Variational approach for complex Monge-Amp`ere equations and geometric applications

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Variational approach for

complex Monge-Amp`ere equations and geometric applications

Jean-Pierre Demailly

Institut Fourier, Universit´e de Grenoble Alpes & Acad´emie des Sciences de Paris

March 19, 2016 S´eminaire Bourbaki Institut Henri Poincar´e, Paris

J.-P. Demailly (Grenoble), S´eminaire Bourbaki, March 19, 2016 Variational approach for complex Monge-Amp`ere equations 1/27

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Abstract and goals

• Recent work by Berman, Berndtsson, Boucksom, Eyssidieux, Guedj, Jonsson, Zeriahi (among others) leads to a new variational approach for the solution of Monge-Amp`ere equations on compact K¨ahler manifolds.

• The method can be made independent of the previous PDE technicalities of Yau’s approach.

• It is based on the study of certain functionals (Ding-Tian, Mabuchi) on the space of K¨ahler metrics, and their geodesic convexity due to X.X. Chen and Berman-Berndtsson in its full generality.

• Applications include the existence and uniqueness of

K¨ahler-Einstein metrics on Q-Fano varieties with log terminal singularities, and a new proof by Berman-Boucksom-Jonsson of a uniform version of the Yau-Tian-Donaldson conjecture solved around 2013 by Chen-Donaldson-Sun.

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Abstract and goals

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Abstract and goals

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Abstract and goals

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K¨ ahler-Einstein metrics

To a K¨ahler metric on a compact complex n fold X ω =i X

1≤j,k≤n

ωjk(z)dzj ∧d zk, dω= 0 one associates its Ricci curvature form

Ricci(ω) = Θ_Λⁿ_T_X_,Λⁿ_ω =−dd^clog det(ω_jk) where d^c = _4iπ¹ (∂−∂),dd^c = _2πⁱ ∂∂.

The K¨ahler metricω is said to be K¨ahler-Einsteinif

(∗) Ricci(ω) = λω for some λ∈R.

This requires λω ∈c₁(X), hence (∗) can be solved only when c₁(X) is positive definite, negative definite or zero, and after rescaling ω by a constant, one can always assume

that λ ∈ {0,1,−1}.

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K¨ ahler-Einstein metrics

1≤j,k≤n

Ricci(ω) = Θ_Λⁿ_T_X_,Λⁿ_ω =−dd^clog det(ω_jk)

where d^c = _4iπ¹ (∂−∂),dd^c = _2πⁱ ∂∂. The K¨ahler metric ω is said to be K¨ahler-Einsteinif

that λ ∈ {0,1,−1}.

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K¨ ahler-Einstein metrics

1≤j,k≤n

Ricci(ω) = Θ_Λⁿ_T_X_,Λⁿ_ω =−dd^clog det(ω_jk)

where d^c = _4iπ¹ (∂−∂),dd^c = _2πⁱ ∂∂. The K¨ahler metric ω is said to be K¨ahler-Einsteinif

that λ ∈ {0,1,−1}.

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K¨ ahler-Einstein ⇐⇒ Monge-Amp`ere equation (1)

Fix a reference K¨ahler metric ω₀ and putω =ω₀+dd^cϕ. The KE condition (∗) is equivalent to

(∗∗) (ω0 +dd^cϕ)ⁿ=e^−λϕ+fωⁿ₀.

• When λ =−1 and c₁(X)<0, i.e. c₁(K_X)>0, Aubin has shown in 1978 that there is always a unique solution, hence a unique K¨ahler metric ω ∈c₁(K_X) such that

Ricci(ω) = −ω.

This is a very natural generalization of the existence of constant curvature metrics on complex algebraic curves, implied by Poincar´e’s uniformization theorem in dimension 1.

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K¨ ahler-Einstein ⇐⇒ Monge-Amp`ere equation (1)

Ricci(ω) =−ω.

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K¨ ahler-Einstein ⇐⇒ Monge-Amp`ere equation (1)

Ricci(ω) =−ω.

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K¨ ahler-Einstein ⇐⇒ Monge-Amp`ere equation (2)

• For λ= 0 and c₁(X) = 0, a celebrated result of Yau (solution of the Calabi conjecture, 1978) states that there exists a unique metric ω =ω₀+dd^cϕin the given cohomology class {ω₀} such that Ricci(ω) = 0.

Moreover, the Monge-Amp`ere equation (ω₀+dd^cϕ)ⁿ=e^fω₀ⁿ

has a unique solution whenever R

X e^fωⁿ₀ =R

X ωⁿ₀.

Equivalently, the Ricci curvature form can be prescribed to be equal any given smooth closed (1,1)-form

Ricci(ω) = ρ, provided that ρ∈c₁(X).

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K¨ ahler-Einstein ⇐⇒ Monge-Amp`ere equation (2)

• For λ= 0 and c₁(X) = 0, a celebrated result of Yau (solution of the Calabi conjecture, 1978) states that there exists a unique metric ω =ω₀+dd^cϕin the given cohomology class {ω₀} such that Ricci(ω) = 0.Moreover, the Monge-Amp`ere equation

(ω₀+dd^cϕ)ⁿ=e^fω₀ⁿ has a unique solution whenever R

X e^fωⁿ₀ =R

X ωⁿ₀.

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K¨ ahler-Einstein ⇐⇒ Monge-Amp`ere equation (2)

• For λ= 0 and c₁(X) = 0, a celebrated result of Yau (solution of the Calabi conjecture, 1978) states that there exists a unique metric ω =ω₀+dd^cϕin the given cohomology class {ω₀} such that Ricci(ω) = 0.Moreover, the Monge-Amp`ere equation

(ω₀+dd^cϕ)ⁿ=e^fω₀ⁿ has a unique solution whenever R

X e^fωⁿ₀ =R

X ωⁿ₀.

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The case of Fano manifolds

For λ= +1, the equation to solve is

(∗∗) (ω₀+dd^cϕ)ⁿ =e^−ϕ+fωⁿ₀.

This is possible only if −K_X ( = ΛⁿT_X) is ample. One then says that X is a Fano manifold.

When solutions exist, it is known by Bando Mabuchi (1987) that they are unique up to the action of the identity component Aut⁰(X) in the complex Lie group of biholomorphisms of X. Berman-Boucksom-Jonsson 2015

Let X be a Fano manifold with finite automorphism group. Then X admits a K¨ahler-Einstein metric if and only if it is uniformly K-stable.

Recently, Chen, Donaldson and Sun got this result under the more general assumption that X is K-stable (Bourbaki/Ph. Eyssidieux, january 2015).

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The case of Fano manifolds

For λ= +1, the equation to solve is

(∗∗) (ω₀+dd^cϕ)ⁿ =e^−ϕ+fωⁿ₀.

This is possible only if −K_X ( = ΛⁿT_X) is ample. One then says that X is a Fano manifold.

When solutions exist, it is known by Bando Mabuchi (1987) that they are unique up to the action of the identity component Aut⁰(X) in the complex Lie group of biholomorphisms of X. Berman-Boucksom-Jonsson 2015

Let X be a Fano manifold with finite automorphism group. Then X admits a K¨ahler-Einstein metric if and only if it is uniformly K-stable.

Recently, Chen, Donaldson and Sun got this result under the more general assumption that X isK-stable (Bourbaki/Ph. Eyssidieux, january 2015).

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The case of log Fano varieties

Definition

A log Fano pairis a klt pair (X,∆) such that X is projective and the Q-divisor A=−(K_X + ∆) is ample.

Here X is a normal compact complex variety X and ∆ an effective Q-divisor such that K_X + ∆ is Q-Cartier. By Hironaka, there exists a log resolution π: ˜X →X of (X,∆), i.e. a modification of X over the complement of the singular loci of X and ∆, such that the pull-back of ∆ and of X_sing consists of simple normal crossing (snc) divisors in ˜X. One writes

π^∗(K_X + ∆) =KX˜ +E, E =P

ja_jE_j

for some Q-divisor E whose push-forward to X is ∆ (sinceX_sing has codimension 2, the components E_j that lie over X_sing yield π∗Ej = 0). The coefficient −aj ∈Q is known as the discrepancy of (X,∆) along E_j.

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The case of log Fano varieties

Definition

Here X is a normal compact complex variety X and ∆ an effective Q-divisor such that K_X + ∆ is Q-Cartier.

By Hironaka, there exists a log resolution π: ˜X →X of (X,∆), i.e. a modification of X over the complement of the singular loci of X and ∆, such that the pull-back of ∆ and of X_sing consists of simple normal crossing (snc) divisors in ˜X. One writes

π^∗(K_X + ∆) =KX˜ +E, E =P

ja_jE_j

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The case of log Fano varieties

Definition

Here X is a normal compact complex variety X and ∆ an effective Q-divisor such that K_X + ∆ is Q-Cartier. By Hironaka, there exists a log resolution π: ˜X →X of (X,∆), i.e. a modification of X over the complement of the singular loci of X and ∆, such that the pull-back of ∆ and of X_sing consists of simple normal crossing (snc) divisors in ˜X.

One writes

π^∗(K_X + ∆) =KX˜ +E, E =P

ja_jE_j

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The case of log Fano varieties

Definition

Here X is a normal compact complex variety X and ∆ an effective Q-divisor such that K_X + ∆ is Q-Cartier. By Hironaka, there exists a log resolution π: ˜X →X of (X,∆), i.e. a modification of X over the complement of the singular loci of X and ∆, such that the pull-back of ∆ and of X_sing consists of simple normal crossing (snc) divisors in ˜X. One writes

π^∗(K_X + ∆) =KX˜ +E, E =P

ja_jE_j

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The klt condition (“Kawamata Log Terminal”)

Definition

(X,∆) is klt if a_j <1 for allj.

Let r be a positive integer such that r(K_X + ∆) is Cartier, and σ a local generator of O(r(K_X + ∆)) on some open set U ⊂X. Then the (n,n) form

|σ|^2/r :=iⁿ²σ^1/r ∧σ^1/r

is a volume form with poles along S =Supp∆∪X_sing.

By the change of variable formula, the local integrability can be checked by pulling back σ to ˜X, in which case it is easily seen that the integrability occurs if and only if a_j <1 for all j, i.e. when (X,∆) is klt.

In local coordinates

|σ|^2/r ∼ volume form Q|zj|^2a^j .

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The klt condition (“Kawamata Log Terminal”)

Definition

|σ|^2/r :=iⁿ²σ^1/r ∧σ^1/r

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The klt condition (“Kawamata Log Terminal”)

Definition

|σ|^2/r :=iⁿ²σ^1/r ∧σ^1/r

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Singular Monge-Amp`ere equation

By definition (X,∆) log Fano =⇒ c₁(X,∆) 3ω₀ K¨ahler.

Every form ω=ω₀+dd^cψ ∈ {ω₀} can be seen as the curvature form of a smooth hermitian metric h on O(−(K_X + ∆)), whose weight is φ =u₀ +ψ whereu₀ is a local potential of ω₀, hence

ω=ω₀+dd^cψ =dd^cφ

where φ is understood as the weight of a global metric formally denoted h=e^−φ on the Q-line bundleO(−(K_X + ∆)).

The inverse e^φ is a hermitian metric onO(K_X + ∆). If σ is a local generator of O(r(K_X + ∆)), the product|σ|^2/re^φ=e^ψ+u⁰ is (locally) a smooth positive function whenever ϕis smooth, hence

e^−φ=|σ|^2/re^−(ψ+u⁰⁾

is an integrable volume form on X with poles along

S :=Supp∆∪ {singularities}. The KE condition can be rewritten (dd^cφ)ⁿ =c e^−φ on X rS ⇔Ricci(ω) =ω+ [∆].

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Singular Monge-Amp`ere equation

Every form ω=ω₀+dd^cψ ∈ {ω₀} can be seen as the curvature form of a smooth hermitian metric h on O(−(KX + ∆)), whose weight is φ =u₀ +ψ whereu₀ is a local potential of ω₀, hence

The inverse e^φ is a hermitian metric onO(K_X + ∆). If σ is a local generator of O(r(K_X + ∆)), the product|σ|^2/re^φ=e^ψ+u⁰ is (locally) a smooth positive function whenever ϕis smooth, hence

e^−φ=|σ|^2/re^−(ψ+u⁰⁾

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Singular Monge-Amp`ere equation

Every form ω=ω₀+dd^cψ ∈ {ω₀} can be seen as the curvature form of a smooth hermitian metric h on O(−(KX + ∆)), whose weight is φ =u₀ +ψ whereu₀ is a local potential of ω₀, hence

The inverse e^φ is a hermitian metric on O(K_X + ∆). If σ is a local generator of O(r(K_X + ∆)), the product|σ|^2/re^φ=e^ψ+u⁰ is (locally) a smooth positive function whenever ϕis smooth, hence

e^−φ=|σ|^2/re^−(ψ+u⁰⁾

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The space of K¨ ahler metrics

Let A∈H^1,1

∂∂(X,R) be a K¨ahler ∂∂-cohomology class, and let ω₀ =α+dd^cψ₀ =dd^cφ₀ ∈A

be a K¨ahler metric.

Here we are mostly interested in the Fano case A=−K_X and the log Fano case A=−(K_X + ∆). Let V =R

X ω₀ⁿ =Aⁿ be the volume of ω₀.

Definition

The space K_A of Kähler metrics (resp. P_A of Kähler potentials) is the set of Kähler metrics ω (resp. functions ψ) such that

ω =ω₀+dd^cψ >0.

Here φ=u₀+ψ is thought intrinsically as a hermitian metric h =e^−φ on A with strictly plurisubharmonic (psh) weightφ. Clearly K_A ' P_A/R.

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The space of K¨ ahler metrics

Let A∈H^1,1

Definition

Here φ=u₀+ψ is thought intrinsically as a hermitian metric h =e^−φ on A with strictly plurisubharmonic (psh) weightφ.

Clearly K_A ' P_A/R.

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The space of K¨ ahler metrics

Let A∈H^1,1

Definition

Here φ=u₀+ψ is thought intrinsically as a hermitian metric h =e^−φ on A with strictly plurisubharmonic (psh) weightφ.

Clearly K_A ' P_A/R.

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The Riemannian structure on P

_A

The basic operator of interest on P_A is the Monge-Amp`ere operator PA → M+, MA(φ) = (dd^cφ)ⁿ= (ω0+dd^cψ)ⁿ

According to Mabuchi the space PA can be seen as some sort of infinite dimensional Riemannian manifold: a “tangent vector” to P_A is an infinitesimal variationδφ∈C^∞(X,R) ofφ (orψ), and the infinitesimal Riemannian metric at a point h =e^−φ is given by

kδφk²₂ = 1 V

Z

X

(δφ)²MA(φ).

X.X. Chen and his collaborators have studied the metric and geometric properties of the space P_A, showing in particular that it is a path metric space (a non trivial assertion in this infinite dimensional setting) of nonpositive curvature in the sense of Alexandrov. A key step has been to produce almost C^1,1-geodesics which minimize the geodesic distance.

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The Riemannian structure on P

_A

The basic operator of interest on P_A is the Monge-Amp`ere operator PA → M+, MA(φ) = (dd^cφ)ⁿ= (ω0+dd^cψ)ⁿ

According to Mabuchi the space PA can be seen as some sort of infinite dimensional Riemannian manifold: a “tangent vector” to P_A is an infinitesimal variationδφ∈C^∞(X,R) ofφ (orψ), and the infinitesimal Riemannian metric at a point h =e^−φ is given by

kδφk²₂ = 1 V

Z

X

(δφ)²MA(φ).

X.X. Chen and his collaborators have studied the metric and geometric properties of the space P_A, showing in particular that it is a path metric space (a non trivial assertion in this infinite dimensional setting) of nonpositive curvature in the sense of Alexandrov. A key step has been to produce almost C^1,1-geodesics which minimize the geodesic distance.

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Basic functionals (1)

Given φ₀, φ ∈ P_A, one defines:

• The Monge-Amp`ere functional Eφ0(φ) = 1

(n+ 1)V

n

X

j=0

Z

X

(φ−φ0)(dd^cφ)^j ∧(dd^cφ0)^n−j

= 1

(n+ 1)V

n

X

j=0

Z

X

ψ(ω₀+dd^cψ)^j ∧ω^n−j₀ , ψ =φ−φ₀. (∗∗∗)

It is a primitive of the Monge-Amp`ere operator in the sense that dE_φ₀(φ) = _V¹ MA(φ), i.e. for any path [T,T⁰]3t 7→φ_t, one has

d

dtE_φ₀(φ_t) = 1 V

Z

X

φ˙_tMA(φ_t) where ˙φ_t = d dtφ_t.

This is easily checked by a differentiation under the integral sign.

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Basic functionals (1)

Given φ₀, φ ∈ P_A, one defines:

• The Monge-Amp`ere functional Eφ0(φ) = 1

(n+ 1)V

n

X

j=0

Z

X

(φ−φ0)(dd^cφ)^j ∧(dd^cφ0)^n−j

= 1

(n+ 1)V

n

X

j=0

Z

X

ψ(ω₀+dd^cψ)^j ∧ω^n−j₀ , ψ =φ−φ₀. (∗∗∗)

It is a primitive of the Monge-Amp`ere operator in the sense that dE_φ₀(φ) = _V¹ MA(φ), i.e. for any path [T,T⁰]3t 7→φ_t, one has

d

dtE_φ₀(φ_t) = 1 V

Z

X

φ˙_tMA(φ_t) where ˙φ_t = d dtφ_t.

This is easily checked by a differentiation under the integral sign.

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Basic functionals (2)

As a consequence E satisfies the cocycle relation Eφ0(φ1) +Eφ1(φ2) =Eφ0(φ2), so its dependence on φ₀ is only up to a constant.

Finally, if φ_t depends linearly on t, one has ¨φ_t = _dt^d²2φ_t = 0 and a further differentiation of (∗∗∗) yields

d²

dt²E_φ₀(φ_t) = n V

Z

X

φ˙_tdd^cφ˙_t∧(dd^cφ_t)ⁿ⁻¹

=−n V

Z

X

dφ˙_t∧d^cφ˙_t∧(dd^cφ_t)ⁿ⁻¹ ≤0. It follows that E_φ₀ isconcave onP_A.

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Basic functionals (2)

d²

Z

X

=−n V

Z

X

dφ˙_t∧d^cφ˙_t∧(dd^cφ_t)ⁿ⁻¹ ≤0.

It follows that Eφ0 isconcave onPA.

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Basic functionals (2)

d²

Z

X

=−n V

Z

X

dφ˙_t∧d^cφ˙_t∧(dd^cφ_t)ⁿ⁻¹ ≤0.

It follows that Eφ0 isconcave onPA.

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The J and J

^∗

functionals

• The concavity of E implies the nonnegativity of

J_φ₀(φ) :=dE_φ₀(φ₀)·(φ−φ₀)−E_φ₀(φ), This quantity is called the Aubin J-energy functional

J_φ₀(φ) = V⁻¹ Z

X

(φ−φ₀)(dd^cφ₀)ⁿ−E_φ₀(φ)≥0.

• By exchanging the roles of φ, φ0 and putting J_φ^∗

0(φ) =J_φ(φ₀)≥0, the cocycle relation for E yields E_φ(−φ₀) = −E_φ₀(φ). The transposed J-energy functional is

J_φ^∗₀(φ) : =E_φ₀(φ)−V⁻¹ Z

X

(φ−φ₀)(dd^cφ)ⁿ

=Eφ0(φ)−V⁻¹ Z

X

ψ(ω0+dd^cψ)ⁿ ≥0, ψ=φ−φ0.

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The J and J

^∗

functionals

• The concavity of E implies the nonnegativity of

J_φ₀(φ) :=dE_φ₀(φ₀)·(φ−φ₀)−E_φ₀(φ), This quantity is called the Aubin J-energy functional

J_φ₀(φ) = V⁻¹ Z

X

(φ−φ₀)(dd^cφ₀)ⁿ−E_φ₀(φ)≥0.

• By exchanging the roles of φ, φ0 and putting J_φ^∗

0(φ) =J_φ(φ₀)≥0, the cocycle relation for E yields E_φ(−φ₀) = −E_φ₀(φ). The transposed J-energy functional is

J_φ^∗₀(φ) : =E_φ₀(φ)−V⁻¹ Z

X

(φ−φ₀)(dd^cφ)ⁿ

=Eφ0(φ)−V⁻¹ Z

X

ψ(ω0+dd^cψ)ⁿ ≥0, ψ=φ−φ0.

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The symmetric I functional

• The I-functional is the symmetric functional defined by I_φ₀(φ) =I_φ(φ₀) := −1

V Z

X

(φ−φ₀) MA(φ)−MA(φ₀)

=

n−1

X

j=0

V⁻¹ Z

X

d(φ−φ₀)∧d^c(φ−φ₀)∧(dd^cφ)^j∧(dd^cφ₀)^n−1−j ≥0.

In fact I_φ₀(φ) =J_φ₀(φ) +J_φ^∗

0(φ), and one can also write I_φ₀(φ) =V⁻¹

Z

X

ψ ωⁿ₀ − Z

X

ψ(ω₀+dd^cψ)ⁿ

.

It satisfies the quasi-triangle inequality: ∃c_n>0 s.t. I_φ₀(φ)≤c_n I_φ₀(φ₁) +I_φ₁(φ)

. ∀ φ₀, φ₁, φ ∈ P_A.

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The symmetric I functional

V Z

X

=

n−1

X

j=0

V⁻¹ Z

X

Z

X

ψ ω₀ⁿ− Z

X

ψ(ω₀+dd^cψ)ⁿ

.

It satisfies the quasi-triangle inequality: ∃c_n>0 s.t. I_φ₀(φ)≤c_n I_φ₀(φ₁) +I_φ₁(φ)

. ∀ φ₀, φ₁, φ ∈ P_A.

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The symmetric I functional

V Z

X

=

n−1

X

j=0

V⁻¹ Z

X

Z

X

ψ ω₀ⁿ− Z

X

ψ(ω₀+dd^cψ)ⁿ

.

It satisfies the quasi-triangle inequality: ∃c_n>0 s.t.

I_φ₀(φ)≤c_n I_φ₀(φ₁) +I_φ₁(φ)

. ∀ φ₀, φ₁, φ ∈ P_A.

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The Ding and Mabuchi functionals (1)

• In the Fano or log Fano setting, the Ding functional is defined by D_φ₀ =L−L(φ₀)−E_φ₀, where L(φ) = −log

Z

X

e^−φ.

Recall: e^−φ is integrable by the klt condition.

• Given probability measures µ, ν on a spaceX, the relative entropy Entr_µ(ν)∈[0,+∞] of ν with respect to µ is defined as the integral

Entr_µ(ν) := Z

X

log dν

dµ

dν,

if ν is absolutely continuous w.r.t. µ; Entr_µ(ν) = +∞otherwise. Pinsker inequality: for all proba measuresµ, ν one has

Entr_µ(ν)≥ 1

2kµ−νk² ≥0. In particular, µ=ν ⇐⇒Entr_µ(ν) = 0.

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The Ding and Mabuchi functionals (1)

• In the Fano or log Fano setting, the Ding functional is defined by D_φ₀ =L−L(φ₀)−E_φ₀, where L(φ) = −log

Z

X

e^−φ. Recall: e^−φ is integrable by the klt condition.

• Given probability measures µ, ν on a spaceX, the relative entropy Entr_µ(ν)∈[0,+∞] of ν with respect to µ is defined as the integral

Entr_µ(ν) :=

Z

X

log dν

dµ

dν,

if ν is absolutely continuous w.r.t. µ; Entr_µ(ν) = +∞otherwise.

Pinsker inequality: for all proba measuresµ, ν one has Entr_µ(ν)≥ 1

2kµ−νk² ≥0.

In particular, µ=ν ⇐⇒Entr_µ(ν) = 0.

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The Ding and Mabuchi functionals (2)

In the Fano or log Fano situation, the entropy functional H_φ₀(φ) is defined to be the entropy of the probability measure _V¹(dd^cφ)ⁿ with respect to e^L(φ⁰⁾e^−φ⁰, namely

H_φ₀(φ) = Z

X

log

(dd^cφ)ⁿ/V e^L(φ⁰⁾e^−φ⁰

(dd^cφ)ⁿ V ≥0.

• The Mabuchi functional is then defined by M_φ₀ =H_φ₀−J_φ^∗₀. One gets the more explicit expression

M_φ₀(φ) = Z

X

log

e^φ(dd^cφ)ⁿ V

(dd^cφ)ⁿ

V −E_φ₀(φ)−L(φ₀).

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The Ding and Mabuchi functionals (2)

H_φ₀(φ) = Z

X

log

(dd^cφ)ⁿ V ≥0.

• The Mabuchi functional is then defined by M_φ₀ =H_φ₀−J_φ^∗₀.

One gets the more explicit expression M_φ₀(φ) =

Z

X

log

e^φ(dd^cφ)ⁿ V

(dd^cφ)ⁿ

V −E_φ₀(φ)−L(φ₀).

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The Ding and Mabuchi functionals (2)

H_φ₀(φ) = Z

X

log

(dd^cφ)ⁿ V ≥0.

• The Mabuchi functional is then defined by M_φ₀ =H_φ₀−J_φ^∗₀. One gets the more explicit expression

M_φ₀(φ) = Z

X

log

e^φ(dd^cφ)ⁿ V

(dd^cφ)ⁿ

V −E_φ₀(φ)−L(φ₀).

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Comparison properties

Observation

If c is a constant, then

E_φ₀(φ+c) =E_φ₀(φ) +c and L(φ+c) =L(φ) +c. On the other hand, the functionals I_φ₀,J_φ₀,J_φ^∗

0,D_φ₀,H_φ₀,M_φ₀ are invariant by φ7→φ+c and therefore descend to the quotient space K_A =P_A/R of K¨ahler metricsω =dd^cφ∈A.

Comparison between I, J, J^∗

The functionals I,J, J^∗ are essentially growth equivalent: 1

nJ_φ(φ₀)≤J_φ₀(φ)≤ n+ 1

n J_φ₀(φ)≤I_φ₀(φ)≤(n+ 1)J_φ₀(φ).

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Comparison properties

Observation

If c is a constant, then

E_φ₀(φ+c) =E_φ₀(φ) +c and L(φ+c) =L(φ) +c. On the other hand, the functionals I_φ₀,J_φ₀,J_φ^∗

0,D_φ₀,H_φ₀,M_φ₀ are invariant by φ7→φ+c and therefore descend to the quotient space K_A =P_A/R of K¨ahler metricsω =dd^cφ∈A.

Comparison between I, J, J^∗

The functionals I,J, J^∗ are essentially growth equivalent:

1

nJ_φ(φ₀)≤J_φ₀(φ)≤ n+ 1

n J_φ₀(φ)≤I_φ₀(φ)≤(n+ 1)J_φ₀(φ).

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Comparison between Ding and Mabuchi functionals

Proposition

Let (X,∆) be a log Fano manifold. ThenMφ0(φ)≥Dφ0(φ) and, in case of equality, φ must be K¨ahler-Einstein.

Proof. From the definitions one gets

M−D = (H−J^∗)(L−L(φ₀)−E), E_φ₀(φ)−J_φ^∗₀(φ) = V⁻¹

Z

X

(φ−φ₀)(dd^cφ)ⁿ, M_φ₀(φ)−D_φ₀(φ)

= Z

X

log

+ (φ−φ0)

(dd^cφ)ⁿ

V +L(φ0)−L(φ)

= Z

X

log

(dd^cφ)ⁿ/V e^L(φ)e^−φ

(dd^cφ)ⁿ V ≥0.

In case of equality, Pinsker implies KE condition: ^(dd_V^c^φ)ⁿ =e^L(φ)e^−φ

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Non pluripolar products

• Bedford-Taylor Monge-Amp`ere products : for u_j ∈L^∞_loc, one sets inductively

dd^cu₁∧dd^cu₂∧. . .∧dd^cu_k :=dd^c(u₁dd^cu₂∧. . .∧dd^cu_k)

• Non pluripolar products (Guedj-Zeriahi)

Let P(X, ω₀) be the set ofω₀-psh potentials, i.e. φ=φ₀ +ψ such that dd^cφ =ω₀ +dd^cψ ≥0.

The functions ψ_ν := max{ψ,−ν} are again ω₀-psh and bounded for all ν ∈N. The Monge-Amp`ere measures (ω₀+dd^cψ_ν)ⁿ are therefore well-defined in the sense of Bedford-Taylor, and one defines for any bidegree (p,p) a positive current

T =h(ω₁+dd^cψ₁)∧...∧(ω_p+dd^cψ_p)i= lim

ν→+∞

1^T{ψ_j>−ν}(ω1+dd^cmax{ψ1,−ν})∧...∧(ωp+dd^cmax{ψp,−ν}) Basic fact: T is still closed [Proof uses ideas of Skoda & Sibony].