Jean-Pierre Demailly

Hoàng Hiê

A sharp lower bound for the log canonical threshold

Keywords: 2000 Mathematics Subject Classification: 14B05, 32S05, 32S10, 32U25 Lelong number, Monge-Ampère operator, log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function ϕ with an isolated singularity at 0 in an open subset of C n . This threshold is defined as the supremum of constants c > 0 such that e -2cϕ is integrable on a neighborhood of 0. We relate c(ϕ) to the intermediate multiplicity numbers e j (ϕ), defined as the Lelong numbers of (dd c ϕ) j at 0 (so that in particular e 0 (ϕ) = 1). Our main result is that c(ϕ) ≥ e j (ϕ)/e j+1 (ϕ), 0 ≤ j ≤ n -1. This inequality is shown to be sharp; it simultaneously improves the classical result c(ϕ) ≥ 1/e 1 (ϕ) due to Skoda, as well as the lower estimate c(ϕ) ≥ n/e n (ϕ) 1/n which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.

Notation and main results

Here we put d c = i 2π (∂ -∂), so that dd c = i π ∂∂. The normalization of the d c operator is chosen so that we have precisely (dd c log |z|) n = δ 0 for the Monge-Ampère operator in C n . The Monge-Ampère operator is defined on locally bounded plurisubharmonic functions according to the definition of Bedford-Taylor [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]; it can also be extended to plurisubharmonic functions with isolated or compactly supported poles by [START_REF] Demailly | Monge-Ampère operators, Lelong numbers and intersection theory, Complex Analysis and Geometry[END_REF]. If Ω is an open subset of C n , we let PSH(Ω) (resp. PSH -(Ω)) be the set of plurisubharmonic (resp. psh ≤ 0) functions on Ω.

Definition 1.1. Let Ω be a bounded hyperconvex domain (i.e. a domain possessing a negative psh exhaustion). Following Cegrell [START_REF] Cegrell | The general definition of the complex Monge-Ampère operator[END_REF], we introduce certain classes of psh functions on Ω, in relation with the definition of the Monge-Ampère operator :

F (Ω) = {ϕ ∈ PSH -(Ω) : ∃ E 0 (Ω) ∋ ϕ p ց ϕ, sup p≥1 Ω (dd c ϕ p ) n < +∞}, (c) E(Ω) = {ϕ ∈ PSH -(Ω) : ∃ ϕ K ∈ F (Ω) such that ϕ K = ϕ on K, ∀K ⊂⊂ Ω}.
It is proved in [START_REF] Cegrell | The general definition of the complex Monge-Ampère operator[END_REF] that the class E(Ω) is the biggest subset of PSH -(Ω) on which the Monge-Ampère operator is well-defined. For a general complex manifold X, after removing the negativity assumption of the functions involved, one can in fact extend the Monge-Ampère operator to the class

(1.2) E(X) ⊂ PSH(X)
of psh functions which, on a neighborhood Ω ∋ x 0 of an arbitrary point x 0 ∈ X, are equal to a sum u + v with u ∈ E(Ω) and v ∈ C ∞ (Ω); again, this is the biggest subclass of functions of PSH(X) on which the Monge-Ampère operator is locally well defined. It is easy to see that E(X) contains the class of psh functions which are locally bounded outside isolated singularities.

For ϕ ∈ PSH(Ω) and 0 ∈ Ω, we introduce the log canonical threshold at 0

(1.3) c(ϕ) = sup c > 0 : e -2cϕ is L 1 on a neighborhood of 0 ,
and for ϕ ∈ E(Ω) we introduce the intersection numbers

(1.4) e j (ϕ) = {0} (dd c ϕ) j ∧ (dd c log z ) n-j
which can be seen also as the Lelong numbers of (dd c ϕ) j at 0. Our main result is the following sharp estimate. It is a generalization and a sharpening of similar inequalities discussed in [START_REF] Corti | Factoring birational maps of threefolds after Sarkisov[END_REF], [START_REF] Corti | Singularities of linear systems and 3-fold birational geometry[END_REF], [START_REF] De Fernex | Bounds for log canonical thresholds with applications to birational rigidity[END_REF], [START_REF] De Fernex | Multiplicities and log canonical thresholds[END_REF]; such inequalities have fundamental applications to birational geometry (see [START_REF] Iskovskikh | Three-dimensional quartics and counterexamples to the Lüroth problem[END_REF], [START_REF] Pukhlikov | Birational automorphisms of a four-dimensional quintic[END_REF], [START_REF] Pukhlikov | Birationally rigid Fano hypersurfaces[END_REF], [START_REF] Iskovskikh | Birational rigidity and Mori theory[END_REF], [START_REF] Chel | 'tsov, Birationally rigid Fano manifolds[END_REF]).

Theorem 1.5. Let ϕ ∈ E(Ω) and 0 ∈ Ω. Then c(ϕ) = +∞ if e 1 (ϕ) = 0, and otherwise

c(ϕ) ≥ n-1 j=0
e j (ϕ) e j+1 (ϕ) .

Remark 1.6. By Lemma 2.1 below, we have (e 1 (ϕ), . . . , e n (ϕ)) ∈ D where D = t = (t 1 , . . . , t n ) ∈ [0, +∞) n : t 2 1 ≤ t 2 , t 2 j ≤ t j-1 t j+1 , ∀j = 2, . . . , n -1 , i.e. log e j (ϕ) is a convex sequence. In particular, we have e j (ϕ) ≥ e 1 (ϕ) j , and the denominators do not vanish in 1.5 if e 1 (ϕ) > 0. On the other hand, a well known inequality due to Skoda [START_REF] Skoda | Sous-ensembles analytiques d'ordre fini ou infini dans C n[END_REF] tells us that 1 e 1 (ϕ) ≤ c(ϕ) ≤ n e 1 (ϕ) , hence c(ϕ) < +∞ iff e 1 (ϕ) > 0. To see that Theorem 1.5 is optimal, let us choose

ϕ(z) = max a 1 ln |z 1 |, . . . , a n ln |z n | with 0 < a 1 ≤ a 2 ≤ . . . ≤ a n
. Then e j (ϕ) = a 1 a 2 . . . a j , and a change of variable

z j = ζ 1/a j j on C R -easily shows that c(ϕ) = n j=1 1 a j .
Assume that we have a function f : D → [0, +∞) such that c(ϕ) ≥ f (e 1 (ϕ), . . . , e n (ϕ)) for all ϕ ∈ E(Ω). Then, by the above example, we must have

f (a 1 , a 1 a 2 , . . . , a 1 . . . a n ) ≤ n j=1 1 a j
for all a j as above. By taking a j = t j /t j-1 , t 0 = 1, this implies that

f (t 1 , . . . , t n ) ≤ 1 t 1 + t 1 t 2 + . . . + t n-1 t n , ∀t ∈ D,
whence the optimality of our inequality.

Remark 1.7. Theorem 1.5 is of course stronger than Skoda's lower bound c(ϕ) ≥ 1/e 1 (ϕ). By the inequality between the arithmetic and geometric means, we infer the main inequality of [START_REF] De Fernex | Bounds for log canonical thresholds with applications to birational rigidity[END_REF], [START_REF] De Fernex | Multiplicities and log canonical thresholds[END_REF] and [START_REF] Demailly | Estimates on Monge-Ampère operators derived from a local algebra inequality[END_REF] (1.8) c(ϕ) ≥ n e n (ϕ) 1/n . By applying the arithmetic-geometric inequality for the indices 1 ≤ j ≤ n -1 in our summation n-1 j=0 e j (ϕ)/e j+1 (ϕ), we also infer the stronger inequality

(1.9) c(ϕ) ≥ 1 e 1 (ϕ) + (n -1) e 1 (ϕ) e n (ϕ) 1 n-1 .

Log convexity of the multiplicity sequence

The log convexity of the multiplicity sequence can be derived from very elementary integration by parts and the Cauchy-Schwarz inequality, using an argument from [START_REF] Cegrell | The general definition of the complex Monge-Ampère operator[END_REF].

Lemma 2.1. Let ϕ ∈ E(Ω) and 0 ∈ Ω. We have e j (ϕ) 2 ≤ e j-1 (ϕ)e j+1 (ϕ), ∀j = 1, . . . , n-1.

Proof. Without loss generality, by replacing ϕ with a sequence of local approximations ϕ p (z) = max(ϕ(z) -C, p log |z|) of ϕ(z) -C, C ≫ 1, we can assume that Ω is the unit ball and ϕ ∈ E 0 (Ω). Take also h, ψ ∈ E 0 (Ω). Then integration by parts and the Cauchy-Schwarz inequality yield

Ω -h(dd c ϕ) j ∧ (dd c ψ) n-j 2 = Ω dϕ ∧ d c ψ ∧ (dd c ϕ) j-1 ∧ (dd c ψ) n-j-1 ∧ dd c h 2 ≤ Ω dψ ∧ d c ψ ∧ (dd c ϕ) j-1 ∧ (dd c ψ) n-j-1 ∧ dd c h Ω dϕ ∧ d c ϕ ∧ (dd c ϕ) j-1 ∧ (dd c ψ) n-j-1 ∧ dd c h = Ω -h(dd c ϕ) j-1 ∧ (dd c ψ) n-j+1 Ω -h(dd c ϕ) j+1 ∧ (dd c ψ) n-j-1 . Now, as p → +∞, take h(z) = h p (z) = max -1, 1 p log z ր 0 if z ∈ Ω {0} -1 if z = 0.
By the monotone convergence theorem we get in the limit

{0} (dd c ϕ) j ∧ (dd c ψ) n-j 2 ≤ {0} (dd c ϕ) j-1 ∧ (dd c ψ) n-j+1 {0} (dd c ϕ) j+1 ∧ (dd c ψ) n-j-1 .
For ψ(z) = ln z , this is the desired estimate.

Corollary 2.2. Let ϕ ∈ E(Ω) and 0 ∈ Ω. We have the inequalities

e j (ϕ) ≥ e 1 (ϕ) j , ∀j = 0, 1, . . . ≤ n e k (ϕ) ≤ e j (ϕ) l-k l-j e l (ϕ) k-j l-j , ∀0 ≤ j < k < l ≤ n.
In particular e 1 (ϕ) = 0 implies e k (ϕ) = 0 for k = 2, . . . , n -1 if n ≥ 3.

Proof. If e j (ϕ) > 0 for all j, Lemma 2.1 implies that j → e j (ϕ)/e j-1 (ϕ) is increasing, at least equal to e 1 (ϕ)/e 0 (ϕ) = e 1 (ϕ), and the inequalities follow from the log convexity. The general case can be proved by considering ϕ ε (z) = ϕ(z) + ε log z , since 0 < ε j ≤ e j (ϕ ε ) → e j (ϕ) when ε → 0. The last statement is obtained by taking j = 1 and l = n.

Proof of the main theorem

We start with a monotonicity statement.

Lemma 3.1. Let ϕ, ψ ∈ E(Ω) be such that ϕ ≤ ψ (i.e. ϕ is "more singular" than ψ). Then Proof. As in Remark 1.6, we set

D = {t = (t 1 , . . . , t n ) ∈ [0, +∞) n : t 2 1 ≤ t 2 , t 2 j ≤ t j-1 t j+1 , ∀j = 2, . . . , n -1}. Then D is a convex set in R n ,
as can be checked by a straightforward application of the Cauchy-Schwarz inequality. We consider the function

f : int D → [0, +∞) (3.2) f (t 1 , . . . , t n ) = 1 t 1 + t 1 t 2 . . . + t n-1 t n .
We have (3.3) Proof of the main theorem in the "toric case".

∂f ∂t j (t) = - t j-1 t 2 j + 1 t j+1 ≤ 0, ∀t ∈ D.
It will be convenient here to introduce Kiselman's refined Lelong numbers (cf. [START_REF] Kiselman | Un nombre de Lelong raffiné[END_REF], [START_REF] Kiselman | Attenuating the singularities of plurisubharmonic functions[END_REF]):

Definition 3.4. Let ϕ ∈ PSH(Ω). Then the function

ν ϕ (x) = lim t→-∞ max{ϕ(z) : |z 1 | = e x 1 t , . . . , |z n | = e xnt } t
is called the refined Lelong number of ϕ at 0. This function is increasing in each variable x j and concave on R n + . By "toric case", we mean that ϕ(z 1 , . . . , z n ) = ϕ(|z 1 |, . . . , |z n |) depends only on |z j | for all j; then ϕ is psh if and only if (t 1 , . . . , t n ) → ϕ(e t 1 , . . . , e tn ) is increasing in each t j and convex. By replacing ϕ with ϕ(λz) -ϕ(λ, ..., λ), 0 < λ ≪ 1, we can assume that Ω = ∆ n is the unit polydisk, ϕ(1, . . . , 1) = 0 (so that ϕ ≤ 0 on Ω), and we have e 1 (ϕ) = n ν ϕ ( 1 n , . . . , 1 n ). By convexity, the slope max{ϕ(z) : |z j |=e x j t } t is increasing in t for t < 0. Therefore, by taking t = -1 we get

ν ϕ (-ln |z 1 |, . . . , -ln |z n |) ≤ -ϕ(z 1 , . . . , z n ).
Notice also that ν ϕ (x) satisfies the 1-homogeneity property ν ϕ (λx) = λν ϕ (x) for λ ∈ R + . As a consequence, ν ϕ is entirely characterized by its restriction to the set

Σ = x = (x 1 , . . . , x n ) ∈ R n + : n j=1 x j = 1 .
We choose

x 0 = (x 0 1 , . . . , x 0 n ) ∈ Σ such that ν ϕ (x 0 ) = max{ν ϕ (x) : x ∈ Σ} ∈ e 1 (ϕ) n , e 1 (ϕ) .
By Theorem 5.8 in [START_REF] Kiselman | Attenuating the singularities of plurisubharmonic functions[END_REF] (see also [START_REF] Howald | Multiplier ideals of monomial ideals[END_REF] for similar results in an algebraic context) we have the formula

c(ϕ) = 1 ν ϕ (x 0 ) . Set ζ(x) = ν ϕ (x 0 ) min x 1 x 0 1 , . . . , x n x 0 n , ∀x ∈ R n + .
Then ζ is the smallest nonnegative concave 1-homogeneous function on R n + that is increasing in each variable x j and such that ζ(x 0 ) = ν ϕ (x 0 ). Therefore we have ζ ≤ ν ϕ , hence

ϕ(z 1 , . . . , z n ) ≤ -ν ϕ (-ln |z 1 |, . . . , -ln |z n |) ≤ -ζ(-ln |z 1 |, . . . , -ln |z n |) ≤ ν ϕ (x 0 ) max ln |z 1 | x 0 1 , . . . , ln |z n | x 0 n := ψ(z 1 , . . . , z n ).
By Lemma 3.1 and Remark 1.6 we get f (e 1 (ϕ), . . . , e n (ϕ)) ≤ f (e 1 (ψ), . . . , e n (ψ)) = c(ψ) = 1 ν ϕ (x 0 ) = c(ϕ).

(3.5) Reduction to the case of psh functions with analytic singularities.

In the second step, we reduce the proof to the case ϕ = log(|f [START_REF] Demailly | Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds[END_REF], mainly based on to the Ohsawa-Takegoshi L 2 extension theorem [OT87] (see also [START_REF] Demailly | Regularization of closed positive currents and Intersection Theory[END_REF]), there are constants C 1 , C 2 > 0 independent of m such that

ϕ(z) - C 1 m ≤ ψ m (z) ≤ sup |ζ-z|<r ϕ(ζ) + 1 m log C 2 r n
for every z ∈ Ω and r < d(z, ∂Ω) and

ν(ϕ) - n m ≤ ν(ψ m ) ≤ ν(ϕ), 1 c(ϕ) - 1 m ≤ 1 c(ψ m ) ≤ 1 c(ϕ)
.

By Lemma 3.1, we have f (e 1 (ϕ), . . . , e n (ϕ)) ≤ f (e 1 (ψ m ), . . . , e n (ψ m )), ∀m ≥ 1.

The above inequalities show that in order to prove the lower bound of c(ϕ) in Theorem 1.5, we only need to prove it for c(ψ m ) and let m tend to infinity. Also notice that since the Lelong numbers of a function ϕ ∈ E(Ω) occur only on a discrete set, the same is true for the functions ψ m .

(3.6) Reduction of the main theorem to the case of monomial ideals. The final step consists of proving the theorem for ϕ = log(|f 1 | 2 +. . . .+|f N | 2 ), where f 1 , . . . , f N are germs of holomorphic functions at 0 [this is because the ideals (g m,k ) k∈N in the Noetherian ring O C n ,0 are always finitely generated]. Set J = (f 1 , . . . , f N ), c(J ) = c(ϕ), e j (J ) = e j (ϕ), ∀j = 0, . . . , n. By the final observation of 3.5, we can assume that J has an isolated zero at 0. Now, by fixing a multiplicative order on the monomials z α = z α 1 1 . . . z αn n (see [START_REF] Eisenbud | Commutative algebra with a view toward algebraic geometry[END_REF] Chap. 15 and [START_REF] De Fernex | Multiplicities and log canonical thresholds[END_REF]), it is well known that one can construct a flat family (J s ) s∈C of ideals of O C n ,0 depending on a complex parameter s ∈ C, such that J 0 is a monomial ideal, J 1 = J and dim(O C n ,0 /J t s ) = dim(O C n ,0 /J t ) for all s and t ∈ N; in fact J 0 is just the initial ideal associated to J with respect to the monomial order. Moreover, we can arrange by a generic rotation of coordinates C p ⊂ C n that the family of ideals J s | C p is also flat, and that the dimensions dim(O C p ,0 /(J (notice, in the analytic setting, that the Lelong number of the (p, p)-current (dd c ϕ) p at 0 is the Lelong number of its slice on a generic C p ⊂ C n ); in particular e p (J 0 ) = e p (J ) for all p.

The semicontinuity property of the log canonical threshold (see for example [START_REF] Demailly | Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds[END_REF]) now implies that c(J 0 ) ≤ c(J s ) for s small. As c(J s ) = c(J ) for s = 0 (J s being a pull-back of J by a biholomorphism, in other words O C n ,0 /J s ≃ O C n ,0 /J as rings, see again [START_REF] Eisenbud | Commutative algebra with a view toward algebraic geometry[END_REF], chap. 15), the lower bound is valid for c(J ) if it is valid for c(J 0 ).

  (a) E 0 (Ω) = {ϕ ∈ PSH -(Ω) : lim z→∂Ω ϕ(z) = 0, Ω (dd c ϕ) n < +∞},(b)

For

  a, b ∈ int D such that a j ≥ b j , ∀j = 1, . . . , n, [0, 1] ∋ λ → f (b+λ(a-b)) is thus a decreasing function. This implies that f (a) ≤ f (b) for a, b ∈ int D, a j ≥ b j , ∀j = 1, . . . , n. On the other hand, the hypothesis ϕ ≤ ψ implies e j (ϕ) ≥ e j (ψ), ∀j = 1, . . . , n, by the comparison principle (see e.g.[START_REF] Demailly | Nombres de Lelong généralisés, théorèmes d'intégralité et d'analyticité[END_REF]). Therefore f (e 1 (ϕ), . . . , e n (ϕ)) ≤ f (e 1 (ψ), . . . , e n (ψ)).

  1 | 2 + . . . + |f N | 2 ), where f 1 , . . . , f N are germs of holomorphic functions at 0. Following the technique introduced in [Dem92], we let H mϕ (Ω) be the Hilbert space of holomorphic functions f on Ω such that Ω |f | 2 e -2mϕ dV < +∞, and let ψ m = 1 2m log |g m,k | 2 where {g m,k } k≥1 is an orthonormal basis of H mϕ (Ω). Thanks to Theorem 4.2 in

  s | C p ) t ) = dim(O C p ,0 /(J | C p )

t ) compute the intermediate multiplicities e p (J s ) = lim t→+∞ p! t p dim(O C p ,0 /(J s | C p ) t ) = e p (J )