A sharp lower bound for the log canonical threshold

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A sharp lower bound for the log canonical threshold

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p

Institut Fourier, Université de Grenoble I, France NORDAN 12, Kiruna (Sweden) dediated to the memory of Mikael Passare in honor of Urban Cegrell, on the occasion of his retirement

May 12, 2012

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log canonical threshold of psh functions

Singularities of psh (plurisubharmonic) functions can be measured by Lelong numbers. Another useful invariant is thelog canonical threshold.

Definition

LetX be a complex manifold,p∈X, andϕ be a plurisubharmonic function defined onX.

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log canonical threshold of psh functions

Definition

LetX be a complex manifold,p∈X, andϕ be a plurisubharmonic function defined onX. Thelog

canonical threshold orcomplex singularity exponentofϕ atp is defined by

cp(ϕ) =sup

c ≥0 : e^−2cϕ isL¹ on a neighborhood ofp ,

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log canonical threshold of psh functions

Definition

LetX be a complex manifold,p∈X, andϕ be a plurisubharmonic function defined onX. Thelog

canonical threshold orcomplex singularity exponentofϕ atp is defined by

cp(ϕ) =sup

c ≥0 : e^−2cϕ isL¹ on a neighborhood ofp , Here we will takep =0 and denotec(ϕ) =c0(ϕ).

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log canonical threshold of coherent ideals

The log canonical threshold is a subtle invariant. A special interesting case is

ϕ(z) = 1

2log(|g₁|²+. . .+|gN|²)

associated to some ideal J = (g1, . . . ,gN)of polynomials or holomorphic functions on some complex manifold X.

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log canonical threshold of coherent ideals

ϕ(z) = 1

2log(|g₁|²+. . .+|gN|²)

associated to some ideal J = (g1, . . . ,gN)of polynomials or holomorphic functions on some complex manifold X. Then by Hironaka, ∃modificationµ:Xe →X such that

µ^∗J = (g1◦µ, . . . ,gN ◦µ) = O(−X ajEj) for some normal crossing divisor.

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log canonical threshold of coherent ideals

ϕ(z) = 1

2log(|g₁|²+. . .+|gN|²)

µ^∗J = (g1◦µ, . . . ,gN ◦µ) = O(−X ajEj) for some normal crossing divisor.LetO(P

bjEj)be the divisor of Jac(µ).

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log canonical threshold of coherent ideals

ϕ(z) = 1

2log(|g₁|²+. . .+|gN|²)

µ^∗J = (g1◦µ, . . . ,gN ◦µ) = O(−X ajEj) for some normal crossing divisor.LetO(P

bjEj)be the divisor of Jac(µ). We have

c(ϕ) = min

E_j, µ(E_j)∋0

1+bj

aj

∈Q^∗₊.

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Proof of the formula for the log canonical threshold

In fact, we have to find the supremum of c >0 such that

I = Z

V∋0

dλ(z)

|g1|²+. . .+|gN|²c <+∞.

Let us perform the change of variablez =µ(w). Then

dλ(z) =|Jac(µ)(w)|²∼ Y

w_j^b^j

2

dλ(w) with respect to coordinates on the blow-up Ve ofV, and

I ∼ Z

Ve

Qw_j^b^j

²dλ(w) Qw_j^a^j

^2c

so convergence occurs ifcaj−bj <1 for allj.

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Notations and basic facts

A domainΩ⊂Cⁿ is calledhyperconvexif∃ψ ∈ PSH(Ω), ψ ≤0, such that{z :ψ(z)<c}⋐Ωfor all c <0.

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Notations and basic facts

E0(Ω)=

ϕ∈PSH ∩L^∞(Ω) : lim

z→∂Ωϕ(z)=0,Z

Ω

(dd^cϕ)ⁿ<+∞

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Notations and basic facts

E0(Ω)=

z→∂Ωϕ(z)=0,Z

Ω

(dd^cϕ)ⁿ<+∞

F(Ω) =n

ϕ ∈ PSH(Ω) : ∃ E0(Ω)∋ϕp ցϕ, and sup

p≥1

Z

Ω

(dd^cϕp)ⁿ<+∞o , Ee(X) ={ϕ ∈ PSH(X)locally inF(Ω) modC^∞(Ω)}

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Notations and basic facts

E0(Ω)=

z→∂Ωϕ(z)=0,Z

Ω

(dd^cϕ)ⁿ<+∞

F(Ω) =n

ϕ ∈ PSH(Ω) : ∃ E0(Ω)∋ϕp ցϕ, and sup

p≥1

Z

Ω

(dd^cϕp)ⁿ<+∞o ,

Ee(X) ={ϕ ∈ PSH(X)locally inF(Ω) modC^∞(Ω)}

Theorem (U. Cegrell)

Ee(X)is the largest subclass of psh functionsdefined on a complex manifold X for which the complex

Monge-Ampère operator is locally well-defined.

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Intermediate Lelong numbers

Set hered^c = _2πⁱ (∂−∂)so thatdd^c = _πⁱ∂∂.

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Intermediate Lelong numbers

Set hered^c = _2πⁱ (∂−∂)so thatdd^c = _πⁱ∂∂. Ifϕ ∈Ee(Ω) and 0∈Ω, the products(dd^cϕ)^j are well defined and one can consider the Lelong numbers

ej(ϕ) =ν (dd^cϕ)^j,0 . In other words

ej(ϕ) = Z

{0}

(dd^cϕ)^j∧(dd^clogkzk)^n−j. One hase₀(ϕ) =1 ande₁(ϕ) =ν(ϕ,0)(usual Lelong number).

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Intermediate Lelong numbers

Set hered^c = _2πⁱ (∂−∂)so thatdd^c = _πⁱ∂∂. Ifϕ ∈Ee(Ω) and 0∈Ω, the products(dd^cϕ)^j are well defined and one can consider the Lelong numbers

ej(ϕ) =ν (dd^cϕ)^j,0 . In other words

ej(ϕ) = Z

{0}

(dd^cϕ)^j∧(dd^clogkzk)^n−j. One hase₀(ϕ) =1 ande₁(ϕ) =ν(ϕ,0)(usual Lelong number). When

ϕ(z) = 1

2log(|g₁|²+. . .+|gN|²), one hasej(ϕ)∈N.

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The main result

Main Theorem (Demailly & Pha.m)

Letϕ ∈Ee(Ω). Ife1(ϕ) =0, thenc(ϕ) =∞.

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The main result

Otherwise, we have

c(ϕ)≥ Xn−1

j=0

ej(ϕ) e_j+1(ϕ).

The lower bound improves a classical result of H. Skoda (1972), according to which

1

e1(ϕ) ≤c(ϕ)≤ n e1(ϕ).

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The main result

Otherwise, we have

c(ϕ)≥ Xn−1

j=0

ej(ϕ) e_j+1(ϕ).

The lower bound improves a classical result of H. Skoda (1972), according to which

1

e1(ϕ) ≤c(ϕ)≤ n e1(ϕ).

Remark: The above theorem is optimal, with equality for ϕ(z) = log(|z₁|^a¹+. . .+|zn|^aⁿ), 0<a₁≤a₂ ≤. . .≤an. Thenej(ϕ) = a1. . .aj, c(ϕ) = 1

a₁ +. . .+ 1 an

.

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Geometric applications

The log canonical threshold has a lot of applications. It is essentially a local version of Tian’s invariant, which determines a sufficient condition for theexistence of Kähler-Einstein metrics.

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Geometric applications

Another important application is tobirational rigidity.

Theorem (Pukhlikov 1998, Corti 2000, de Fernex 2011) LetX be a smooth hypersurface of degreed inCPⁿ⁺¹. Then ifd =n+1,Bir(X)≃Aut(X)

It was first shown by Manin-Iskovskih in the early 70’s that a 3-dim quartic in CP⁴(n=3,d =4) is not rational.

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Geometric applications

Another important application is tobirational rigidity.

Theorem (Pukhlikov 1998, Corti 2000, de Fernex 2011) LetX be a smooth hypersurface of degreed inCPⁿ⁺¹. Then ifd =n+1,Bir(X)≃Aut(X)

It was first shown by Manin-Iskovskih in the early 70’s that a 3-dim quartic in CP⁴(n=3,d =4) is not rational.

Question

For 3 ≤d ≤n+1, when is it true thatBir(X)≃Aut(X) (birational rigidity) ?

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Lemma 1

Letϕ ∈Ee(Ω)and 0∈Ω. Then we have that ej(ϕ)²≤e_j−1(ϕ)ej+1(ϕ),

for all j =1, . . . ,n−1.

In other words j 7→logej(ϕ)is convex, thus we have ej(ϕ)≥e1(ϕ)^j and the ratiosej+1(ϕ)/ej(ϕ)are increasing.

Corollary

Ife1(ϕ) = ν(ϕ,0) =0, thenej(ϕ) =0 for j =1,2, . . . ,n−1.

A hard conjecture by V. Guedj and A. Rashkovskii (∼1998) states thatϕ ∈Ee(Ω),e₁(ϕ) =0 also implies en(ϕ) =0.

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Proof of Lemma 1

Without loss generality we can assume that Ωis the unit ball andϕ ∈ E0(Ω).

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Proof of Lemma 1

Without loss generality we can assume that Ωis the unit ball andϕ ∈ E0(Ω). For h, ψ ∈ E0(Ω) an integration by parts and the Cauchy-Schwarz inequality yield

Z

Ω

−h(dd^cϕ)^j ∧(dd^cψ)^n−j 2

=

" Z

Ω

dϕ∧d^cψ∧(dd^cϕ)^j−1∧(dd^cψ)^n−j−1∧dd^ch

#2

≤ Z

Ω

dψ ∧d^cψ∧(dd^cϕ)^j−1∧(dd^cψ)^n−j−1∧dd^ch Z

Ω

dϕ∧d^cϕ∧(dd^cϕ)^j−1∧(dd^cψ)^n−j−1∧dd^ch

= Z

Ω

−h(dd^cϕ)^j−1∧(dd^cψ)^n−j+1 Z

Ω

−h(dd^cϕ)^j+1∧(dd^cψ)^n−j−1,

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Proof of Lemma 1, continued

Now, as p→+∞, take

h(z) =hp(z) = max

−1,1

plogkzk ր

0 ifz ∈Ω\{0}

−1 ifz =0.

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Proof of Lemma 1, continued

Now, as p→+∞, take

h(z) =hp(z) = max

−1,1

plogkzk ր

0 ifz ∈Ω\{0}

−1 ifz =0.

By the monotone convergence theorem we get in the limit that

Z

{0}(dd^cϕ)^j ∧(dd^cψ)^n−j 2

≤ Z

{0}(dd^cϕ)^j−1∧(dd^cψ)^n−j+1 Z

{0}(dd^cϕ)^j+1∧(dd^cψ)^n−j−1. For ψ(z) =lnkzk, this is the desired estimate.

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Lemma 2

Letϕ, ψ ∈Ee(Ω)be such thatϕ≤ψ (i.eϕ is ”more singular” thanψ).

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Lemma 2

Letϕ, ψ ∈Ee(Ω)be such thatϕ≤ψ (i.eϕ is ”more singular” thanψ). Then we have

Xn−1 j=0

e_j(ϕ) ej+1(ϕ) ≤

Xn−1 j=0

e_j(ψ) ej+1(ψ).

The argument if based on the monotonicity of Lelong numbers with respect to the relationϕ ≤ψ, and on the monotonicity of the right hand side in the relevant range of values.

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Proof of Lemma 2

Set

D ={t=(t1, ...,tn)∈[0,+∞)ⁿ :t₁²≤t2,t_j²≤tj−1tj+1,∀j =2, ...,n−1}.

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Proof of Lemma 2

Set

ThenD is a convex set inRⁿ, as can be checked by a straightforward application of the Cauchy-Schwarz inequality.

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Proof of Lemma 2

Set

ThenD is a convex set inRⁿ, as can be checked by a straightforward application of the Cauchy-Schwarz

inequality. Next, consider the functionf :intD→[0,+∞) defined by

f(t1, . . . ,tn) = 1 t1 +t₁

t2 . . .+t_n−1 tn

.

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Proof of Lemma 2

Set

ThenD is a convex set inRⁿ, as can be checked by a straightforward application of the Cauchy-Schwarz

inequality. Next, consider the functionf :intD→[0,+∞) defined by

f(t1, . . . ,tn) = 1 t1 +t₁

t2 . . .+t_n−1 tn

.

We have

∂f

∂tj

(t) = −t_j−1 t_j² + 1

t_j+1 ≤0, ∀t ∈D.

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Proof of Lemma 2, continued

For a,b∈intD such thataj ≥bj,j =1, . . . ,n, the function [0,1]∋λ →f(b+λ(a−b))

is decreasing.

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Proof of Lemma 2, continued

is decreasing. Hence,

f(a)≤f(b) for alla,b∈intD, aj ≥bj, j =1, . . . ,n.

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Proof of Lemma 2, continued

is decreasing. Hence,

f(a)≤f(b) for alla,b∈intD, aj ≥bj, j =1, . . . ,n. On the other hand, the hypothesis ϕ ≤ψ implies that ej(ϕ)≥ej(ψ),j =1, . . . ,n, by the comparison principle.

Therefore we have that

f(e1(ϕ), . . . ,en(ϕ))≤f(e1(ψ), . . . ,en(ψ)).

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Proof of the Main Theorem

It will be convenient here to introduce Kiselman’s refined Lelong number.

Definition

Letϕ ∈ PSH(Ω). Then the function defined by

νϕ(x) = lim

t→−∞

max

ϕ(z) :|z1|=e^x¹^t, . . . ,|zn|=e^xⁿ^t t

is called therefined Lelong numberofϕ at 0.

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Proof of the Main Theorem

It will be convenient here to introduce Kiselman’s refined Lelong number.

Definition

Letϕ ∈ PSH(Ω). Then the function defined by

νϕ(x) = lim

t→−∞

max

ϕ(z) :|z1|=e^x¹^t, . . . ,|zn|=e^xⁿ^t t

is called therefined Lelong numberofϕ at 0.

The refined Lelong number of ϕat 0 is increasing in each variable xj, and concave onRⁿ₊.

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Proof of the Main Theorem

The proof is divided into the following steps:

Proof of the theorem in the toric case, i.e.

ϕ(z1, . . . ,zn) = ϕ(|z1|, . . . ,|zn|)depends only on|zj| and therefore we can without loss of generality assume thatΩ = ∆ⁿ is the unit polydisk.

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Proof of the Main Theorem

Reduction to the case of plurisubharmonic functions with analytic singularity, i.e. ϕ =log(|f1|²+. . .+|fN|²), wheref₁, . . . ,f_N are germs of holomorphic functions at 0.

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Proof of the Main Theorem

Reduction to the case of plurisubharmonic functions with analytic singularity, i.e. ϕ =log(|f1|²+. . .+|fN|²), wheref₁, . . . ,f_N are germs of holomorphic functions at 0.

Reduction to the case of monomial ideals, i.e. for ϕ=log(|f1|²+. . . .+|fN|²), wheref1, . . . ,fN are germs of monomial elements at 0.

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Proof of the theorem in the toric case

Set

Σ =



x = (x1, . . . ,x_n)∈Rⁿ₊ : Xn

j=1

x_j =1



 .

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Proof of the theorem in the toric case

Set

Σ =



x = (x1, . . . ,x_n)∈Rⁿ₊ : Xn

j=1

x_j =1



 . We choose x⁰ = (x₁⁰, . . . ,x_n⁰)∈Σsuch that

νϕ(x⁰) =max{ν_ϕ(x) : x ∈S}.

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Proof of the theorem in the toric case

Set

Σ =



x = (x1, . . . ,x_n)∈Rⁿ₊ : Xn

j=1

x_j =1



 .

We choose x⁰ = (x₁⁰, . . . ,x_n⁰)∈Σsuch that νϕ(x⁰) =max{ν_ϕ(x) : x ∈S}.

By Theorem 5.8 in [Kis94] we have the following formula

c(ϕ) = 1 νϕ(x⁰).

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Proof of the theorem in the toric case, continued

Set ζ(x) =νϕ(x⁰)min x₁

x₁⁰, . . . , x_n x_n⁰

, ∀x ∈Σ.

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Proof of the theorem in the toric case, continued

x₁⁰, . . . , x_n x_n⁰

, ∀x ∈Σ.

Thenζ is the smallest nonnegative concave increasing function onΣsuch thatζ(x⁰) =νϕ(x⁰), henceζ ≤νϕ.

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Proof of the theorem in the toric case, continued

x₁⁰, . . . , x_n x_n⁰

, ∀x ∈Σ.

Thenζ is the smallest nonnegative concave increasing function onΣsuch thatζ(x⁰) =νϕ(x⁰), henceζ ≤νϕ. This implies that

ϕ(z1, . . . ,zn)≤ −νϕ(−ln|z1|, . . . ,−ln|zn|)

≤ −ζ(−ln|z1|, . . . ,−ln|zn|)

≤νϕ(x⁰)max

ln|z1|

x₁⁰ , . . . ,ln|zn| x_n⁰

:=ψ(z1, . . . ,zn).

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Proof of the theorem in the toric case, continued

x₁⁰, . . . , x_n x_n⁰

, ∀x ∈Σ.

Thenζ is the smallest nonnegative concave increasing function onΣsuch thatζ(x⁰) =νϕ(x⁰), henceζ ≤νϕ. This implies that

ϕ(z1, . . . ,zn)≤ −νϕ(−ln|z1|, . . . ,−ln|zn|)

≤ −ζ(−ln|z1|, . . . ,−ln|zn|)

≤νϕ(x⁰)max

ln|z1|

x₁⁰ , . . . ,ln|zn| x_n⁰

:=ψ(z1, . . . ,zn).

By Lemma 2 we get that

f(e1(ϕ), ...,en(ϕ))≤f(e1(ψ), ...,en(ψ)) = c(ψ) = 1

νϕ(x⁰) =c(ϕ).

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Reduction to the case of plurisubharmonic functions with analytic singularity

LetHmϕ(Ω)be the Hilbert space of holomorphic functions f onΩsuch that

Z

Ω

|f|²e^−2mϕdV <+∞,

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Reduction to the case of plurisubharmonic functions with analytic singularity

Z

Ω

|f|²e^−2mϕdV <+∞, and letψm = _2m¹ logP

|gm,k|² where{gm,k}k≥1be an orthonormal basis for Hmϕ(Ω).

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Reduction to the case of plurisubharmonic functions with analytic singularity

Z

Ω

|f|²e^−2mϕdV <+∞, and letψm = _2m¹ logP

|gm,k|² where{gm,k}k≥1be an orthonormal basis for Hmϕ(Ω). Using∂¯-equation with L²-estimates (D-Kollár), there are constantsC1,C2 >0 independent ofmsuch that

ϕ(z)− C1

m ≤ψm(z)≤ sup

|ζ−z|<r

ϕ(ζ) + 1

mlogC2

rⁿ for everyz ∈Ωandr <d(z, ∂Ω).

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Reduction to the case of plurisubharmonic functions with analytic singularity, continued

and

ν(ϕ)− n

m ≤ν(ψm)≤ν(ϕ), 1

c(ϕ)− 1

m ≤ 1

c(ψm) ≤ 1 c(ϕ).

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Reduction to the case of plurisubharmonic functions with analytic singularity, continued

and

ν(ϕ)− n

m ≤ν(ψm)≤ν(ϕ), 1

c(ϕ)− 1

m ≤ 1

c(ψm) ≤ 1 c(ϕ). By Lemma 2, we have that

f(e1(ϕ), . . . ,en(ϕ))≤f(e1(ψm), . . . ,en(ψm)), ∀m ≥1.

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Reduction to the case of plurisubharmonic functions with analytic singularity, continued

and

ν(ϕ)− n

m ≤ν(ψm)≤ν(ϕ), 1

c(ϕ)− 1

m ≤ 1

c(ψm) ≤ 1 c(ϕ). By Lemma 2, we have that

f(e1(ϕ), . . . ,en(ϕ))≤f(e1(ψm), . . . ,en(ψm)), ∀m ≥1.

The above inequalities show that in order to prove the lower bound of c(ϕ)in the Main Theorem, we only need prove it forc(ψm)and then letm→ ∞.

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Reduction to the case of monomial ideals

For j =0, . . . ,n set

J = (f1, . . . ,fN), c(J) =c(ϕ), andej(J) =ej(ϕ).

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Reduction to the case of monomial ideals

For j =0, . . . ,n set

J = (f1, . . . ,fN), c(J) =c(ϕ), andej(J) =ej(ϕ). Now, by fixing a multiplicative order on the monomials

z^α =z₁^α¹. . .z_n^αⁿ

it is well known that one can construct a flat family(Js)_s∈C of ideals ofOCⁿ,0 depending on a complex parameter s ∈C, such thatJ0is a monomial ideal, J1=J and

dim(O_Cⁿ_,0/J_s^t) =dim(O_Cⁿ_,0/J^t) for alls,t ∈N.

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Reduction to the case of monomial ideals

For j =0, . . . ,n set

J = (f1, . . . ,fN), c(J) =c(ϕ), andej(J) =ej(ϕ). Now, by fixing a multiplicative order on the monomials

z^α =z₁^α¹. . .z_n^αⁿ

it is well known that one can construct a flat family(Js)_s∈C of ideals ofOCⁿ,0 depending on a complex parameter s ∈C, such thatJ0is a monomial ideal, J1=J and

dim(O_Cⁿ_,0/J_s^t) =dim(O_Cⁿ_,0/J^t) for alls,t ∈N. In fact J0is just the initial ideal associated to J with respect to the monomial order.

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Reduction to the case of monomial ideals, continued

Moreover, we can arrange by a generic rotation of

coordinates C^p ⊂Cⁿthat the family of idealsJs∩ OC^p,0is also flat,

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Reduction to the case of monomial ideals, continued

coordinates C^p ⊂Cⁿthat the family of idealsJs∩ OC^p,0is also flat, and that the dimensions

dim Ob^p,0/(Js∩ OC^p,0)^t

=dim OC^p,0/(J ∩ OC^p,0)^t

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Reduction to the case of monomial ideals, continued

=dim OC^p,0/(J ∩ OC^p,0)^t compute the intermediate multiplicities

ep(Js) = lim

t→+∞

p!

t^p dim OC^p,0/(Js∩ OC^p,0)^t

=ep(J),

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Reduction to the case of monomial ideals, continued

ep(Js) = lim

t→+∞

p!

=ep(J), in particular, ep(J0) =ep(J)for all p.

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Reduction to the case of monomial ideals, continued

ep(Js) = lim

t→+∞

p!

=ep(J), in particular, ep(J0) =ep(J)for all p. The semicontinuity property of the log canonical threshold implies that

c(J0)≤c(Js) =c(J)for alls, so the lower bound is valid for c(J)if it is valid forc(J0).

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About the continuity of Monge-Ampère operators

Conjecture

Letϕ ∈Ee(Ω)andΩ∋0. Then the analytic approximations ψm satisfye_j(ψm)→e_j(ϕ)asm →+∞, in other words, we have “strong continuity” of Monge-Ampère operators and higher Lelong numbers with respect to Bergman kernel approximation.

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About the continuity of Monge-Ampère operators

Conjecture

In the 2-dimensional case, e₂(ϕ)can be computed as follows (at least whenϕ ∈Ee(ω)has analytic singularities).

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About the continuity of Monge-Ampère operators

Conjecture

In the 2-dimensional case, e₂(ϕ)can be computed as follows (at least whenϕ ∈Ee(ω)has analytic singularities).

Letµ:Ωe →Ωbe the blow-up ofΩat 0. Take local coordinates (w1,w2)onΩe so that the exceptional divisor E can be writtenw₁=0.

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About the continuity of Monge-Ampère operators (II)

Withγ =ν(ϕ,0), we get that

ϕ(we ) =ϕ◦µ(w)−γlog|w1|

is psh with generic Lelong numbers equal to 0 alongE, and therefore there are at most countably many points xℓ ∈E at whichγℓ =ν(ϕ,e xℓ)>0. SetΘ =dd^cϕ,

Θ =e dd^cϕe=µ^∗Θ−γ[E]. SinceE²=−1 in cohomology, we have{Θ}e ²={µ^∗Θ}²−γ²inH²(E,R), hence

(∗)

Z

{0}(dd^cϕ)²=γ²+ Z

E

(dd^cϕ)e ².

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About the continuity of Monge-Ampère operators (II)

Withγ =ν(ϕ,0), we get that

ϕ(we ) =ϕ◦µ(w)−γlog|w1|

is psh with generic Lelong numbers equal to 0 alongE, and therefore there are at most countably many points xℓ ∈E at whichγℓ =ν(ϕ,e xℓ)>0. SetΘ =dd^cϕ,

Θ =e dd^cϕe=µ^∗Θ−γ[E]. SinceE²=−1 in cohomology, we have{Θ}e ²={µ^∗Θ}²−γ²inH²(E,R), hence

(∗)

Z

{0}(dd^cϕ)²=γ²+ Z

E

(dd^cϕ)e ².

Ifϕeonly has ordinary logarithmic poles at thexℓ’s, then R

E(dd^cϕ)e ²=P

γ_ℓ², but in general the situation is more complicated. Let us blow-up any of the points x_ℓand repeat the process k times.

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About the continuity of Monge-Ampère operators (III)

We set ℓ=ℓ1 in what follows, as this was the first step, and at stepk =0 we omit any indices as 0 is the only point we have to blow-up to start with. We then get inductively (k+1)-iterated blow-ups depending on multi-indices ℓ= (ℓ1, . . . , ℓk) = (ℓ^′, ℓk)with

ℓ^′ = (ℓ1, . . . , ℓk−1),

µℓ :Ωeℓ →Ωeℓ^′, k ≥1, µ_∅ =µ:Ωe∅ =Ωe →Ω, γ∅ =γ and exceptional divisorsE_ℓ ⊂Ωeℓ lying over points x_ℓ ∈E_ℓ′ ⊂Ωeℓ^′, where

γℓ =ν(ϕeℓ^′,xℓ)>0,

ϕeℓ(w) =ϕeℓ^′ ◦µℓ(w)−γℓlog|w₁^(ℓ)|,

(w₁^(ℓ) =0 an equation ofEℓ in the relevant chart).

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About the continuity of Monge-Ampère operators (IV)

Formula (∗)implies

(∗∗) e₂(ϕ)≥

X+∞

k=0

X

ℓ∈N^k

γ_ℓ²

with equality when ϕhas an analytic singularity at 0. We conjecture that (∗∗)is always an equality wheneverϕ ∈Ee(Ω).

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About the continuity of Monge-Ampère operators (IV)

(∗∗) e₂(ϕ)≥

X+∞

k=0

X

ℓ∈N^k

γ_ℓ²

This would imply the Guedj-Rashkovskii conjecture.

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About the continuity of Monge-Ampère operators (IV)

(∗∗) e₂(ϕ)≥

X+∞

k=0

X

ℓ∈N^k

γ_ℓ²

This would imply the Guedj-Rashkovskii conjecture.

Notice that the currents Θℓ=dd^cϕeℓ satisfy inductively Θℓ =µ^∗_ℓΘℓ^′ −γℓ[Eℓ], hence the cohomology class ofΘℓ

restricted toEℓ is equal toγℓ times the fundamental generator of E_ℓ. As a consequence we have

X

ℓk+1∈Nγℓ,ℓ_k+1 ≤γℓ,

in particular γℓ =0 for allℓ∈N^k ifγ =ν(ϕ,0) =0.

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References I

[ÅCCP] Åhag P., Cegrell U., Czy˙z R. and Pha.m H. H., Monge-Ampère measures on pluripolar sets, J. Math.

Anal. Appl. 92 (2009), 613-627.

[ÅCKPZ]Åhag P., Cegrell U., Kołodziej S., Pha.m H.H.

and Zeriahi A.,Partial pluricomplex energy and integrability exponents of plurisubharmonic functions, Adv. Math. 222 (2009), 2036-2058.

[AGZV85]Arnol’d V. I., Guse˘ın-Zade S. M. and

Varchenko A. N.,Singularities of differentiable maps. Vols.

I–II, Monographs in Mathematics, vol. 82, Birkhäuser Boston Inc., Boston, MA, 1985, Translated from the Russian by Ian Porteous and Mark Reynolds.

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References II

[Ati70]Atiyah M. F.,Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145-150.

[Ber71]Bernšte˘ın I. N.,Modules over a ring of differential operators. An investigation of the fundamental solutions of equations with constant coefficients, Funkcional. Anal. i Priložen. 5 (1971), 1-16.

[Ceg04]Cegrell U.,The general definition of the complex Monge-Ampère operator, Ann. Inst. Fourier (Grenoble) 54 (2004), 159-179.

[DK01]Demailly J.-P. and Kollár J.,Semi-continuity of complex singularity exponents and Kähler-Einstein

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References III

metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), 525-556.

[dFEM03] de Fernex T, Ein L. and Musta¸tˇa,Bounds for log canonical thresholds with applications to birational rigidity, Math. Res. Lett. 10 (2003) 219-236.

[dFEM04] de Fernex T., Ein L. and Musta¸tˇa M., Multiplicities and log canonical threshold, J. Algebraic Geom. 13 (2004), 603-615.

[dFEM09] de Fernex T., Ein L. and Musta¸tˇa M.,

Shokurov’s ACC Conjecture for log canonical thresholds on smooth varieties, Duke Math. J., 152 (2010) 93-114.

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References IV

[Eis95]Eisenbud D.,Commutative algebra with a view toward algebraic geometry, Grad. Texts in Math. 150, Springer, New York, 1995.

[GŠ58]Gel^′fand I. M. and Šilov G. E.,Obobshchennye funksii i deistviya iad nimi, Obobšˇcennye funkcii, Vypusk 1. Generalized functions, part 1, Gosudarstv. Izdat.

Fiz.-Mat. Lit., Moscow, 1958.

[Hör58]Hörmander L.,On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555-568.

[Kis94]Kiselman C.O., Attenuating the singularities of plurisubharmonic functions, Ann. Polon. Math. 60 (1994) 173-197.

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References V

[Łoj58]Łojasiewicz S.,Division d’une distribution par une fonction analytique de variables réelles, C. R. Acad. Sci.

Paris 246 (1958), 683-686.

[PS00] Phong D. H. and Sturm J.,On a conjecture of Demailly and Kollár. Kodaira’s issue, Asian J. Math. 4 (2000), 221-226.

[Sch50] Schwartz L.,Théorie des distributions. Tome I, Actualités Sci. Ind., no. 1091 = Publ. Inst. Math. Univ.

Strasbourg 9, Hermann & Cie., Paris, 1950.

[Sko72] Skoda H.,Sous-ensembles analytiques d’ordre fini ou infini dans Cⁿ, Bull. Soc. Math. France 100 (1972), 353-408.