Residue formula for Morita–Futaki–Bott invariant on orbifolds

C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1109–1113

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Algebraic geometry/Differential geometry

Residue formula for Morita–Futaki–Bott invariant on orbifolds ✩ Une formule résiduelle pour l’invariant de Morita–Futaki–Bott sur une orbifold Maurício Corrêa a , Miguel Rodríguez b a b

Dep. Matemática ICEx, UFMG, Campus Pampulha, 31270-901 Belo Horizonte, Brazil Dep. Matemática, UFSJ, Praça Frei Orlando, 170, Centro, 36307-352 São João Del Rei, MG, Brazil

a r t i c l e

i n f o

Article history: Received 9 June 2016 Accepted after revision 7 October 2016 Available online 17 October 2016

a b s t r a c t In this work, we prove a residue formula for the Morita–Futaki–Bott invariant with respect to any holomorphic vector field, with isolated (possibly degenerated) singularities in terms of Grothendieck’s residues. © 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Presented by Claire Voisin

r é s u m é On obtient, en utilisant les résidus de Grothendieck, une formule résiduelle pour l’invariant de Morita–Futaki–Bott par rapport à un champ de vecteurs holomorphes avec singularités isolées, dégénérées ou non. © 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

0. Introduction Let X be a compact complex orbifold of dimension n. That is, X is a complex space endowed with the following property: each point p ∈ X possesses a neighborhood, which is the quotient  U /G p , where  U is a complex manifold, say of dimenU , so that locally we have a quotient map sion n, and G p is a properly discontinuous finite group of automorphisms of  πp ( U , p˜ ) −→ ( U /G p , p ). See [1]. Let η( X ) be the complex Lie algebra of all holomorphic vector fields of X . Choose any Hermitian metric h on X and let ∇ and  be the Hermitian connection and the curvature form with respect to h, respectively. Let ξ be a global holomorphic vector field on X and consider the operator

L (ξ ) := [ξ, · ] − ∇ξ ( · ) : T 1,0 X −→ T 1,0 X . ✩

This work was partially supported by CNPq, CAPES, FAPEMIG and FAPESP-2015/20841-5. E-mail addresses: [email protected] (M. Corrêa), [email protected] (M. Rodríguez).

http://dx.doi.org/10.1016/j.crma.2016.10.006 1631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

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M. Corrêa, M. Rodríguez / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1109–1113

Let φ be an invariant polynomial of degree n + k; the Futaki–Morita integral invariant is defined by



f φ (ξ ) = X

 i i φ¯ L (ξ ), ..., L (ξ ), , ...,  ,    2π 2 π    k times n times

where φ¯ denotes the polarization of φ . Morita and Futaki proved in [6] that the definition of f φ (ξ ) does not depend on the choice of the Hermitian metric h. It is well known that the Futaki–Morita integral invariant can be calculated via a Bott-type residue formula for non-degenerated holomorphic vector fields, see [5–7] and [4] in the orbifold case. In this work, we prove a residue formula for holomorphic vector fields with isolated and possibly degenerated singularities in terms of Grothendieck’s residues (see [8, Chapter 5]). Theorem 1. Let ξ ∈ η( X ) a holomorphic vector field with only isolated singularities, then



n+k n

 k

f φ (ξ ) = (−1)



  φ J ξ˜ d z˜ 1 ∧ · · · ∧ d z˜ n Res p˜ , #G p ξ˜1 . . . ξ˜n 1

p ∈Sing(ξ )



πp

where, given p such that ξ( p ) = 0 and ( U , p˜ ) −→ ( U /G p , p ) denotes the projection: ξ˜ = π∗p ξ , J ξ˜ =

∂ ξ˜i ∂ z˜ j

 and 1≤i , j ≤n

  φ J ξ˜ dz˜ 1 ∧ · · · ∧ dz˜ n Res p˜ is Grothendieck’s point residue and (˜z1 , . . . , z˜ n ) is a germ of the coordinate system on ( U , p˜ ). ξ˜1 . . . ξ˜n

Wenote that such residue can be calculated using Hilbert’s Nullstellensatz and Martinelli’s integral formula. In fact, if a n ˜ z˜ i i = j =1 b i j ξ j , then (see [11])

     φ J ξ˜ d z˜ 1 ∧ · · · ∧ d z˜ n 1 ∂n ˜ Res p˜ Det(b i j )φ( J ξ ) ( p˜ ). = n ∂ z˜ 1a1 , ..., z˜ nan ξ˜1 . . . ξ˜n i =1 (ai − 1)!

(1)

Moreover, note that if p ∈ Sing(ξ ) is a non-degenerated singularity of ξ , then

   φ J ξ˜ d z˜ 1 ∧ · · · ∧ d z˜ n φ J ξ˜ ( p˜ ) Res p˜ =  . Det J ξ˜ ( p˜ ) ξ˜1 . . . ξ˜n

Theorem 1 allows us to calculate the Morita–Futaki invariant for holomorphic vector fields with possible degenerated singularities. For instance, in a recent work [9], the Futaki–Bott residue for vector fields with degenerated singularities, on the blowup of Kähler surfaces, was calculated by Li and Shi. Compare the equation (1) with Lemma 3.6 of [9]. Futaki showed in [5] that if X admits a Kähler–Einstein metric, then f C n+1 ≡ 0, where C 1 = T r denotes the trace, i.e., the 1

first elementary symmetric polynomial. Taking φ = C 1n+1 = T r n+1 , we obtain the following corollary of Theorem 1. Corollary 2. Let ξ ∈ η( X ) with only isolated singularities, then

−1 f C n+1 (ξ ) = 1 (n + 1)2

p ∈Sing(ξ )

1 #G p

Res p˜





T r n+1 J ξ˜ d z˜ 1 ∧ · · · ∧ d z˜ n

ξ˜1 . . . ξ˜n

 .

This result generalizes the Proposition 1.2 of [4]. It is well known that projective planes are Kähler–Einstein. However, the non-existence of Kähler–Einstein metrics on singular weighted projective planes was shown in previous works; see, for example, [12]. As an application of Theorem 1, we will give, in Section 1, a new proof of this fact. 1. Non-existence of Kähler–Einstein metric on weighted projective planes Here we consider weighted complex projective planes with only isolated singularities, which we briefly recall. Let w 0 , w 1 , w 2 be positive integers two by two co-primes, set w := ( w 0 , w 1 , w 2 ) and | w | := w 0 + w 1 + w 2 . Define an action of C∗ in C3 \ {0} by

C∗ × C3 \ {0} −→ C3 \ { 0 } w0 λ.(z0 , z1 , z2 ) −→ (λ z0 , λ w 1 z1 , λ w 2 z2 ) and let P2w := C3 \{0}/ ∼. The weights are chosen to be pairwise co-primes in order to assure a finite number of singularities and to give P2w the structure of an effective, Abelian, compact orbifold of dimension 2. The singular locus is:

M. Corrêa, M. Rodríguez / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1109–1113

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Sing(P2w ) = [1 : 0 : 0]ω , [0 : 1 : 0]ω , [0 : 0 : 1]ω . We have the canonical projection

π : C3 \ {0} −→ P2w w ( z0 , z1 , z2 ) −→ [ z0 0 : z1w 1 : z2w 2 ] w and the natural map

ϕ w : Pn

−→ Pnw w0 [z0 : z1 : z2 ] −→ [z0 : z1w 1 : z2w 2 ] w of degree deg ϕ w = w 0 w 1 w 2 . The map ϕ w is good in the sense of [1, section 4.4], which means, among other things, that V-bundles behave well under pullback. It is shown in [10] that there is a line V-bundle OP2 (1) on P2w , unique up to w isomorphism, such that

ϕ w∗ OP2w (1) ∼ = OP2 (1) ∗ O (1)) = c (O (1)) = ϕ ∗ c (O (1)), from which we obtain the Chern number and, by naturality, c 1 (ϕ w 1 P2 w 1 P2 P2 w

n   2 [P2w ] c 1 (OP2w (1)) = c 1 (OP2 (1)) = w 

Pnw

since

 1=



2

c 1 (OP2 (1))

 =

P2



2

ϕ w∗ c1 (OP2w (1))

w

1 w0 w1 w2

= (deg ϕ w )

P2

 

2

c 1 (OP2 (1)) w

.

P2w

There exist an Euler type sequence on Pnw 2 

0 −→ C −→

i =0

OP2w ( w i ) −→ T P2w −→ 0,

where (i) 1 −→ ( w 0 z0 , w 1 z1 , w 2 z2 ).

(ii) ( P 0 , P 1 , P 2 ) −→ π∗

2

i =0

P i ∂∂z

 i

.

It is well known that the non-singular weighted projective planes admit Kähler–Einstein metrics. On the other side, singular weighted projective spaces do not admit Kähler–Einstein metrics, see [12]. We give a simple proof of the non-existence of Kähler–Einstein metrics on singular P2ω by using Corollary 2. Theorem 3. The singular weighted projective space P2ω does not admit any Kähler–Einstein metric. Proof. Choose a0 , a1 , a2 ∈ C∗ such that ai w j = a j w i , for all i = j. Suppose, without loss of generality, that 1 ≤ w 0 ≤ w 2 < w 1 . Consider the holomorphic vector field on P2ω given by

ξa =

2

ak Z k

k =0

∂ ∈ H 0 (P2ω , T P2ω ), ∂ Zk

where ( Z 0 , Z 1 , Z 3 ) denotes the homogeneous coordinate system. The local expression of ξ over U i = {[ Z 0 : Z 1 : Z 3 ] ∈ P2 ; Z i = 0} is given by

ξa |U i =

2



ak − ai

k =0

wk



wi

Zk

∂ . ∂ Zk

k=i

Therefore, the singular set Sing(ξ |U i ) is reduced to {0} and it is nondegenerate. In general,





Sing(ξa ) = [1 : 0 : 0]ω , [0 : 1 : 0]ω , [0 : 0 : 1]ω = Sing(P2ω ). It follows from Corollary 2 that

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M. Corrêa, M. Rodríguez / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1109–1113



−1 1 2

f (ξa ) =

32

i =0

w 2i

k=i (ak w i

3 − ai w k )

k=i (ak w i

− ai w k )



.

Thus



ζ (a0 , a1 , a2 ) = −32 w 20 w 21 w 22

(ai w j − a j w i ) f (ξa ) =

0≤ i < j ≤ 2

(3w 51 w 22 w 0 − 3w 41 w 32 w 0 + 3w 31 w 42 w 0 + 3w 21 w 52 w 0 − 3w 40 w 22 w 21 + 3w 30 w 32 w 21 + 6w 20 w 42 w 21 + + 3w 40 w 21 w 22 − 3w 30 w 31 w 22 − 6w 20 w 41 w 22 ) · a1 a2 a20 + · · · is a homogeneous polynomial of degree 4 in the variables a0 , a1 , a2 . Suppose by contradiction that ζ (a0 , a1 , a2 ) ≡ 0. In particular, the coefficient of the monomial a20 a1 a2 is zero. Thus, we have the following equation

w 2 ( w 1 w 2 + w 22 + w 20 + 2w 0 w 2 ) = w 1 ( w 1 w 2 + w 21 + w 20 + 2w 0 w 1 ). This contradicts 1 ≤ w 0 ≤ w 2 < w 1 . Thus the non-vanishing of ζ (a0 , a1 , a2 ) implies that f (ξa ) is not zero. Therefore, P2ω does not admit Kähler–Einstein metrics. 2 2. Proof of Theorem 1 For the proof, we will use Bott–Chern’s transgression method, see [2] and [3]. Let p 1 , . . . , pm be the zeros of ξ . Let {U β } be an open cover orbifold of X ( ϕβ :  U β → U β ⊂ X coordinate map ). Suppose that {U β } is a trivializing neighborhood for the holomorphic tangent orbibundle T X (see [1, section 2.3]) of X and that we have disjoint neighborhoods coordinates U α with p α ∈ U α and p α ∈ U β if α = β . On each  U α , take local coordinates z˜ α = (˜z1α , . . . , z˜ nα ) and the holomorphic frame { ∂ ∂z˜ α , . . . , ∂ ∂z˜ α } of T X . Thus, we have a local representation 1

ξ˜ α =



ξ˜iα

n

∂ , ∂ z˜ αi

where ξ˜iα are holomorphic functions in  U α , 1 ≤ i ≤ n. Let h˜ α the Hermitian metric in  U α defined by ∂/∂ z˜ α , ∂/∂ z˜ αj  = δ ji . i       Also consider U α ⊂ U α and U α = ϕα (U α ) for each α . Take a Hermitian metric h0 in any X \ ∪α { p α } and {ρ0 , ρα } a partition  of unity subordinate to the cover { X \ ∪α U α , U α }α . Define a Hermitian metric h = ρ0 h0 + ρα hα in X . Then we have that for every α , the metric curvature  ≡ 0 in U α .

 βj β Consider the matrix of the metric connection ∇ in the open  U β given by θ β = ( k ik d z˜ k ). β The local expression of L (ξ ) is given by E˜ β = ( E˜ i j ) such that

E˜ i j = − β

∂ ξ˜i

β

β

∂ z˜ j

−



 js ξ˜s , βi β

s

see [2] and [8]. We indicate by A p , q ( X ) the vector space of complex-valued ( p + q)-forms on X of type ( p , q). Define



φr :=

n+k r



¯ E , ..., E , , ..., ) ∈ Ar , r ( X ) r = 0, ..., n. φ(       n+k−r

r

Let ω ∈ A1,0 ( X ) in X \Sing(ξ ), with ω(ξ ) = 1. Following idea (see [2]), it is sufficient to show that there exists ψ such n−Bott’s 1 that i (ξ )(∂¯ ψ + φn ) = 0 on X \Sing(ξ ). We take ψ = r =0 ψr such that

ψr = ω ∧ (∂¯ ω)n−r −1 ∧ φr ∈ An, n−1 ( X ) r = 0, ..., n − 1. The following formulas hold (see [2] or [8]): a) ∂¯  = 0, ∂¯ E = i (ξ ); b) ∂¯ φr = i (ξ )φr +1 , r = 0, ..., n + 1; c) i (ξ )∂¯ ω = 0. Let us prove b): since ∂¯  = 0 and ∂¯ E = i (ξ ), we have

 n+k−r

n+k ¯ E , ..., i (ξ ), ..., E , , ..., ) = i (ξ )φr +1 . ∂¯ φr = φ( r

i =1

M. Corrêa, M. Rodríguez / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1109–1113

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Therefore, a), b) and c) implies that on X \Sing(ξ ) we get

i (ξ )(∂¯ ψ + φn ) = 0.

¯ = −φn on X \Sing(ξ ). Thus, by the Satake–Stokes Theorem, we have Therefore, dψ = ∂ψ



n+k





f φ (ξ ) =

n

n 

i 2π

X

=−



φn = n

i

i 2π

n



lim

 lim

2π

→ 0

φn

→ 0

X \∪α B  ( p α )

X \∪α B  ( p α )



dψ =

i 2π

n lim

→ 0



 ψα,

(2)

α ∂ B (p )  α

where is B  ( p α ) = B  ( p˜ α )/G p α and B  ( p˜ α ) is an Euclidean ball centered at p˜ α such that B  ( p˜ α ) ⊂ U α . Since our metric is Euclidean in B  ( p˜ α ), its connection is zero and

E˜ iαj = −

∂ ξ˜iα . ∂ z˜ αj

Now, by our choice of metric,  and hence φr , for r > 0, vanishes identically in B  ( p˜ α ). Then, we have n −1 n −1 ψ˜ α = ψ˜ 0α = ω ∧ (∂ ω) φ( E˜ α ) = (−1)n+k ω ∧ (∂ ω) φ( J ξ˜ α )

on B  ( p˜ α ). Therefore, n −1 ψ˜ α = (−1)k ω ∧ (∂ ω) φ( J ξ˜ α ).

(3)

given by (˜z) = (˜z + ξ˜ (˜z), z˜ ). There is a (2n, 2n − 1) closed form βn in C2n \{0} (the

Consider the map  : C → C Bochner–Martinelli kernel) such that n

∗ βn =



i

2n

n

2π

n −1

ω ∧ (∂ ω)

.

(4)

Finally, if we substitute (3) and (4) into (2), and by using Martinelli’s formula ([8, p. 655])



φ( J ξ˜ α ) ∗ βn = Res p˜ α

∂ B  ( p˜ α )

we obtain



n+k n

 k

f φ (ξ ) = (−1)

α

  φ J ξ˜ α d z˜ 1 ∧ · · · ∧ d z˜ n ξ˜1 . . . ξ˜n

  φ J ξ˜ α d z˜ 1 ∧ · · · ∧ d z˜ n Res p˜ α . #G p α ξ˜1 . . . ξ˜n 1

Acknowledgements We are grateful to Marcio Soares and Marcos Jardim for many stimulating conversations on this subject. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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