THE LOG CANONICAL THRESHOLD OF HOLOMORPHIC ...

THE LOG CANONICAL THRESHOLD OF HOLOMORPHIC FUNCTIONS LE MAU HAI, PHAM HOANG HIEP AND VU VIET HUNG Abstract. In this paper we give the relation between the log canonical threshold c0 (f ) and the geometry of the zero set {f = 0} of a holomorphic function f . Applying the above relation we give a simple proof for the ascending chain condition in dimension 2.

1. Introduction Let f be a holomorphic function on a neighbourhood of 0 ∈ Cn . Define Z dV2n < +∞}, c0 (f ) = sup{c > 0 : ∃ δ > 0, |f |2c B(0,δ) n

where B(0, δ) = {z ∈ C : kzk < δ} is a ball with centered at 0 and radius δ > 0 and dV2n denotes the Lebesgue measure in Cn . Note that (z)| 0 : ∃ δ > 0 with V2n ({z∈B(0,δ):|f is bounded r2c as r → 0}. Relying on [Lin] we have known that c(f ) ∈ (0, 1] ∩ Q. We set C(n) = {c(f ) : f is holomorphic in a neighbourhood of 0 ∈ Cn }. In [DK], Demailly and Kollár gave the following conjecture about the ascending chain condition. Let {fj }∞ j=1 be a sequence of holomorphic functions n on a neighbourhood of 0 in C satisfying a condition: c0 (f1 ) 6 c0 (f2 ) 6 · · · .

(1.1)

Then there exists j0 such that c0 (fj0 ) = c0 (fj0 +1 ) = · · · . The above conjecture in algebraic geometry has been formulated earlier by Shokurov (see [Sho] and also [K1],[K2]). By using method of algebraic geometry Shokurov has proved this conjecture for dimension n = 2 and n = 3 by Alexeev in [Al]. Next, in 2000, by relying on the methods of [PS1], D. H. Phong and J. Sturm proved the conjecture of Demailly and Kollár in dimension n = 2 (see [PS2]). Recently, T. Fernex, L. Ein and 2010 Mathematics Subject Classification. Primary 32S05; Secondary 14B05, 32U25. Key words and phrases. log canonical threshold, ascending chain condition, level sets of holomorphic functions. 1

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LE MAU HAI, PHAM HOANG HIEP AND VU VIET HUNG

M. Mustata (see [FEM]) extended the result of Shokurov and Alexeev for arbitrary dimension. In this paper, we will show the relation between the log canonical threshold c0 (f ) and the geometry of the zero set {f = 0} of a holomorphic function f . After that, by applying the obtained relation we give a simple and analytic proof for the ascending chain condition in dimension 2.

2. Area of level sets of holomorphic functions of one variable Let a = (a1 , . . . , am ) ∈ Cm be given. For each r > 0, we set E = E(a, r) = {z ∈ C : |z − a1 |........|z − am | < r}, Φj,k = Φj,k (a) =

max

16i1 · · · > |z − am | > β(m)t. On the other hand, we have |z − ai | + |z − aj | > |ai − aj | and it follows that |z − ai | >

|ai − aj | ∀ 1 6 i < j 6 m. 2

THE LOG CANONICAL THRESHOLD OF HOLOMORPHIC FUNCTIONS

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Thus this yields the following |z − a1 | . . . |z − am | > |z − a1 | . . . |z − ak |(β(m)t)m−k 1 > k max |a1 − ai | . . . max |ak − ai |(β(m)t)m−k 2 i=2,m i=k+1,m β(m)m−k max |a1 − ai | . . . max |ak − ai |tm−k i>1 i>k 2k m−k β(m) > Φmk tm−k 22k β(m) > 2(m−1) Φmk tm−k , ∀ k = 0, m − 1. 2 >

Thus we infer that |z − a1 | . . . |z − am | > max tm , Φm1 tm−1 , . . . , Φm,m−1 t > r, where in this case we choose β(m) = 22(m−1) and the desired conclusion follows. ii) Choose j0 such that 1 1 r m−1 r m t = min r , . ,..., Φj0 1 Φj0 ,m−1 Next take z ∈ 4(aj0 , α(m)t). Then we have |z − aj | 6 |z − aj0 | + |aj0 − aj | 6 α(m)t + |aj0 − aj |, ∀ j = 1, m. Thus it follows that |z − a1 | . . . |z − am | 6

Yh

i α(m)t + |aj0 − aj |

j=1,m

= (α(m)t)m + (α(m)t)m−1

X

|aj0 − aj | + · · ·

j=1,m m

6 (α(m)t) + (m − 1)(α(m)t)m−1 Φj0 1 + · · · + α(m)tΦj0 ,m−1 m i hX m−1 6 (α(m))k r k−1 k=1 = α(m)(α(m) + 1)m−1 r < r. where by choosing α(m)(α(m)+1)m−1 6 1 and, hence, the proof is complete. From the above theorem we obtain the following corollary.

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h i Corollary 2.2. V2 (E(a, r)) ∈ α(m)2 πt2 , mβ(m)2 πt2 . We also have the following. Remark 2.3. Φ2j1 > Φj2 , Φ2j2 > Φj1 Φj3 , . . . , Φ2j,m−2 > Φj,m−3 Φj,m−1 , ∀ j = 1, m. We need the following. Definition 2.4. Given the two systems of functions {ai (z)}i=1,m and {bi (z)}i=1,m defined on a domain D ⊂ Cn . We write {ai (z)}i=1,m {bi (z)}i=1,m on D if there exists a constant C > 0 such that for each z ∈ D we can find a bisection τ : {1, . . . , m} −→ {1, . . . , m} with |ai (z) − aj (z)| > C|bτ (i) (z) − bτ (j) (z)|, ∀1 6 i, j 6 m. From Definition 2.4 we arrive at the following. Proposition 2.5. Let {ai (z)}i=1,m and {bi (z)}i=1,m be the two systems of functions as in Definition 2.4 defined on a domain D ⊂ Cn . Then there exists M > 0 such that the following holds. V2 (E(a(z), r)) 6 M V2 (E(b(z), r)), ∀z ∈ D and ∀ r > 0, where a(z) = (a1 (z), . . . , am (z)) and b(z) = (b1 (z), . . . , bm (z)). Proof. We have Φik (a(z)) > αk Φτ (i)k (b(z)), ∀ i = 1, m and ∀ k = 1, m − 1. By using the definition of Φik and Definition 2.4 it follows that 1 t(a(z), r) 6 m−1 t(b(z), r). α mβ(m) Thus by choosing M = α(m)αm−1 the desired conclusion follows.

3. Volume of the level set of holomorphic functions of several variables In this section we estimate volume of the level set of holomorphic functions of several variables and, as consequently, we prove the decreasing of the sequence of log canonical threshold c(fj ). Let f be a holomorphic function on a neighbourhood of 0 ∈ Cn . By the Weierstrass preparation theorem we write f (z) = (zn − a1 (z 0 )) . . . (zn − am (z 0 ))g(z), where g(z) is a holomorphic function in a neighbourhood of 0 with g(0) 6= 0, z 0 = (z1 , . . . , zn−1 ) and m is the Lelong number of log|f | at 0. We need the


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following estimate for volume of the level set of Weierstrass polynomial P (z) = (zn − a1 (z 0 )) . . . (zn − am (z 0 )). By Fubini’s theorem we have V2n ({z ∈ B(0, δ) : |P (z)| < r}) Z = V2n−2 {zn ∈ 4(0, δ) : |P (z 0 , zn )| < r} dV2n−2 (z 0 ) B0 (0,δ)

Z =

V2n−2 E(a(z 0 ), r) dV2n−2 (z 0 ).

B0 (0,δ)

Definition 3.1. Let f and g be holomorphic functions on a neighbourhood of 0 ∈ Cn and {ai (z 0 )}i=1,m and {bi (z 0 )}i=1,m are roots of f (z 0 , .) and g(z 0 , .) respectively. We write f g if {ai (z 0 )}i=1,m {bi (z 0 )}i=1,m . The following result is an extension of Proposition 2.5 for higher dimension. Theorem 3.2. Assume that f g. Then V2n {|f | < r} 6 constant.V2n {|g| < r} , ∀ r > 0 and c0 (f ) > c0 (g). Now relying on Theorem 2.1 we establish the decreasing of the sequence of log canonical threshold c(fj ) in dimension 2. Namely we have the following. Theorem 3.3. Assume that {fj }∞ j=1 be a sequence of holomorphic functions of two variables in a neighbourhood of 0 ∈ C2 . Then there exists a subsequence {jk } such that the sequence of log canonical threshold c(fjk ) is decreasing. Proof. Put mj = γlog|fj | (0), the Lelong number of fj at 0. Without loss of generality we may assume that m1 6 m2 6 · · · .

(4.1)

We consider the following cases. a) Case 1. Let mj % +∞. Since we have 2 1 6 c0 (fj ) 6 . mj mj Hence, in the case it follows that c(fj ) → 0. Thus we can choose a subsequence {jk } such that c0 (fjk ) decreases.

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b) Case 2. Assume that sup mj < +∞. Then we may assume that j>1

m1 = m2 = · · · = m. Puiseux’s theorem (see [No]) implies that there exists holomorphic functions pj1 , . . . , pjm such that fj (z1m! , z2 ) = (z2 − pj1 (z1 )) . . . (z2 − pjm (z1 )). Put pj = {pji (z)}i=1,m . Then we have V4 {z ∈ B(0, δj ) : |fj | < r} Z = V2 {z2 ∈ 4(0, δj ) : |fj (z1 , z2 )| < r} dV2 (z1 ) 4(0,δj )

Z

2(m!−1)

|z1 |

=

V2 {z2 ∈ 4(0, δj ) :

|fj (z1m! , z2 )|

< r} dV2 (z1 )

4(0,δj )

Z =

|z1 |2(m!−1) V2 (E(pj (z), r))dV2 (z1 ),

4(0,δj )

when r is sufficiently small. We have |pji (z1 )−pjl (z1 )| ≈ |z1 |sjil with sjil ∈ N. Hence, we can choose a subsequence {jk } such that sj1 il 6 sj2 il 6 ... for all i, l = 1, m. It follows that pjk (z1 ) pjk+1 (z1 ), on 4(0, δk ), ∀k > 1. Proposition 2.5 implies that c0 (fjk ) > c0 (fjk+1 ) and the desired conclusion follows. References [Al] V. Alexeev, Two two-dimensional terminations, Duke Math. J. 69 (1993), 527-545. [DK] J-P. Demailly and J. Kollar, Semi- continuity of complex singularity exponents and K¨ ahler-Einstein metrics on Fano orbifolds, Ann. Ec. Norm. Sup. 34 (2001), 525-556. [FEM] T. Fernex, L. Ein and Mircea Mustata, Shokurov’s ACC Conjecture for log canonical thresholds on smooth varieties, Duke Math. J. 152, (2010), 93-114. [I] J. I. Igusa, On the first terms of certain asymptotic expansions, in Complex and Algebraic Geometry, Iwanami Shoten, 1977, pp. 357-368. [K1] J . Kollar(with 14 coauthors), Flips and Abundance for Algebraic Threefolds, Asterique, 211 (1992). [K2] J. Kollar, Singilarities of pairs, in Proceedings of Symposia in Pure Mathematics 62, Americal Mathematical Society, 1997, 221- 285. [Lin] B. Lichtin, Poles of |f (z, w)|2s and roots of the B-function, Ark för Math, 27 (1989), 283-304. [No] K. J. Nowak, Some elementary proofs of Puiseux’s theorems, Univ. Iagel. Acta. Math., 38 (2000), 279-282. [PS1] D. H. Phong and J. Sturm, Algebraic estimetes, stability of local zeta functions, and uniform estimates for distribution functions, Ann. of Math, Second Series, 152 (2000), 277-329.


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[PS2] D. H. Phong and J. Sturm, On a Conjecture of Demailly and Kollar, Asian J. Math. 4 (2000), 221-226. [Sho] V. Shokurov, 3-fold log flips, Isv. Russ. A. N. Ser. Math., 56(1992), 105-203. Department of Mathematics, Hanoi National University of Education (Dai hoc Su Pham Ha Noi), 136 Xuan Thuy Street, Caugiay District, Ha Noi, TayBac University( Dai hoc Tay Bac), Vietnam E-mail address: [email protected], phhiep− [email protected],[email protected]