collect the coordinate expressions for objects associated with f γ , leaving proofs to the reader.

Lemma 3.2

(Objects associated with f γ in coordinates) In the w coordinates, the eigenfunction f γ is given by

(3.8) f γ ( w ) = ‖ w ‖ 2 − n 1 + ‖ w ‖ 2 ;

the derivative ∂ f γ is given by

(3.9) ( ∂ f γ ) k = ( n + 1 ) w ¯ k ( 1 + ‖ w ‖ 2 ) 2 ;

the Hessian of f γ is given in complex coordinates by

(3.10) ∇ k ∇ l ¯ f γ = n + 1 ( 1 + ‖ w ‖ 2 ) 2 δ k l − 2 w ¯ k w l 1 + ‖ w ‖ 2 = n + 1 1 + ‖ w ‖ 2 2 g k l ¯ − δ k l 1 + ‖ w ‖ 2 ,

(3.11) ∇ k ∇ l f γ = 0 ;

the covariant derivative of the Hessian is given by

(3.12) ∇ q ¯ ∇ k ∇ l ¯ f γ = − ( g k l ¯ ( ∂ f γ ) q ¯ + g k q ¯ ( ∂ f γ ) l ¯ ) ;

the four derivative term needed in Koiso’s obstruction is given by

(3.13) ∇ p ∇ q ¯ ∇ k ∇ l ¯ f γ = − ( g k l ¯ ∇ p ∇ q ¯ f γ + g k q ¯ ∇ p ∇ l ¯ f γ ) .

Finally, in this section, we note that we can write the terms in Koiso’s obstruction (2.7) in complex coordinates but this will introduce some conversion factors in the terms. The tangent space at a point is an inner product space ( V 2 n , g ) with an almost complex structure J such that g ( J ⋅ , J ⋅ ) = g ( ⋅ , ⋅ ) . We complexify V and extend the tensors g and h C -linearly in both arguments to obtain a 2 n -complex-dimensional space V C and tensors g C and h C . The space V C splits into the ± − 1 -eigenspaces for J and we write V C = V C ( 1 , 0 ) ⊕ V C ( 0 , 1 ) .

Given a g -orthonormal basis of V of the form v 1 , J v 1 , v 2 , J v 2 , … , v n , J v n , we can form the basis { e i } i = 1 n of V C ( 1 , 0 ) where

e i = 1 2 ( v i − − 1 J v i ) .

This is an orthogonal basis of ( V C ( 1 , 0 ) , g C ) with ‖ e i ‖ = 1 ⁄ 2 . The set of conjugates

e ¯ i = 1 2 ( v i + − 1 J v i )

form a basis of V C ( 0 , 1 ) . As the tensors g and h are J -invariant, the only non-vanishing terms of the extensions g C and h C are those of the form

g k l ¯ ≔ g C ( e k , e ¯ l ) and h k l ¯ = h C ( e k , e ¯ l ) .

We consider Koiso’s quantities but in complex coordinates, e.g.

⟨ h k p h p l ¯ , h k l ¯ ⟩ = g k q ¯ g r l ¯ h k p h p l ¯ h r q ¯ = H k p H p r H r k = tr ( H 3 ) .

As the metric is Kähler, the Chern connection is the same as the C -linear extension of the Levi-Civita connection. We also note that

( ∇ ⋅ ∇ ⋅ h ) ( J X , J Y ) = ( ∇ ⋅ ∇ ⋅ h ) ( X , Y ) .

We refer the reader to [3] for a proof of the following.

Lemma 3.3

Let h ∈ s 2 ( V ∗ ) be J-invariant and let T ∈ ( V ∗ ) ⊗ 4 satisfy

T ( − , − , X , Y ) = T ( − , − , Y , X ) a n d T ( − , − , J X , J Y ) = T ( − , − , X , Y ) ,

for all X , Y ∈ V . Then, with the notation defined previously,

(3.14) ⟨ h k p h l p , h k l ⟩ = 2 ⟨ h k p ¯ h l ¯ p ¯ , h k l ¯ ⟩ ,

(3.15) ⟨ T k l r s , h k l h r s ⟩ = 4 Re ( ⟨ T k l ¯ r s ¯ , h k l ¯ h r s ¯ ⟩ ) ,

(3.16) ⟨ T k r s l , h k l h r s ⟩ = 2 Re ( ⟨ T k r ¯ s l ¯ , h k l ¯ h s r ¯ ⟩ ) .

4 Computing integral terms

Throughout this section, we will consider the Fubini-Study metric g on CP n with the scaling of the previous section such that the Einstein constant is ( n + 1 ) . All of the integrals we need to calculate are of a similar form; more specifically, for r ∈ { 0 , 1 , 2 , 3 } , we will need formulae for the integrals

(4.1) I n r ≔ ∫ C n ‖ w ‖ 2 r ( 1 + ‖ w ‖ 2 ) n + 4 − 1 2 n d w 1 ∧ d w ¯ 1 ∧ ⋯ ∧ d w n ∧ d w ¯ n .

Lemma 4.1

(Standard integral values) Let I n r be as in (4.1), then

I n 0 = 6 ⋅ π n ( n + 3 ) ! , I n 1 = 2 n ⋅ π n ( n + 3 ) ! , I n 2 = 2 n + 2 n 2 ⋅ π n ( n + 3 ) ! , I n 3 = 6 n + 12 n 2 + 6 n 3 ⋅ π n ( n + 3 ) ! .

As explained in Section 1, the terms in the second-order obstruction can be seen as S U n + 1 -invariant cubic polynomials and therefore multiples of

P 0 ( η ) ≔ ∫ CP n f η 3 d V g ,

where η ∈ su n + 1 .

Remark

The integral is a fixed (non-zero) multiple of the cubic polynomial

− 1 tr ( η 3 ) .

The fact that the multiple is non-zero was proved in [4].

The explicit form of the eigenfunction and the volume form in equations (3.8) and (3.3) as well as the values of the integrals in Lemma 4.1 yields

(4.2) ∫ CP n f γ 3 d V g = I n 3 − 3 n I n 2 + 3 n 2 I n 1 − n 3 I n 0 = 2 n ( 1 − n 2 ) π n ( n + 3 ) ! .

We can now compute the key quantities that Koiso needed.

Lemma 4.2

(Identities (6.8.2), (6.8.10), and (6.8.12) from Lemma 6.8 in [7]) For any eigenfunction f ∈ E 1 , we have

(4.3) ⟨ ∣ Hess ( f ) ∣ 2 , f ⟩ L 2 = 0 ,

(4.4) ⟨ ∇ i ∇ j f ⋅ ∇ j ∇ k f , ∇ i ∇ k f ⟩ L 2 = ( n + 1 ) 3 ∫ CP n f 3 d V g .

For the eigenfunction f γ ∈ E 1 , we have

(4.5) Δ Hess ( f γ ) = 2 ( Hess ( f γ ) − ( n + 1 ) f γ ⋅ g ) .

For any eigenfunction f ∈ E 1 , we have

(4.6) ⟨ Δ ( Hess ( f ) + ( n + 1 ) f g ) , f Hess ( f ) ⟩ L 2 = − 4 n ( n + 1 ) 2 ∫ CP n f 3 d V g .

Proof

Using equation (3.11), we find, up to a constant that we need not worry about,

∣ Hess ( f ) ∣ 2 = ∇ k ∇ l ¯ f ⋅ ∇ k ∇ l ¯ f .

We now choose the eigenfunction to be f γ and calculate using (3.10) and (3.2)

∣ Hess ( f γ ) ∣ 2 = ( n + 1 ) 2 ( 1 + ‖ w ‖ 2 ) 2 ( n − 2 ‖ w ‖ 2 + ‖ w ‖ 4 ) .

Thus, by (3.8) and the values of the integrals in Lemma 4.1

⟨ ∣ Hess ( f γ ) ∣ 2 , f γ ⟩ L 2 = ( n + 1 ) 3 ( I n 3 − ( n + 2 ) I n 2 + 3 n I n 1 − n 2 I n 0 ) = 0

and an application of the standard argument proves the first identity.

For the second, we again use (3.10) and (3.2) to prove

H l k ≔ ∇ k ∇ l f γ = n + 1 1 + ‖ w ‖ 2 ( δ k l − w ¯ k w l ) .

Thus, the (complex coordinate version of the) integrand of the second identity is

⟨ ∇ k ∇ p ¯ f γ ⋅ ∇ p ¯ ∇ l ¯ f γ , ∇ k ∇ l ¯ f γ ⟩ = H l k H p l H k p = ( n + 1 ) 3 ( 1 + ‖ w ‖ 2 ) 3 ( n − 3 ‖ w ‖ 2 + 3 ‖ w ‖ 4 − ‖ w ‖ 6 ) .

Using the values in Lemma 4.1 yields

⟨ ∇ k ∇ p ¯ f γ ⋅ ∇ p ¯ ∇ l ¯ f γ , ∇ k ∇ l ¯ f γ ⟩ L 2 = ( n + 1 ) 3 ( I n 0 − 3 I n 1 + 3 I n 2 − I n 3 ) = ( n + 1 ) 3 n ( 1 − n 2 ) π n ( n + 3 ) ! .

The identity follows from (4.2), multiplying by the appropriate factor from Lemma 3.3, and the standard argument.

The third identity follows by noting that

Δ Hess ( f γ ) = − 2 g p q ¯ ∇ p ∇ q ¯ ∇ k ∇ l ¯ f γ = − 2 ( n + 1 ) f γ g k l ¯ + 2 ∇ k ∇ l ¯ f γ ,

where we have used equation (3.13) in the final equality. Integration, as well as an application of the first identity (4.3), and the standard argument yields the final result.□

We can now give a fairly easy proof of the following.

Lemma 4.3

(Koiso, Lemma 6.9 in [7]) For any eigenfunction f ∈ E 1 , we have

(4.7) ⟨ ∇ i ∇ j ∇ k ∇ l f , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 = − 2 ( n + 1 ) 3 ∫ CP n f 3 d V g ,

(4.8) ⟨ ∇ i ∇ k ∇ j ∇ l f , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 = − ( n + 1 ) 3 ∫ CP n f 3 d V g .

Proof

For the first identity, we use (3.13) and (3.2) to calculate

⟨ ∇ p ∇ q ¯ ∇ k ∇ l ¯ f γ , ∇ p ∇ q ¯ f γ ⋅ ∇ k ∇ l ¯ f γ ⟩ = ( n + 1 ) ⟨ ∣ Hess ( f γ ) ∣ 2 , f γ ⟩ − ⟨ ∇ k ∇ p ¯ f γ ⋅ ∇ p ¯ ∇ l ¯ f γ , ∇ k ∇ l ¯ f γ ⟩ .

Using the standard argument, the result follows from (4.3) and (4.4) as well as noting the factor of 4 coming from Lemma 3.3.

For the second identity, we note that as the Hessian is symmetric, we can also calculate

⟨ ∇ i ∇ k ∇ l ∇ j f , ∇ i ∇ j f ⋅ ∇ l ∇ k f ⟩ L 2 .

Thus, in complex coordinates

⟨ ∇ p ∇ k ¯ ∇ l ∇ q ¯ f , ∇ p ∇ q ¯ f ⋅ ∇ l ∇ k ¯ f ⟩ = ( n + 1 ) ⟨ ∣ Hess ( f γ ) ∣ 2 , f γ ⟩ − ⟨ ∇ k ∇ p ¯ f γ ⋅ ∇ p ¯ ∇ l ¯ f γ , ∇ k ∇ l ¯ f γ ⟩ .

Using the standard argument regarding cubic polynomials, the result follows from (4.3) and (4.4) as well as noting the factor of 2 coming from Lemma 3.3.□

We will need a final identity not explicitly used by Koiso in order to bypass some of the calculations of [7].

Lemma 4.4

For any eigenfunction f ∈ E 1 , we have

(4.9) ⟨ ∇ i ∇ l ∇ k ∇ j f , f g i j ⋅ ∇ k ∇ l f ⟩ L 2 = − 2 ( n + 1 ) 2 ∫ CP n f 3 d V g .

Proof

We specialise to f γ . Calculating in complex coordinates, we have, pointwise, by (3.13)

⟨ ∇ i ∇ l ¯ ∇ k ∇ j ¯ f γ , f γ g i j ¯ ⋅ ∇ k ∇ l ¯ f γ ⟩ = − ⟨ g k j ¯ ∇ i ∇ l ¯ f γ + g k l ¯ ∇ i ∇ j ¯ f γ , f γ g i j ¯ ∇ k ∇ l ¯ f γ ⟩ = − ⟨ ∣ Hess ( f γ ) ∣ 2 , f γ ⟩ − ( n + 1 ) 2 f γ 3 .

The result follows after integrating, using (4.3), scaling by factor 2 as in Lemma 3.3, and applying the standard argument.□

5 Computing the obstruction integral

With the identities of the previous section in hand, we can now more-or-less follow Koiso’s original calculation of the obstruction.

Proof

Proof (of Koiso’s theorem) As in [7], we write

ψ = Hess ( f ) + ( n + 1 ) f g 1 and φ = ( 1 − n ) ( n + 1 ) f g 0 ,

where f ∈ E 1 . This infinitesimal deformation associated with f is then

h = ψ + φ .

The zeroth-order term is thus

⟨ h i k h j k , h i j ⟩ L 2 = ⟨ ψ i k ψ j k , ψ i j ⟩ L 2 + ⟨ φ i k φ j k , φ i j ⟩ L 2 .

Calculating the term ⟨ ψ i k ψ j k , ψ i j ⟩ L 2 yields

⟨ ∇ i ∇ j f , ∇ i ∇ k f ⋅ ∇ k ∇ j f ⟩ L 2 + 3 ( n + 1 ) ⟨ ∣ Hess ( f ) ∣ 2 , f ⟩ L 2 − 6 ( n + 1 ) 3 ∫ CP n f 3 d V g 1 + 2 n ( n + 1 ) 3 ∫ CP n f 3 d V g 1 .

Using equations (4.4) and (4.3) on the first two terms in the previous expression gives

⟨ ψ i k ψ j k , ψ i j ⟩ L 2 = ( 2 n − 5 ) ( n + 1 ) 3 ∫ CP n f 3 d V g 1 .

It is easy to see that

⟨ φ i k φ j k , φ i j ⟩ L 2 = 2 ( 1 − n ) 3 ( n + 1 ) 3 ∫ CP n f 3 d V g 1 ,

and thus

(5.1) ⟨ h i k h j k , h i j ⟩ L 2 = − ( 2 n 3 − 6 n 2 + 4 n + 3 ) ( n + 1 ) 3 ∫ CP n f 3 d V g 1 .

The second term in Koiso’s obstruction (2.7) splits as

⟨ ∇ i ∇ j h k l , h i j h k l ⟩ L 2 = ⟨ ∇ i ∇ j ψ k l , ψ i j ψ k l ⟩ L 2 + ⟨ ∇ i ∇ j φ k l , ψ i j φ k l ⟩ L 2 .

The first term in the splitting is given by

⟨ ∇ i ∇ j ψ k l , ψ i j ψ k l ⟩ L 2 = ⟨ ∇ i ∇ j ψ k l , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 + ( n + 1 ) ⟨ ∇ i ∇ j ψ k l , ∇ i ∇ j f ⋅ f ( g 1 ) k l ⟩ L 2 + ( n + 1 ) ⟨ ∇ i ∇ j ψ k l , f ( g 1 ) i j ⋅ ∇ k ∇ l f ⟩ L 2 + ( n + 1 ) 2 ⟨ ∇ i ∇ j ψ k l , f 2 ( g 1 ) i j ⋅ ( g 1 ) k l ⟩ L 2 = ⟨ ∇ i ∇ j ∇ k ∇ l f , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 + ( n + 1 ) ⟨ ∇ i ∇ j f ⋅ ( g 1 ) k l , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 − ( n + 1 ) ⟨ Δ ψ k l , f ∇ k ∇ l f ⟩ L 2 − ( n + 1 ) 2 ⟨ Δ tr ( ψ ) , f 2 ⟩ L 2 .

Using (4.7), (4.3) yields

⟨ ∇ i ∇ j ψ k l , ψ i j ψ k l ⟩ L 2 = − 2 ( n + 1 ) 3 ∫ CP n f 3 d V g 1 − ( n + 1 ) ⟨ Δ ψ k l , f ∇ k ∇ l f ⟩ L 2 − 4 ( n − 1 ) ( n + 1 ) 4 ∫ CP n f 3 d V g 1 .

Using equation (4.6), we have

⟨ ∇ i ∇ j ψ k l , ψ i j ψ k l ⟩ L 2 = − 2 ( n + 1 ) 3 ( 2 n 2 − 2 n − 1 ) ∫ CP n f 3 d V g 1 .

The second term can be computed as

⟨ ∇ i ∇ j φ k l , ψ i j φ k l ⟩ L 2 = 2 ( 1 − n ) 2 ( n + 1 ) 2 ⟨ ∇ i ∇ j f , f ∇ i ∇ j f + ( n + 1 ) f 2 ( g 1 ) i j ⟩ L 2 , = − 4 ( 1 − n ) 2 ( 1 + n ) 4 ∫ CP n f 3 d V g 1 ,

where we have used equation (4.3). Hence,

(5.2) ⟨ ∇ i ∇ j h k l , h i j h k l ⟩ L 2 = − 2 ( n + 1 ) 3 ( 2 n 3 − 4 n + 1 ) ∫ CP n f 3 d V g 1 .

The final term in (2.7) splits as

⟨ ∇ i ∇ l h k j , h i j h k l ⟩ L 2 = ⟨ ∇ i ∇ l ψ k j , ψ i j ψ k l ⟩ L 2 .

The fact that ψ is divergence free yields

⟨ ∇ i ∇ l ψ k j , ψ i j ψ k l ⟩ L 2 = ⟨ ∇ i ∇ l ψ k j , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 + ( n + 1 ) ⟨ ∇ i ∇ l ψ k j , f ( g 1 ) i j ⋅ ∇ k ∇ l f ⟩ L 2 .

The first term splits again, so

⟨ ∇ i ∇ l ψ k j , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 = ⟨ ∇ i ∇ l ∇ k ∇ j f , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 + ( n + 1 ) ⟨ ∇ i ∇ l f ⋅ ( g 1 ) k j , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 .

Using identities (4.8) and (4.4) yields

⟨ ∇ i ∇ l ψ k j , ∇ i ∇ j f ⋅ ∇ k ∇ l f ⟩ L 2 = ( − ( n + 1 ) 3 + ( n + 1 ) 4 ) ∫ CP n f 3 d V g 1 .

The second term can be split

⟨ ∇ i ∇ l ψ k j , f ( g 1 ) i j ⋅ ∇ k ∇ l f ⟩ L 2 = ⟨ ∇ i ∇ l ∇ k ∇ j f , f ( g 1 ) i j ⋅ ∇ k ∇ l f ⟩ L 2 + ( n + 1 ) ⟨ ∇ i ∇ l f ⋅ ( g 1 ) k j , f ( g 1 ) i j ⋅ ∇ k ∇ l f ⟩ L 2 .

Hence, from equations (4.9) and (4.3), we obtain

⟨ ∇ i ∇ l ψ k j , f ( g 1 ) i j ⋅ ∇ k ∇ l f ⟩ L 2 = − 2 ( n + 1 ) 2 ∫ CP n f 3 d V g 1 .

Finally,

(5.3) ⟨ ∇ i ∇ l h k j , h i j h k l ⟩ L 2 = ( ( n + 1 ) 4 − 3 ( n + 1 ) 3 ) ∫ CP n f 3 d V g 1 .

We compute Koiso’s obstruction (2.7) using equations (5.1)–(5.3)

ℐ ( h ) = ( − 2 ( n + 1 ) 4 ( 2 n 3 − 6 n 2 + 4 n + 3 ) − 6 ( n + 1 ) 3 ( 2 n 3 − 4 n + 1 ) − 6 ( n + 1 ) 3 ( n − 2 ) ) ∫ CP n f 3 d V g 1 .

Simplifying yields

ℐ ( h ) = − 4 n ( n − 1 ) ( n + 1 ) 5 ∫ CP n f 3 d V g 1 .

The theorem follows for CP 2 n × CP 1 as the integral does not vanish identically and so all the EIDs are obstructed to second order by the argument after Lemma 2.2.□

Remark

We can compare our calculation of ℐ ( h ) to the quantity on page 667 of [7] (written in the notation of [7]),

⟨ E ″ ( h , h ) , h ⟩ = − ( n 1 − 2 ) ( n 1 + n 2 − 2 ) ( n 1 + 2 n 2 − 2 ) n 2 2 ℰ 4 ⟨ f 2 , f ⟩ .

Unpacking this in our notation, ⟨ E ″ ( h , h ) , h ⟩ = 1 2 ℐ ( h ) , n 1 = 2 n , n 2 = 2 , and ℰ = n + 1 . Thus, we see that we recover exactly Koiso’s original constant.

Acknowledgements

The author would like to thank Paul Schwahn and Uwe Semmelmann for useful discussions and the referees for their careful reading and correcting of the article.

Funding information: Author states no funding involved.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. SJH contributed to 100% of the manuscript.
Conflict of interest: The author has no competing interests to declare that are relevant to the content of this article.

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Received: 2024-09-09

Accepted: 2024-12-20

Published Online: 2025-02-20

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/coma-2024-0008

Keywords for this article

Einstein manifolds; deformations

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