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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

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

log canonical threshold of psh functions

Singularities of psh (plurisubharmonic) functions can be

measured by Lelong numbers. Another useful invariant is the log canonical threshold.

Definition

Let X be a complex manifold, p ∈ X, and ϕ be a

plurisubharmonic function defined on X. The log canonical threshold or complex singularity exponent of ϕ at p is defined by

cp(ϕ) = supn

c ≥ 0 : e−2cϕ is L1 on a neighborhood of po , Here we will take p = 0 and denote c(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

2 log(|g1|2 + . . . +|gN|2)

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

Then by Hironaka, ∃ modification µ : Xe → X such that µJ = (g1 ◦µ, . . . ,gN ◦ µ) = O(−X

ajEj) for some normal crossing divisor.Let O(P

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

c(ϕ) = min

Ej, µ(Ej)∋0

1+ bj

aj ∈ Q+.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

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 variable z = µ(w). Then

dλ(z) = |Jac(µ)(w)|2 ∼ Y wjbj

2dλ(w)

with respect to coordinates on the blow-up Ve of V, and

I ∼ Z

Ve

Qwjbj2dλ(w) Qwjaj2c

so convergence occurs if ca − b < 1 for all j.

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Notations and basic facts

A domain Ω ⊂ Cn is called hyperconvex if ∃ψ ∈ PSH(Ω), ψ ≤ 0, such that {z : ψ(z) < c} ⋐ Ω for all c < 0.

E0(Ω)=

ϕ∈PSH ∩L(Ω) : lim

z→∂Ωϕ(z)=0,Z

(ddcϕ)n<+∞

F(Ω) = n

ϕ ∈ PSH(Ω) : ∃ E0(Ω) ∋ ϕp ց ϕ, and supp≥1

Z

(ddcϕp)n < +∞o ,

Ee(X) = {ϕ ∈ PSH(X) locally in F(Ω) mod C(Ω)}

Theorem (U. Cegrell)

Ee(X) is the largest subclass of psh functions defined on a complex manifold X for which the complex Monge-Ampère operator is locally well-defined.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

Intermediate Lelong numbers

Set here dc = i (∂ −∂) so that ddc = πi ∂∂. If ϕ ∈ Ee(Ω) and 0 ∈ Ω, the products (ddcϕ)j are well defined and one can consider the Lelong numbers

ej(ϕ) = ν (ddcϕ)j,0 . In other words

ej(ϕ) = Z

{0}(ddcϕ)j ∧(ddc logkzk)n−j . One has e0(ϕ) = 1 and e1(ϕ) = ν(ϕ,0) (usual Lelong number). When

ϕ(z) = 1

2 log(|g1|2 + . . .+ |gN|2),

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The main result

Main Theorem (Demailly & Pha.m)

Let ϕ ∈ Ee(Ω). If e1(ϕ) = 0, then c(ϕ) = ∞.

Otherwise, we have

c(ϕ) ≥

n−1X

j=0

ej(ϕ) ej+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. Then ej(ϕ) = a1. . .aj, c(ϕ) = 1

a1 +. . . + 1 an.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

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 the existence of Kähler-Einstein

metrics.

Another important application is to birational rigidity.

Theorem (Pukhlikov 1998, Corti 2000, de Fernex 2011)

Let X be a smooth hypersurface of degree d in CPn+1. Then if d = 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 that Bir(X) ≃ Aut(X) (birational rigidity) ?

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Lemma 1

Lemma 1

Let ϕ ∈ Ee(Ω) and 0 ∈ Ω. Then we have that ej(ϕ)2 ≤ ej−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 ratios ej+1(ϕ)/ej(ϕ) are increasing.

Corollary

If e1(ϕ) = ν(ϕ,0) = 0, then ej(ϕ) = 0 for j = 1,2, . . . ,n− 1.

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

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

Proof of Lemma 1

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

Z

−h(ddcϕ)j ∧(ddcψ)n−j 2

=

" Z

dϕ∧dcψ ∧(ddcϕ)j−1∧(ddcψ)n−j−1 ∧ddch

#2

≤ Z

dψ ∧dcψ ∧(ddcϕ)j−1 ∧(ddcψ)n−j−1 ∧ddch Z

dϕ ∧dcϕ∧(ddcϕ)j−1 ∧(ddcψ)n−j−1 ∧ddch

= Z

−h(ddcϕ)j−1∧(ddcψ)n−j+1 Z

−h(ddcϕ)j+1∧(ddcψ)n−j−1,

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Proof of Lemma 1, continued

Now, as p → +∞, take

h(z) = hp(z) = max

− 1, 1

p logkzk ր

0 if z ∈ Ω\{0}

−1 if z = 0.

By the monotone convergence theorem we get in the limit that

"Z

{0}(ddcϕ)j ∧(ddcψ)n−j

#2

≤ Z

{0}(ddcϕ)j−1 ∧(ddcψ)n−j+1 Z

{0}(ddcϕ)j+1 ∧(ddcψ)n−j1. For ψ(z) = lnkzk, this is the desired estimate.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

Lemma 2

Lemma 2

Let ϕ, ψ ∈ Ee(Ω) be such that ϕ ≤ ψ (i.e ϕ is ”more singular”

than ψ). Then we have

n−1X

j=0

ej(ϕ) ej+1(ϕ) ≤

n−1X

j=0

ej(ψ) ej+1(ψ) .

The argument if based on the monotonicity of Lelong numbers with respect to the relation ϕ ≤ 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,tj2≤tj1tj+1,∀j = 2, ...,n−1}.

Then D is a convex set in Rn, as can be checked by a

straightforward application of the Cauchy-Schwarz inequality.

Next, consider the function f : int D → [0,+∞) defined by

f(t1, . . . ,tn) = 1

t1 + t1

t2 . . . + tn−1 tn . We have

∂f

∂tj(t) = −tj−1

tj2 + 1

tj+1 ≤ 0, ∀t ∈ D.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

Proof of Lemma 2, continued

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

is decreasing. Hence,

f(a) ≤ f(b) for all a,b ∈ int D, 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

Let ϕ ∈ PSH(Ω). Then the function defined by

νϕ(x) = lim

t→−∞

max

ϕ(z) : |z1| = ex1t, . . . ,|zn| = exnt t

is called the refined Lelong number of ϕ at 0.

The refined Lelong number of ϕ at 0 is increasing in each variable xj, and concave on Rn+.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

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 that Ω = ∆n is the unit polydisk.

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

Reduction to the case of monomial ideals, i.e. for

ϕ = log(|f1|2 + . . . .+ |fN|2), where f1, . . . ,fN are germs of monomial elements at 0.

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Proof of the theorem in the toric case

Set

Σ =



x = (x1, . . . ,xn) ∈ Rn+ : Xn

j=1

xj = 1



 . We choose x0 = (x10, . . . ,xn0) ∈ Σ such that

νϕ(x0) = max{νϕ(x) : x ∈ S}.

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

c(ϕ) = 1 νϕ(x0) .

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

Proof of the theorem in the toric case, continued

Set ζ(x) = νϕ(x0)min x1

x10, . . . , xn xn0

!

, ∀x ∈ Σ.

Then ζ is the smallest nonnegative concave increasing function on Σ such that ζ(x0) = νϕ(x0), hence ζ ≤ 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| xn0

!

:= 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

Let H(Ω) be the Hilbert space of holomorphic functions f on

Ω such that Z

|f|2e−2mϕdV < +∞, and let ψm = 2m1 logP

|gm,k|2 where{gm,k}k≥1 be an orthonormal basis for H(Ω). Using ∂-equation with¯ L2-estimates (D-Kollár), there are constants C1,C2 > 0 independent of m such that

ϕ(z)− C1

m ≤ ψm(z) ≤ sup

−z|<r

ϕ(ζ) + 1

m log C2 rn for every z ∈ Ω and r < d(z, ∂Ω).

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

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(e1m), . . . ,enm)), ∀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 for c(ψm) and then let m → ∞.

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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

zα = z1α1 . . .znαn

it is well known that one can construct a flat family (Js)s∈C of ideals of OCn,0 depending on a complex parameter s ∈ C, such that J0 is a monomial ideal, J1 = J and

dim(OCn,0/Jst) = dim(OCn,0/Jt) for all s,t ∈ N.

In fact J0 is just the initial ideal associated to J with respect to the monomial order.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

Reduction to the case of monomial ideals, continued

Moreover, we can arrange by a generic rotation of coordinates Cp ⊂ Cn that the family of ideals Js ∩ OCp,0 is also flat, and that the dimensions

dim Obp,0/(Js ∩ OCp,0)t

= dim OCp,0/(J ∩ OCp,0)t compute the intermediate multiplicities

ep(Js) = lim

t→+∞

p!

tp dim OCp,0/(Js ∩ OCp,0)t

= 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 all s, so the lower bound is valid for c(J) if it is valid for c(J0).

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About the continuity of Monge-Ampère operators

Conjecture

Let ϕ ∈ Ee(Ω) and Ω ∋ 0. Then the analytic approximations ψm satisfy ejm) → ej(ϕ) as m → +∞, in other words, we have

“strong continuity” of Monge-Ampère operators and higher

Lelong numbers with respect to Bergman kernel approximation.

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

Let µ : Ωe → Ω be the blow-up of Ω at 0. Take local coordinates (w1,w2) on Ωe so that the exceptional divisor E can be written w1 = 0.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

About the continuity of Monge-Ampère operators (II)

With γ = ν(ϕ,0), we get that

ϕ(we ) = ϕ◦µ(w) −γ log|w1|

is psh with generic Lelong numbers equal to 0 along E, and therefore there are at most countably many points x ∈ E at which γ = ν(ϕ,e x) > 0. Set Θ = ddcϕ,

Θ =e ddcϕe = µΘ− γ[E]. Since E2 = −1 in cohomology, we have {Θ}e 2 = {µΘ}2 − γ2 in H2(E,R), hence

(∗)

Z

{0}

(ddcϕ)2 = γ2 + Z

E

(ddcϕ)e 2.

If ϕe only has ordinary logarithmic poles at the x’s, then R

E(ddcϕ)e 2 = P

γ2, but in general the situation is more

complicated. Let us blow-up any of the points x and 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 step k = 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),

µ : Ωe → Ωe, k ≥ 1, µ = µ : Ωe = Ωe → Ω, γ = γ and exceptional divisors E ⊂ Ωe lying over points

x ∈ E ⊂ Ωe, where

γ = ν(ϕe,x) > 0,

ϕe(w) = ϕe ◦µ(w)− γlog|w1(ℓ)|,

(w1(ℓ) = 0 an equation of E in the relevant chart).

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

About the continuity of Monge-Ampère operators (IV)

Formula (∗) implies

(∗∗) e2(ϕ) ≥

X+∞

k=0

X

ℓ∈Nk

γ2

with equality when ϕ has an analytic singularity at 0. We

conjecture that (∗∗) is always an equality whenever ϕ ∈ Ee(Ω).

This would imply the Guedj-Rashkovskii conjecture.

Notice that the currents Θ = ddcϕe satisfy inductively Θ = µΘ −γ[E], hence the cohomology class of Θ

restricted to E is equal to γ times the fundamental generator of E. As a consequence we have

X

k+1N γℓ,ℓk+1 ≤ γ,

in particular 0 for all k if 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.

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

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

References II

[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 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.

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References III

[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.

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

[GŠ58] Gelfand 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.

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

References IV

[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.

[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.

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References V

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

Jean-Pierre Demailly / Pha.m Hoàng Hiê.p A sharp lower bound for the log canonical threshold

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