LOCALIZING THE DONALDSON--FUTAKI INVARIANT

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LOCALIZING THE DONALDSON–FUTAKI INVARIANT

Eveline Legendre

To cite this version:

Eveline Legendre. LOCALIZING THE DONALDSON–FUTAKI INVARIANT. 2020. �hal-02868048�

EVELINE LEGENDRE

Abstract. We use the equivariant localization formula to prove that the Donaldson–

Futaki invariant of a compact smooth (K¨ahler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone.

1. Introduction

The Yau–Tian–Donaldson conjecture has been a central theme of K¨ahler geometry in the last 30 years. In one side of this conjectured correspondance, one tests the K–stability of a K¨ahler manifold (X,[ω]) using theDonaldson–Futaki invariant of tests configuration over it.

This invariant has a long history, the first candidate was the generalizedFutaki invari- ant of the central fiber as suggested by Tian [39]. The work of Tian was motivated by Yau conjecture about K¨ahler–Einstein metrics [43] and he was working in the Fano context but the definition he suggested does not need this hypothesis. Then Donaldson reinter- preted and generalized this invariant in the polarized case, in terms of the coefficients of the Hilbert series describing the asymptotics expansion of the dimension and weights of the action on the space of sections of the central fiber [12]. Starting from this general- ization, Odaka [32] and Wang [42], see also [12, p.315], exhibited an intersection product formulation of this now called Donaldson–Futaki invariant. One gain of this formulation is a direct interpretation in the non-polarized/transcendental case as exploited in [13, 38].

In this note, we start with the intersection product formulation of the Donaldson–Futaki invariant DF of a compact smooth test configuration (X,[Ω]) over a smooth compact K¨ahler manifold as a definition. We will recall the precise definition of a compact K¨ahler test configuration in Section 4 and prove the following : we observe that DF(X,[Ω]) is the intersection of S¹–equivariant closed forms on X and show that there exists smooth compacts submanifolds Z1, . . . , Zk, of X, all lying in the central fiber such that

(1) DF(X,Ω) =

Xk

i=1

Z

Zi

Aⁿ_i,Ω∧ B_i,Ω e(N_Z^X_i)

Date: June 15, 2020.

The author was partially supported by France ANR project EMARKS No ANR-14-CE25-0010.

1

whereA_i,Ω,B_i,Ω∈ ⊕_mΓ(Vm

T^∗Zi) are closed forms explicitly given in Proposition 5.1 and e(N_Z^X_i) is the equivariant Euler class of the normal bundle of Z inX.

Using this expression, we show that DF(X,Ω) coincides with the classical Futaki in- variant of the induced C^∗–action on the central fiber when the central fiber is smooth or admits orbifold singularities.

Theorem 1.1. Let (Xⁿ⁺¹,[Ω]) be a regular (compact) test configuration over (Xⁿ,[ω]) with π :X →P¹ and the C^∗–action ν :C^∗ ֒→Aut(X). Let V =ν∗(∂θ) be the vector field induced by the underlying S¹–action and µ:X →R be any Hamiltonian function for V. Assume that the central fiber X0 :=π⁻¹(0) inherits a K¨ahler orbifold structure from the inclusion ι0 :X0 ֒→ Xⁿ⁺¹. Then

DF(X,[Ω])

n! =−πFut_(X0,[Ω0])(J0V) (2)

where Ω0 =ι^∗_X0Ωis the pull-back on X0.

Test configurations with smooth central fiber but with π not being a submersion are known to exist see [1, 39]. Examples withπ being a submersion also fulfill the hypothesis of the Theorem, such examples include the so-called ”product tests configuration” see

§5.1.1.

Donaldson gave in [12] a very simple proof of this fact in the polarized case when the central fiber is smooth using the (equivariant) Hirzebruch–Riemann–Roch formula and the definition ofDF as a coefficient of the expansion of the normalized weight of the induced action on H⁰(L^k). Our approach is differential-geometric and does not distinguish the polarized and transcendental cases but doesn’t provide a more direct proof. A feature of our method is that it provides a formulation of the Donaldson–Futaki invariant as the evaluation of some classes on the (compact smooth) manifolds lying in the possibly singular central fiber, see Proposition 5.1. Using this and the conclusion of §4.2, as well as [22, Proposition 6] it is clear that DF is an equivariant class on the central fiber, see §5.0.1. The hypothesis of Theorem 1.1 is used to make sense of that equivariant class as the classical Futaki invariant.

Another asset of our proof is that it should be valid when replacing the K¨ahler structures involved by almost-K¨ahler ones which might be useful in the attemp to extend the Yau–

Tian–Donaldson conjecture to this setting, see eg [26].

We prove Theorem 1.1 using the Equivariant Localization Formula of Atiyah–Bott[3]

and Berline–Vergne [5], recalled in Section 2. The Equivariant Localization Formula has been used before in K¨ahler geometry [14, 16, 29, 39]. In particular, Tian used it success- fully to exhibit examples of Fano K¨ahler manifolds admitting no compatible and smooth K¨ahler–Einstein metric [40]. Also Wang used it, on P¹, to give the intersection formu- lation of the Donadson–Futaki invariant [42]. This latter technic has been used recently successfully by Inoue in [24] where he highlights also the equivariant interection formula (13) of the invariant as we do here.

In Section 6, we use our formula over fixed point set to compute the Donaldson–Futaki invariant of tests configuration obtained by deformation to the normal cone of any sub- variety Y ⊂X and relate it to the Futaki invariant of the exceptional divisor.

Aknowledgement I have been very talkative about this modest project. I thank the following people for their interest in this work: V. Apostolov, P. Carrillo-Rouse, R. Der- van, P. Gauduchon, H. Guenancia, L. Manivel, G. Maschler and D. Witt Nystr¨om.

2. The equivariant cohomology and localization formula

We briefly introduce the tools of equivariant cohomology we will use later. We refer to the books [6, 20] for more details and proof. We express the results for compact torus acting on smooth compact (symplectic or complex) manifold but the theory is developed in more complicated case.

2.1. Equivariant cohomology. Let M be a smooth manifold with an effective and smooth action of a compact torus T, that is ν : T ֒→ diffeo(M). We will mostly work with the induced infinitesimal action of t := Lie T. For a ∈ t, we denote the induced vector field on M by Va =φ∗(a), equivalently

Va(p) = d

dtt=0ν(exp(ta))(p).

An equivariant form is a polynomial map ψ :t→Ω^∗(M)^T where Ω^∗(M)^T is the graded complex of T-invariant forms. We denote Ω^∗_T(M) the set of equivariant forms over M.

There is an appropriate notion of degree for these forms so that the equivariant differential dt : Ω^∗_T(M)−→Ω^∗_T(M) which is defined by

a 7→ (dtψ)a := (d−_Va¬)ψa

increases the degree of 1. Here _Va¬ψ is the contraction ofψ by V_a. An equivariant form ψ ∈Ω^∗_T(M) is said equivariantly closedif dtψ ≡0.

Example 2.1. Consider (M²ⁿ, ω) a symplectic compact manifold admitting a Hamilton- ian action of a torus T, ν : T ֒→ Ham(M, ω) with momentum map µ : M → t^∗, that is

−dhµ, ai=ω(Va,·)

for any a ∈ t. The form a 7→ ω− hµ, ai is equivariantly closed, as well as, (ω− hµ,·i)^k and, thus, e^ω−hµ,·i:=P+∞

k=0

(ω−hµ,·i)^k k! .

It is known that d²_t = 0 and the equivariant cohomology can be defined as the coho- mology of the associated chain complex:

H_T^∗(M) := kerdt

im dt

see [6]. There is another definition of equivariant cohomology introduced by Atiyah and Bott [3] and valid for non connected compact Lie group G, which is the cohomology of the quotient M_G :=EG×_GM where EG→BG is the universal principal G–bundle. In the case of connected compact Lie group both definitions coincide.

2.2. Localization formula. Let Z := FixMT be the fixed points set of T in M. It is known, see eg. [6, 20], thatZ consists in a disjoint union of smooth submanifoldsZ=⊔Z of even codimension. Given a connected component, say Z ⊂ Z of codimension 2ℓZ, the normal bundle p:N_Z^M −→Z bears a complex structure induced by the action [20, §8.5]

and at any given pointz ∈Z, (N_Z^M)zsplits into a sum of (complex) lines (N_Z^M)z =⊕_jN_z,j^Z according to the action of T, acting on N_z,j^Z with weight w_j^Z ∈ t^∗, here j runs from 1 to ℓZ.

Theequivariant Localization Formulaas we use many times in this note is the following.

Theorem 2.2. [Atiyah–Bott [3], Berline–Vergne[5]] Let ψ ∈Ω^∗_T(M) be an equivariently closed form. For any generic a∈t we have

(3)

Z

M

ψa =X

Z

Z

Z

ι^∗_Zψa

e(N_a^Z).

e(N^Z)⁻¹ is the inverse of the equivariant Euler class of N_Z^M in H_S^∗1(Z).

Here, a point of t is said generic if the zero set of Va is the fixed points set of T. Recall that the equivariant Euler (respectively Todd, Thom, Chern...) class of a rank r G–equivariant bundle E → M is the Euler (respectively Todd, Thom, Chern...) class of the corresponding rank r bundle EG → MG. When the splitting into line bundles is global N_Z^M =⊕jN_j^Z the equivariant Euler class of N_Z^M is just

(4) a 7→e(N_a^Z) = 1

(−2π)^ℓZ

ℓZ

Y

j=1

(2πc1(N_j^Z)− hw^Z_j, ai).

Remark 2.3. We use a different convention than most authors (see eg. [6]) in incorporating (−2π)^ℓZ in the notation of e(N_a^Z).

Example 2.4. Consider the Riemann sphere P¹ =C⊔ {∞} with the complex structure induced by C⊂P¹, coordinates z =x+iy =re^iθ with 0≤r≤+∞, metric

gF S = 4

(1 +r²)²(dr²+r²dθ²)

and symplectic form ωF S = _(1+r⁴2)²rdr∧dθ. The total volume is then 4π, the metric is K¨ahler-Einstein with Einstein constant equals 1 and scalar curvature 2. The standard action of S¹ is the one induced by the vector field ∂_θ :=x∂_y −y∂_y, that is

(e^iθ, z)7→e^iθz,

a momentum map is µF S = (r²−1)/(r²+ 1) so that R1

P µF SωF S = 0 and the weight of the action onT0P¹ is 1 and the action onT∞P¹ is −1.

3. Localizing the Futaki invariant

We recall how one can use the equivariant Localization Formula (3) to give an alterna- tive formula for the Futaki invariant [29]. This was done in various setting in [14, 16, 39].

Consider a K¨ahler compact manifold (M²ⁿ, g, ω, J) with K¨ahler class [ω]∈H_dR² (M,R).

The Futaki invariant, see [15], is a character on the space of holomorphic vector fields and can be alternatively defined on the space of real holomorphic vector fieldsF_[ω]:h→R as

F_[ω](V) = Z

M

(V.φ)ωⁿ n!

whereρ^g =ρ^g_H+dd^cφis the Ricci form of (M²ⁿ, g, ω, J) andρ^g_H ∈2πc1(M) is the harmonic representative. We have the decomposition, see eg [18, Lemma 2.1.1],

V^♭ =ξH +df^V +d^ch^V

where ξH is a harmonic 1–form and f^V, h^V ∈ C^∞(M). The Futaki invariant is alterna- tively given by

F_[ω](V) = Z

M

(f^V −f^V)sg

ωⁿ n!

wheresg is the scalar curvature and f^V =R

Mf^Vωⁿ/R

Mωⁿ, see [18, §4.11]. In particular, for any a∈t we have

(5) − F_[ω](JVa) = Z

M

(hµ, ai − hµ, ai)s_gωⁿ n! = 2

Z

M

(hµ, ai − hµ, ai)ρ^g ∧ωⁿ⁻¹ (n−1)! . Equivalently,

(6) − F_[ω](JVa) = 2

Z

M

hµ, ai(ρ^g−cω)∧ωⁿ⁻¹ (n−1)!

for c=c_[ω]:=R

Mρ^g∧ωⁿ⁻¹/R

Mωⁿ. Recall, see eg [18, Lemma 1.23.4], that the Bochner formula gives in the K¨ahler setting

Va

¬ρ=−1 2d∆^gµa

where we put µ_a :=hµ, ai. We get two equivariantly closed forms αω,a:=a7→(ω−µa) anda 7→βω,a =

nc

n+ 1(ω−µa)−

ρ^g −1 2∆^gµa

such that

(7) F[ω](JVa) = 2

n![αω,a]ⁿ∪[βω,a] ([M])

in equivariant cohomology.

Remark 3.1. The equivariant class of (ω −µa) only depends on the de Rham class of [ω]. Indeed, if ω−ω^′ = dd^cφ then (ω^′−(µa−d^cφ(Va))) is equivariently closed and the difference is equivariantly exact

(ω−µ_a)−(ω^′−(µ_a−d^cφ(V_a))) =dt(d^cφ)_a. Using the same argument [βω,a] = [βω^′,a] in equivariant cohomology.

Recall that the momentum map µ is a constant on any connected component Z ⊂ Fix_T(M). Moreover, for any a ∈ t the set of values {µ_a(Z)}_Z depends on the choice of the momentum map and on the representative in the de Rham class [ω] only up to an overall constant. We put

h^Z_[ω](a) :=µa(Z) and recall that for any z ∈Z,

(8) (∆^ghµ, bi)_z =−2h

n−nXZ

j=1

w^Z_j, bi

for a proof of this see eg. [10]. Therefore, using Theorem 2.2 and (13) for f = ¹₂∆^gµa, we get that

F[ω](JVa) = X

Z

nc[ω]([ω]−h^Z_[ω](a))ⁿ⁺¹ (n+ 1)!e(N_a^Z) ([Z])

−2X

Z

(2πc₁(M) +Pn−nZ

i=1 hw_i, ai)∪([ω]−h^Z_[ω](a))ⁿ n!e(N_a^Z) ([Z]) (9)

Remark 3.2. The first line of (9) vanishes if we take the normalization R

Mµωⁿ= 0 which turns F_[ω](JVa) into a homogenous polynomial of order n in the variable [ω].

Formulation (9) has been used in [14] to extend the notion of Futaki invariant to singular K¨ahler manifolds, in particular orbifolds. Recall that K¨ahler geometry can be extended to orbifolds since the analysis is essentially the same as explained in [14] and many other works. Be aware, as highlighted in the first section of [35], there is a difference between complex orbifold singularity and a (K¨ahler) metric orbifold singularity. The analysis is similar to smooth manifold setting in the second case which is the case we consider here (this is also the case considered in the aforementioned works). Indeed, an orbifold metric is a metric which pulls back to a smooth invariant metric on the uniformizing charts, then it makes sense to define the curvature, Ricci curvature and so on.

Using the localization principle for orbifolds, see [31], we get F[ω](JVa) = X

Z

nc([ω]−h^Z_[ω](a))ⁿ⁺¹ dZ(n+ 1)!e(N_a^Z) ([Z])

−2X

Z

(2πc1(M) +Pn−nZ

i=1 hw_i, ai)∪([ω]−h^Z_[ω](a))ⁿ d_Zn!e(N_a^Z) ([Z]) (10)

where dZ is the number of elements in the isotropy group of generic points of Z and N^Z is the equivariant normal bundle of Z in the uniformizing charts.

4. Regular compact test configurations

4.1. Compact regular test configurations and the Donaldson–Futaki invariant.

Following [13, 38], we call a (regular) test configuration for a K¨ahler manifold (X, ω) a following set of data

(1) A smooth compact K¨ahler manifold (X,Ω);

(2) C^∗–actionν :C^∗ ֒→Aut(X) such that [ν(t)^∗Ω] = [Ω];

(3) a C^∗–equivariant surjective map π : X −→ P¹ for the standard action on P¹ :=

C∪ {+∞};

(4) aC^∗–equivariant biholomorphism

ψ :X^∗ :=X \π⁻¹(0)−→^∼ X×P¹\{0}

for the trivial action onX times the restriction of the standard action on P¹ and pr₂(ψ(x)) =π(x).

Moreover, for t ∈ P¹, we denote X_t := π⁻¹(t) and the inclusion ι_t : X_t ֒→ X. We have, for t6= 0, Xt

ψt

≃X, that isψt= pr₁◦ψ◦ιt. Finally we denote Ωt=ι^∗_tΩ and assume that

(11) [Ωt] = [ψ_t^∗ω].

In this context, the Donaldson–Futaki invariant of (X,Ω), when dimCX = n, can be defined to be:

(12) DF(X,Ω) = n

n+ 1c[ω]

Z

X

Ωⁿ⁺¹− Z

X

(ρ^Ω−π^∗ρ^{F S})∧Ωⁿ where c[ω]=R

Xρ^ω∧ωⁿ⁻¹/R

Xωⁿ.

The quantity (12) coincides with the invariant introduced by Donaldson in [12] when the test configuration is the compactification of a polarized test configuration as shown in Odaka [32] and Wang [42], see also [12, p.315].

Remark 4.1. In the K¨ahler setting [13, 38] the test configurations over (X,[ω]) considered are more generally (compact) K¨ahler analytical spaces. Then the Donadson-Futaki invari- ant as (12) is defined for a smooth resolution over it. It is proved then that the definition is independant of the resolution and that to test K-semistability (i.e thatDF(X,[Ω]) ≥0

for all test configurations over (X,[ω])) and even uniform K-stability (see eg.[13]) it is enough to consider smooth test configurations, see [13, Proposition 2.23].

Remark 4.2. The definition of a test configuration and of its Donaldson–Futaki invariant is independant of the representative Ω chosen in [Ω] ∈ H_dR² (X) and is usually denoted DF(X,[Ω]). For the techniques of equivariant cohomology we apply in this paper it is convenient to pick a S¹–invariant representative in [Ω] where we understand S¹ ֒→C^∗ in the standard way. By abuse of notation we will often avoid the brackets.

4.2. The Donaldson–Futaki invariant as an equivariant classes product. From now on, without loss of generality, we pick a S¹-invariant K¨ahler metric Ω on X.

We denote V := ν∗(∂θ) the real holomorphic vector field on X induced from the gen- erator of S¹ via the action. By assumption this is also a Killing vector field with zeros and thus it is a Hamiltonian vector field on (X,Ω). We pick a Hamiltonian function µ:X → Rthat is

−dµ= Ω(V,·).

We get again two equivariantly closed forms AΩ := (Ω−µ) and B_Ω = nc

n+ 1(Ω−µ)−

ρ^Ω− 1 2∆^Ωµ

+ (π^∗ωF S−π^∗µF S)

where µF S is a Hamiltonian for the standard S¹ action on (P¹, ωF S) see Example 2.4.

Since the integration picks up only the 2(n+ 1)–degree terms we have (13) DF(X,Ω) = [AΩ]ⁿ∪[BΩ] ([X])

in equivariant cohomology. As in Remark 3.1, the equivariant classes [A_Ω], [B_Ω]∈H_S²1(X) are independant of the chosen representative in [Ω]∈H_dR² (X,R).

Many times in this paper we will use the following facts and notation: sinceπ :X →P¹ is equivariant the fixed points set of the S¹ action on X, Fix_S¹X, lies in X₀ ⊔X_∞. Following a classical argument of Riemannian geometry, Fix_S¹X is a disjoint union of smooth submanifolds, Fix_S¹X = X_∞⊔(⊔_lZ_l). One of the component is X_∞ because of the condition (4) defining a test configuration. The remaining components lie in X0. We denote then

Fix_S¹X = (⊔Z)⊔X∞

where Z denotes a generic component of FixS¹X inX0.

5. Localizing the Donaldson–Futaki invariant

Applying Theorem 2.2 we will prove the next proposition where the following notation is used: given any subset S ⊂ X we denote the inclusion ι_S : S → X and, for any form/function/tensor.. α onX the pull-back to S is denoted α_S :=ι^∗_Sα.

Proposition 5.1. Let(X,[Ω], ν, π, ψ)be a regular (compact) test configuration over(Xⁿ,[ω]).

We pick Ω ∈ [Ω] a S¹–invariant K¨ahler metric. Let V = ν_∗(∂θ) be the vector field in- duced by the underlying S¹–action, µ: X → R be any Hamiltonian function for V. The Donaldson–Futaki invariant of (X,[Ω], ν, π, ψ) is

DF(X,Ω) n! =X

Z

Z

Z

nc[ω](ΩZ−µZ)ⁿ⁺¹ (n+ 1)!e(N_Z^X)(V)

−X

Z

Z

Z

(ρ^Ω_Z+Pn−nZ

i=0 hwi, ∂θi)∧(ΩZ−µZ)ⁿ n!e(N_Z^X)(V)

+X

Z

Z

Z

(ΩZ−µZ)ⁿ n!e(N_Z^X)(V). (14)

where Z runs into the set of connected components of the fixed points set of the S¹– action lying in the central fiber X0 = π⁻¹(0). The equivariant vector bundle N_Z^X over Z is the normal bundle of Z in X and w^Z₀, . . . ,w_n−n^Z

Z ∈ (Lie S¹)^∗ are the weights (with multiplicity) of the induced action of S¹ on N_Z^X.

Remark 5.2. A more concise way to express the last claim using the notation introduced in§4.2 is the following

DF(X,Ω) =X

Z

Z

Z

ι^∗_ZAⁿ_Ω∧ι^∗_ZB_Ω,o e(N_Z^X)(V) where B_Ω,o= ^nc_n+1^[ω]A_Ω−(ρ^Ω+Pn−nZ

i=0 hw_i, ∂θi) + 1 and ιZ :Z ֒→ X is the inclusion.

Proof. We use the notation of the last paragraph and the observation and notation lay down in the last subsection. In particular, the normal bundle of X∞ inX, denoted N^∞, is trivial and the weight of the induced action is −1. Consequently the Euler equivariant form (with the normalisation used here (4)) is

e(N_V^X∞) =−(2πc₁(N^X∞)− hw^∞, ∂θi)/2π =−1/2π.

Now, as explained in§4.2, see formula (13) we can write the Donaldson–Futaki invariant as the integral of an equivariantly closed form. We apply Theorem 2.2, usinge(N_V^X∞) = 1 and writtingf = ¹₂∆^Ωµ, we get

DF(X,Ω)

n! =X

Z

Z

Z

nc_[ω](Ω_Z−µ_Z)ⁿ⁺¹ (n+ 1)!e(N_Z^X)V

− (ρ^Ω_Z−fZ)∧(ΩZ−µZ)ⁿ n!e(N_Z^X)V

+ Z

Z

ι^∗_Z(π^∗ρ^{F S}−π^∗µ_{F S})∧(Ω_Z−µ_Z)ⁿ n!e(N_Z^X)V

−2π Z

X∞

nc[ω](Ω∞−µ∞)ⁿ⁺¹

(n+ 1)! − (ι^∗_∞ρ^Ω−f_X∞)∧(Ω_∞−µ_∞)ⁿ n!

−2π Z

X∞

ι^∗_∞(π^∗ρ^{F S}−π^∗µ_{F S})∧(Ω_∞−µ_∞)ⁿ n!

where c[ω] = c(X,[ω]). We will see that the integral over X∞ vanishes up to one term.

Note that ι^∗_∞π^∗ρ^{F S} = 0 and on X∞, ι^∗_∞(∆^Ωµ) = −2hw^∞, ∂θi = 2 and µF S(∞) = 1 see Example 2.4. The splitting ι^∗_∞TX =T X_∞⊕π^∗T_∞P¹ implies that [ι^∗_∞ρ^Ω] = 2πc₁(X_∞) = [ρ^Ω∞] in cohomology. The integral over X_∞ is then

Z

X∞

−nc_[ω]µ∞Ωⁿ_∞

n! − (ρ^Ω∞−f_∞)∧(Ω_∞−µ_∞)ⁿ

n! − (Ω_∞−µ_∞)ⁿ n!

= Z

X∞

−nc[ω]µ∞Ωⁿ_∞

n! − ρ^Ω∞∧(Ω∞−µ∞)ⁿ

n! + (f∞−1)(Ω∞−µ∞)ⁿ n!

= Z

X∞

−nc[ω]µ∞Ωⁿ_∞

n! +nµ∞ρ^Ω∞∧Ωⁿ⁻¹_∞

n! + (f∞−1)(Ω∞−µ∞)ⁿ n! . (15)

The first two terms cancel each others because c[ω]=c(X,[ω]) = c(X∞,[Ω∞]).

The remaining componentsZ of the fixed points set lies inX0 =π⁻¹(0) soι^∗_Zπ^∗ρ^{F S} = 0 and ι^∗_Zπ^∗µF S = −1. Moreover since f∞ = 1 and fZ =−Pn−nZ

i=0 hw_i, ∂θi, we end up with

(14).

5.0.1. Singular central fiber. Proposition 5.1 gives the DF invariant as an intersection number on compact submanifolds lying in the possible singular central fiber. In the next two subsections we show that using back the equivariant localization formula (3) the right hand side of (14) is nothing but the Futaki invariant of (the induced S¹ action on) the central fiber under some strong conditions ensuring this last invariant to be defined as it is classically [15] and recalled in §3. As one can check in the subsequent section, the issue is to relate the equivariant Euler forms e(N_Z^X) and e(N_Z^X0). To get a valid statement when X₀ is singular one needs to refer to equivariant cohomology for singular variety. This theory has been developed in details by Edidin and Graham in the context of algebraic spaces and schemes, in terms of cycles and equivariant Chow groups [21].

They also proved a localization formula [22] which fits well in our setting, namely the singular variety has to be embedded in a smooth one. This embedding allows them to use the normal bundle as we get in the right hand side of (14). Therefore, starting with a test configuration bearing an algebraic structure (eg. polarized scheme see [12]) and up to translating statements about equivariant forms into statement involving equivariant cycles, we can apply directly Edidin-Graham result [22, Proposition 6] to state that in this case

DF(X,[Ω]) =ι^∗₀[A_Ω]ⁿ∪ι^∗₀[B_Ω]([X0]).

Indeed, as an equivariant (S¹–stable) subvariety X0 ⊂ X the forms ι^∗₀AΩ and ι^∗₀BΩ are also equivariantly closed forms on X0 (seen as a K¨ahler space). Moreover, in anycase (even when X0 is singular), Proposition 5.1 tells us that the contribution to DF coming fromX∞ is trivial.

5.1. Smooth central fiber.

Theorem 5.3. Let (X,[Ω], ν, π) be a regular (compact and smooth) test configuration and let X₀ be the central fiber on which the S¹ action induced by ν gives the vector field V =ν_∗∂_θ ∈Γ(T X₀). Assume that the central fiberX₀ is a smooth submanifold ofX, then

(16) DF(X,[Ω])

n! =−πF_[Ω0](J₀V)

where (Ω0, J0) denotes the K¨ahler structure of X0 induced from X.

Proof. We use the notation introduced in §2 and denote (Ω, J, G) a S¹–invariant K¨ahler structure on X. We will apply Proposition 5.1, more precisely formula (14), with a Hamiltonian function µ:X →R for V =ν∗(∂θ) normalized by

(17)

Z

X0

µ0Ωⁿ₀ = 0 where, as usual, we denoteι^∗_X0µ=µ0.

Fix a connected component Z ⊂FixXS¹ lying in X0. The sequence of S¹–equivariant embeddings Z ⊂X0 ⊂ X gives a short exact sequence of vector bundles over Z

0−→N_Z^X0 −→N_Z^X −→N_X^X₀ −→0, which, in turn, gives

e(N_Z^X) =e(N_Z^X0)∪ι^∗_Ze(N_X^X₀).

We have a decomposition into G–orthogonalS¹–equivariant bundles

(18) ι^∗₀TX =T X0⊕E0

where E0 ≃ N_X^X₀ equivariantly. We put E₀^Z := ι^∗_ZE0. Observe that this line bundle corresponds to one summand (E₀^Z)z in the equivariant decomposition of TzX for any z ∈Z. We will show that the weight associated to this summand is 1.

Pick A ⊂ E₀^Z a S¹–invariant tubular neighbourhood of the zero section in E₀^Z → Z.

SinceX₀ is smooth andZ compact we may assume that the restriction of the Riemannian exponential map exp : A → X is injective on A. Moreover, each fiber Az := A ∩TzX is sent to a submanifold tranversal to X₀. Taking a smaller tubular neighbourhood if necessary, the map π◦exp is S¹–equivariant and its image contains some t∈P¹\{0,∞}.

From this we deduce easily that hw0, ∂θi=±1.

The pull back (π◦exp_z)^∗(µF S) is a smooth S¹–invariant function along the fiber Az. Thus, by a classical result of Schwarz [37] it is a smooth function of the fiberwise squared norm induced by G on E₀^Z, which coincides, up to the weight and an additive constant with exp^∗µ, see [19] and [4, §IV.1.b]. Now, sinceG(∇^Gµ,∇^Gπ^∗(µF S)) =π^∗gF S(V, V)>0 on π⁻¹(P¹\{0,∞}) we have that (π ◦ exp_z)^∗(µF S) is an increasing function of exp^∗µ along each fiber of E₀^Z. Therefore the weight of the S¹-action on E₀^Z is positive and thus hw₀, ∂θi= 1.

A key observation is that, because π is equivariant, P¹\{0,∞}are all regular values of π and then the normal bundles N_t:=N_X^X_t ofX_t inX are all trivial. This implies that on the regular part of X₀ (which is the whole of X₀ under the hypothesis here) the normal

bundle E0 ≃N_X^X₀ →X0 is, at least topologically, trivial. Indeed on the regular part any open setU ⊂ X, U∩(X0)reg can be approximated smoothly byU∩(Xt). Another way to be convinced of this is to build a section of N_(X^X_0)reg as follow. For a point x∈(X0)reg an equivariant neighbourhoodUx ⊂ X of the orbit S¹·x={ν(λ)x|λ∈S¹}can be described via the symplectic slice Theorem, see [19, 28]. So that, as Hamiltonian S¹–space, Ux is isomorphic to an open equivariant neighbourhood of the zero section inS¹×S_x¹(R×Vx) or TxS¹ depending ifS_x¹ is finite or not and where Vx = (T(S¹·x))^⊥Ω/(T(S¹·x)^⊥Ω∩T S¹·x).

Therefore the level sets of exp^∗_x|π|² (an increasing function ofexp^∗_xµF S) and ofexp^∗_xµcan be compared in a neighbourhood of 0∈TxN(X0)reg as we did above, to conclude that for t∈P¹ with |t| small enough there is only one single point in exp_x(Ux∩(N(X0)reg)x)∩Xt. With this we can build the desirable section σt∈Γ((X0)reg, N_(X^X_0)reg).

Putting all that together we have:

a) ι^∗_Z[ρ^Ω] =ι^∗_Z[ρ^Ω0]∈H_dR² (Z);

b) hw₀^Z, ∂θi= 1;

c) e(N_Z^X) =−e(N_Z^X0)/2π.

Condition c) allows us to consider the terms of (14) as integrals of equivariantly closed forms on X0 using the Localization Formula (2.2). Indeed the first line of (14) is the integral of the equivalently closed form 2π(Ω0−µ0)ⁿ⁺¹ onX0. Then the first line of (14) is canceled by the normalization (17).

Thanks to condition b) the middle term in the Formula (14) is canceled with the last term (the third line). We are left with

DF(X,Ω)

n! =−X

Z

Z

Z

(ρ^Ω_Z+Pn−nZ

i=1 hw^Z_i , ∂θi)∧(ΩZ −µZ)ⁿ n!(−e(N_Z^X0))(V) .

Condition a) together with the fact that the integrant above are constant or closed forms onZ gives that

DF(X,Ω)

n! =X

Z

Z

Z

(ι^∗_Zρ^Ω0 +Pn−nZ

i=1 hw^Z_i , ∂θi)∧(Ω_Z−µ_Z)ⁿ n!e(N_Z^X0)(V) . Now, we have (∆^Ω0µ₀)_z =−2Pn−nZ

i=1 hw_i, ∂θi for any z ∈Z and thus DF(X,Ω)

n! =X

Z

Z

Z

(ι^∗_Zρ^Ω0 − ¹₂ι^∗_Z(∆^Ω0µ0))∧(ΩZ−µZ)ⁿ n!e(N_Z^X0)(V)

where ρ^Ω0 − ¹₂(∆^Ω0µ0) is an equivariantly closed form on X0. This allows us to use the Localization formula again and we get

DF(X,Ω)

n! =−2π Z

X0

(ρ^Ω0 − ¹₂∆^Ω0µ0)∧(Ω0−µ0)ⁿ

n! =−πF[Ω0](J0V)

thanks to the Formula (13) and using (17) again.

5.1.1. Product test configurations. When a K¨ahler manifold (X, ω) admits a holomorphic action ν : C^∗ ֒→ Aut(X) restricting to a Hamiltonian isometric S¹–action ν : S¹ ֒→ Aut(X, ω), then one can build a test configuration (X_ν,Ω_ν) where the total space X_ν is obtained as the clutching construction from X and the action ν. One way to define (Xν,Ων) is via a K¨ahler reduction, see [8] for the projective case.

First consider the K¨ahler manifold (X×C², ω+ωstd) with a C^∗ action whose induced S¹ action admits the moment map

˜

µ(x, z, w) =µ(x)− 1

2(|z|²+|w|²)

where µ : X → R is the moment map picked for the action on (X, ω). Note that any c < minµ is a regular value of ˜µ and the S¹ action on ˜µ⁻¹(c) is free. Therefore X_ν = ˜µ⁻¹(c)/S¹ is a manifold which inherits of a K¨ahler structure (Jν,Ων,c), see eg.

Futaki[25, 17], so that the map π([x, z, w]) = [z : w] from X_ν to P¹ is holomorphic and equivariant. The underlying complex manifold is independant of the choice of c chosen into a connected set of regular values of ˜µthanks to [25]. The K¨ahler class [Ω_ν,c] depends on the value c∈(−∞,minµ) but in any case ι^∗_t[Ων,c] = [ω] as one can show following the ideas in [30]. Then (Xν,Ων, π, ν) satisfies the conditions of a test configuration enumerated in §4.1. Such a test configuration satisfies the hypothesis of Theorem 5.3 and thus we have DF(Xν,Ων) =−n!πFΩ(Jν∗(∂θ)).

5.2. Orbifold central fiber.

5.2.1. The equivariant geometry of the orbifold central fiber. Assume now that the central fiberX0 of the (smooth, compact) test configuration (X,Ω, π, ν) is not smooth but inherits from (X,Ω) of a K¨ahler orbifold structure. That means that for each singular pointx∈X0

there is an open set U ⊂ X such that Ω0 = ι^∗_X0Ω defines a distance on the set U ∩X0, which is isometric with an orbifold metric on some ˜U/Γ where Γ ⊂GL(Cⁿ) is finite and ˜U is a Γ invariant open subset ofCⁿ see eg [28, 35]. In particular, each connected component of the singular set has codimension at least 2 in X0 and X0 is irreductible as a complex space.

Moreover, the Futaki invariant of the induced C^∗ action on (X0,Ω0 = ι^∗_X0Ω) makes sense, see [14] and §3, and coincides with (10).

We first give some consequences of the assumptions above. Each connected component Z of the S¹–fixed point set is a smooth K¨ahler submanifold of (X,Ω) as recalled in Section 2. A connected (smooth) compact metric manifold cannot be isometric to a metric orbifold with more than one type of isotropy group, see [27, Lemma 2.1]. Therefore, Γ_z ≃Γ_Z for all z ∈ Z. Moreover, Z is compact as the connected component of the zero set of a smooth vector field on a compact manifold X, and thus Z is covered by a finite number of open sets of the form U ∩Z for open set U ⊂ X such that U ∩X0 satisfies the orbifold condition recalled above and that q⁻¹_U (z)∈Ue is a single point for all z ∈Z.

The subset q⁻¹_U (Z∩U) is fixed by Γ_Z, smooth in ˜U, and mapped homeomorphically onto Z ∩U via q_U. The collection of the uniformizing charts Ue together with the transition

maps condition define a differential manifold UeZ with an action of the abstract group ΓZ

so that Z ⊂ UeZ is a smooth submanifold fixed by ΓZ and there is a homeomorphism q_Z :UeZ/ΓZ → ∪U ∩X0. That is, there is a neighbourhood of Z inside X0 which is the global quotient of a manifold by a finite group Γ_Z acting freely on Ue_Z\Z and fixing Z.

We sum up the discussion in the following claim.

Lemma 5.4. Let (X,Ω, ν, π) be a regular (compact and smooth) test configuration and let (X0,Ω0) be the central fiber. Assume that X0 inherits of a structure of metric orbifold from (X,Ω) then each connected component of the singular locus either coincide with a component of the fixed points set or is disjoint of the fixed points sets.

The normal bundle of Z in UeZ inherits of ΓZ action, ˜p : N_Z^Ue → Z and the normal orbibundle of Z in X0 is the quotient N_Z^Ue/ΓZ. We denote p: N_Z^X0 → Z, the orbibundle map.

These open setsU ⊂ X, coveringZ, can be chosen equivariant with respect to the S¹– action since Z is fixed by S¹ and thus U ∩X0 are also equivariant. Then, following [28], there is a group extension ΓZ ֒→ HU

։q S¹ and an action ˜ν :HU ֒→Isom( ˜U,g) covering˜ the initial one via q_U. Since Γz is constant on Z, the extension is also constant, say H_U ≃ H_Z, and we get an isometric (and biholomorphic) action of H_Z defined globally on the manifold UeZ. There are two possibilities: either HZ is a product of S¹ with a finite group (ΓZ or a quotient of it) or HZ ≃ S¹ if HZ is connected. The example of Ding–Tian [14] falls into the first category and any toric examples would give the second [28]. In either case the connected component to the identity is isomorphic to S¹ and the normal bundle N_Z^UeZ is then a S¹–equivariant bundle over Z with weights

˜

w₁, . . . ,w˜_ℓZ ∈(LieS¹)^∗ ≃R(the weights might not lie in the (same) weights lattice of our original representation ofS¹ but it is not a problem in terms of localization formula [31]).

5.2.2. The Donaldson-Fuaki invariant and the Futaki invariant of the orbifold central fiber.

Theorem 5.5. Let (X,Ω, ν, π) be a regular (compact and smooth) test configuration and let (X0,Ω0) be the central fiber on which the S¹ action induced by ν gives the vector field V =ν∗∂θ. Assume that X0 inherits of a structure of metric orbifold from (X,Ω). Then

(19) DF(X,Ω)

n! =−πF_[Ω0](J0V)

where (Ω₀, J₀) denotes the K¨ahler structure of X₀ induced from X.

Ding and Tian produced examples of such test configurations in the polarized set- ting [14].

Proof. Observe from the proof of Theorem 5.3 that it is suffisant to show that these three following conditions hold on each connected component Z of the fixed point set:

a) ι^∗_Z[ρ^Ω] =ι^∗_Z[ρ^Ω0]∈H_dR² (Z);

b) PℓZ

i=0hw_i^Z, ∂θi = 1 +PℓZ

i=1hw˜_i^Z, ∂θi;

c) e(N_Z^X) =−d_Ze(N_Z^X0)/2π;

hered_Zis the order of the isotropy group of a generic point inZ,N_Z^X0 is theS¹–equivariant bundle over Z constructed above and weights ˜w₁, . . .w˜_ℓZ.

We fix a connected component Z ⊂ X0 ∩(FixS¹) for the rest of the discussion and denoteℓZ its complex codimension in X.

A key observation is that, because π is equivariant, P¹\{0,∞} are all regular values of π. Therefore the normal bundles Nt := N_X^X_t of Xt in X are all trivial. This implies that ι^∗_Xt[ρ^Ω] = ι^∗_Xt[ρ^Ωt] by the Adjunction formula. Then ρ^Ωt −→ ρ^Ω0 since it holds on the regular part and do not blow-up along singular locus thanks to the orbifold condition.

Hence, we get condition a).

Pick a point z ∈Z and consider the decomposition (20) TzX =TzZ⊕(⊕^ℓ_j=0^Z Ej,z)

induced by the isometric and HamiltonianS¹-action. This decomposition isG–orthogonal, Ω–orthogonal andJ–invariant as explained for example in [4, §IV.1.b]. Also the induced action ofS¹ is Hamiltonian and isometric and provides a vector fieldWz ∈Γ(T TzX) with Hamiltonian function H ∈C^∞(TzX) whose Laplacian at 0 is

−1 2

ℓZ

X

j=0

w_j

which is equivalent to Formula (8). Let consider the Riemannian exponential map exp_z : TzX −→ X associated to G. This map is S¹ equivariant and is a diffeomorphism onto its image when restricted to an open S¹-equivariant neighbourhood, say B, of 0 ∈ TzX. Pick a pointv ∈ B such that exp_zv /∈X0, such point exists by an argument involving the dimension of X0 ∩U\(Z ∩U). Since exp_z and π are S¹-equivariant and π(exp_zv) 6= 0, the stabilizor of exp_zv in S¹ must be trivial. Comparing the gradients of H = exp^∗_zµ and exp^∗_z|π|² one can argue, as in the proof of Theorem 5.3, that the punctured disk D^∗ ={λ∈C^∗| |λ| ≤1} is sent intoB via the map

λ7→d_zν(λ)v ∈T_zX and dzν(λ)v −→0 when λ→0∈C.

Taking UeZ the uniformizing chart of Z with the notation in§5.2.1, the S¹-equivariant decomposition of

TzUe =TzZ⊕(⊕^ℓ_j=1^Z Eej,z)

also behaves nicely with respect to the K¨ahler structure ( ˜Ωz,G˜z,J) induced by the (orb-˜ ifold) mapι0◦q_Z :Ue →U ⊂ X. We take anS¹-equivariant neighbourhood ˜B of 0∈TzUe such that the Riemannian exponential map expg_z : ˜B → Ue is a diffeomorphism over its image and consider the map

ψ_z(w, λ) = exp⁻¹_z ◦q_U◦expg_z(w) +d_zν(λ)v ∈T_zX.

Up to taking a smaller S¹-equivariant set ˜B this map is well defined and continous on B ט D^∗ and extend smoothly on ˜B ×D. First, observe that ψ_z is a local diffeomorphism when restricted to ( ˜B\(T_zZ∩B)ט D^∗ therefore the image is open. Alsoψz : ˜B ×D →TzX is S¹-equivariant for the action ˜ν times the standard action on D. This is the crucial feature of this map. We can consider theS¹–invariant metric structure induced by ψz on ( ˜B\(TzZ∩B))ט D^∗, this coincides with the previous one on ˜Btimes the standard structure on D^∗ ⊂C (we don’t mean here that the Riemannian exponential is an isometry but we use it as a S¹–equivariant diffeomorphism to identify subsets of the bundles and the fact that exp_z◦d_zq_U should coindice with expg_z by definition of the metric structure on Ue).

Therefore ψ^∗_zH is also the Hamiltonian function for the Killing and Hamiltonian vector field induced by the S¹-action on TzU˜ ×C and

∆^G˜z+ωstd(ψ_z^∗H)(0,0) = (∆^GzH)0

which means, using Formula (8), that we get condition b) above.

It remains to prove that condition c) holds with the hypothesis of the Theorem. One idea would be to argue that even if the normal and tangent bundles of X0 are not well defined as bundles, they exist as sheaves and therefore the Adjunction formula together with a) givesc1(N_X^X₀) = 0 and the weight of theS¹ action onN_X^X₀ is 1. Then the extension of the equivariant cohomology theory to sheaves should ensure that the sequence of S¹- equivariant maps

N_Z^X0 −→^dq N_Z^X −→N_X^X₀ −→0

implies thate(N_Z^X) =dZe(N_X^X₀)e(N_Z^X0) and therefore we get condition c). Another, more pedestrian, approach is to define a S¹–equivariant map, say Ψ, from a tubular neighbour- hood,Ne×Dof the zero section of ˜p:N_Z^X0×C→Z toN, a tubular neighbourhood of the zero section of p:N_Z^X →Z. To be useful here this map should pull back the (compactly supported) equivariant Thom form of N_Z^X todZ times the one ofN_Z^X0×C. Indeed, recall that an equivariant Thom formτ(E) of a rank 2requivariant bundleE →Z is any equiv- ariantly closed form that integrates to (2π)^r over each fiber, see eg. [6, 20]. There are compactly supported version of these forms. An important feature of a Thom form (which all lie in the same cohomology class of course) is that it pulls back to Z, seen as the zero section, to give a representative of the equivariant Euler class of the bundle. Therefore it is sufficiant to construct a map Ψ with domain and target described above such that

1

dZΨ^∗(τ(N_Z^X)) satisfies the properties of a Thom form and Ψ(Z) =Z. We now construct this map. First consider the bundle L = N_(X^X_0)reg −→U\Z over U\Z which we identify with a subbundle ofT(U\Z) using the metric. This bundle is trivial as explained before.

We pick a S¹–invariant nowhere vanishing section σ∈Γ(U\Z, L) such thatσx →0 when x → Z. The set U is a tubular neighbourhood of Z in X and L is a trivial bundle on U\Z so it is clear that such a section exists. We can take σ lying into the subset where the exponential map is injective. Then we define

Ψ(w, λ) := exp⁻¹_p(w)˜ ◦q_U◦exp^_p(w)˜ (w) +dp(w)˜ ν(λ) exp⁻¹_p(w)˜ exp_x(σx)

where x=q_U ◦exp^_p(w)˜ (w)∈X0

This map is continuous, S¹ equivariant, local diffeomorphism away from Z (seen as a zero section ofN_Z^X0) and is surjective. Combined with the fact thatR

U0α = _d¹

Z

R

q⁻¹_U (U0)q^∗_Uα for any formα∈Ω^∗(U0) on any open set U0 ⊂U∩X0 we check easily that _d¹

ZΨ^∗(τ(N_Z^X))

is a Thom form on N_Z^X0 ×C and that Ψ(Z) =Z.

6. Deformation to the normal cone 6.1. Deformation to the normal cone as a test configuration.

6.1.1. Construction and fixed points set. Let Y^m ⊂Xⁿ be a smooth compact connected subvariety of complex codimension k=n−m. We consider the blow-up of X×P¹ along Y × {0} and call it X_Y or X. As explained in [33], see also [13] for the adaptation to compact non-projective test configurations, this gives a test configuration over (X,[ω]) for a suitable K¨ahler class [A] on X. We now recall various aspects of this test configuration, called the deformation to the normal cone.

We denote the blow-down map b : X −→ X ×P¹ and π = pr₂ ◦b : X → P¹ the test configuration surjective map. The C^∗–action on X is the only one that covers the standard one on P¹ via π, we call it φ : C^∗ ֒→ Aut(X). The S¹–action given via φ gives Killing vector field φ∗(∂θ) =V on (X,Ω) and we pick a Hamiltonianµ:X →R.

The central fiber of π is

(21) π⁻¹(0) = b⁻¹(X\Y × {0})⊔ E

where E = b⁻¹(Y × {0}) ⊂ X is the exceptional divisor. Recall that E ≃ P(N_Y^X_×{0}) = P(N_Y^X⊕π^∗T0P¹) whereN_Y^X =T X/T Y (respectivelyN_Y^X =TX/T Y) is the normal bundle of Y in X (respectively in X). Moreover, the closure of b⁻¹(X\Y × {0}) in X will be denoted Z, we have Z ≃ BlY(X) the blow-up of X along Y. We denote the blow-down map ˇb:Z →X and ˇb⁻¹(Y) = E ≃P(N_Y^X) the exceptional divisor. Of course, E ∩Z =E inX.

In the regular test configuration (X,[Ωs], π, φ), theS¹–invariant metric Ωscan be chosen to satisfy the requirements of a test configuration, see [13, 33] as well as

(22) Ωs,E :=ι^∗_EΩs =b^∗ι^∗_Yω−sδE

where δE is the curvature of an S¹–invariant hermitian bundle metric on O_E(−1) → E and s >0 is some positive constant see eg. [41].

6.1.2. Equivariant Euler classes of the normal bundles. The fixed points set of the action of φ:C^∗ ֒→Aut(X) is

X∞⊔Z0⊔Z

where Z₀ =P(0⊕T₀P¹)≃Y and Z ≃Bl_Y(X) are defined above.