In this note, we present a relationship between Lelong numbers and complex
singularity exponents. As an application, we obtain a new proof of Siu's
semicontinuity theorem for Lelong numbers.Comment: 5 pages, revised versio

Guan, Qi'an

Zhou, Xiangyu

Comptes Rendus Mathematique

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arXiv

In this note, we describe a relation between Lelong numbers and complex singularity exponents. As an application, we obtain a new proof of Siu&apos;s semicontinuity theorem for Lelong numbers. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.[NSFC-11431013];  [NSFC-11522101]SCI(E)ARTICLE4415-41935

Guan, Qi&apos

Institutional Repository of Peking University

Lelong numbers, complex singularity exponents, and Siu&apos;s semicontinuity   theorem

Qi'an Guan

Xiangyu Zhou

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Lelong numbers, complex singularity exponents, and Siu's semicontinuity theorem

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C. R. Acad. Sci. Paris, Ser. I 355 (2017) 415–419Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comComplex analysis/Analytic geometryLelong numbers, complex singularity exponents, and Siu’s semicontinuity theorem ✩Nombres de Lelong, exposants de singularités complexes et théorème de semi-continuité de SiuQi’an Guan a, Xiangyu Zhou ba School of Mathematical Sciences, and Beijing International Center for Mathematical Research, Peking University, Beijing 100871, Chinab Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 14 October 2015Accepted after revision 10 March 2017Available online 22 March 2017Presented by Jean-Pierre DemaillyIn this note, we describe a relation between Lelong numbers and complex singularity exponents. As an application, we obtain a new proof of Siu’s semicontinuity theorem for Lelong numbers.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cette note, nous décrivons une relation entre les nombres de Lelong et les exposants de singularités complexes. Comme application, nous obtenons une nouvelle preuve du théorème de semi-continuité de Siu pour les nombres de Lelong.© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet ϕ be a plurisubharmonic function near the origin o ∈ Cn . The Lelong number is classically defined asDefinition 1.1. ν(ϕ, o) := sup{c ≥ 0 : ϕ ≤ c log |z| + O (1)}.In [22], Siu established the semicontinuity theorem for Lelong numbers, namely, that the upper level sets of Lelong numbers of any closed positive current are analytic sets. A few years later, Kiselman [17] (see also [18]) generalized Siu’s semicontinuity theorem to directional Lelong numbers. In [3], Demailly introduced generalized Lelong numbers and ex-✩ The authors were partially supported by NSFC-11431013. The second author would like to thank NTNU for offering him Onsager Professorship. The first author was partially supported by NSFC-11522101.E-mail addresses: guanqian@amss.ac.cn (Q. Guan), xyzhou@math.ac.cn (X. Zhou).http://dx.doi.org/10.1016/j.crma.2017.03.0061631-073X/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.416 Q. Guan, X. Zhou / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 415–419tended the above result of Kiselman in this context. Later, Demailly (see [6]) gave a simple and completely new proof of Siu’s theorem by using the Ohsawa–Takegoshi L2 extension theorem.In this note, relying on our previous works [12–16], we present a new proof of Siu’s theorem by establishing a relation between Lelong numbers and complex singularity exponents.Let us first recall the definition of complex singularity exponents (also called log canonical thresholds by algebraic ge-ometers, see [21,19] et al.); for this, it is convenient to use the concept of multiplier ideal sheafI(ϕ)z0 :=⎧⎨⎩ f ∈ OCn,z0 | ∃V open  z0,∫V| f (z)|2e−2ϕ(z)dλ(z) < +∞⎫⎬⎭ ,introduced by Nadel [20] (see [4,6,23,25] et al.).Definition 1.2. The complex singularity exponent of ϕ at z0 is defined to becz0(ϕ) := sup{c ≥ 0 : I(cϕ)z0 = OCn,z0}.Our main result can be stated as follows:Theorem 1.3. Let ϕ be a plurisubharmonic function on an open set D ⊂Cn. Then for any k ∈N, k ≥ 1, there exists a plurisubharmonic function ϕk defined on a neighborhood of D × {o} ⊂Cn ×Ck with coordinates (z, w), such that(1) ϕk(z, o) = ϕ(z),(2) ν(ϕk, (z, o)) = ν(ϕ, z),(3) kν(ϕ,z) ≤ c(z,o)(ϕk) ≤ n+kν(ϕ,z)for any z ∈ D, where o is the origin in Ck. One can take for instance(4) ϕk(z, w) = supζ∈B(z,|w|)ϕ(ζ ).The reader will observe that the second inequality in (3) can be directly deduced by Skoda’s well-known estimate (see [24]) cz(ϕ) ≤ nν(ϕ,z) applied to ϕk at (z, o), and combined with (2).Remark 1.4. It is clear that property (3) in Theorem 1.3 is equivalent tokn + kν−1(ϕ, z) ≤ c(z,o)((n + k)ϕk) ≤ ν−1(ϕ, z),which implies{ν(ϕ, z) ≥ kn + k c}⊇{c(z,o)((n + k)ϕk) ≤ 1c}⊇ {ν(ϕ, z) ≥ c}.Since limk→+∞ kn+k = 1, we obtain{z|ν(ϕ, z) ≥ c} = ⋂k{z|c(z,o)((n + k)ϕk) ≤ 1c}.It is however well known that the sublevel sets {z|cz(ψ) ≤ a} of complex singularity exponents of any plurisubharmonic function ψ are analytic. This follows, e.g., from Berndtsson’s solution [1] of the openness conjecture (the conjecture was posed by Demailly and Kollár [7]; for a proof of the two-dimensional case, see [10,9,8]). In fact, this had been known since a long time as a consequence of the Hörmander–Bombieri theorem [11,2]. We conclude that the set {z|c(z,o)((n + k)ϕk) ≤ 1c }is analytic for any k ∈ N and c > 0, whence Siu’s semicontinuity theorem for Lelong numbers [22]:Corollary 1.5. (See [22].) {z|ν(ϕ, z) ≥ c > 0} is an analytic set.We refer to [6] and [5] for alternative proofs by Demailly. We would like to thank the referee for pointing out the Hörmander–Bombieri theorem.Q. Guan, X. Zhou / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 415–419 4172. Preparation2.1. Restriction formula for complex singularity exponents and Lelong numbersLet ϕ be a plurisubharmonic function on a neighborhood of the origin o ∈ Cn . In [7], the following restriction formula (“important monotonicity result”) about complex singularity exponents is obtained by using the Ohsawa–Takegoshi L2 ex-tension theorem.Proposition 2.1. (See [7].) For any regular complex submanifold (H, o) ⊂ (Cn, o), the inequalityco(ϕ|H ) ≤ co(ϕ) (2.1)holds whenever ϕ|H ≡ −∞.We recall the following (much easier to prove) restriction property of Lelong numbers.Lemma 2.2. (See [6].) For any regular complex submanifold (H, o) ⊂ (Cn, o), the inequalityν(ϕ|H ,o) ≥ ν(ϕ,o) (2.2)holds whenever ϕ|H ≡ −∞.2.2. Lelong number and complex singularity exponent for U (n) invariant plurisubharmonic functions on CnWe recall the following characterization of U (n) invariant plurisubharmonic function (see, e.g., Lemma III.7.10 in [6]).Lemma 2.3. Let ϕ be a plurisubharmonic function on a ball B(0, r) ⊂ Cn which is U (n) invariant. Then ϕ(z) = χ(log |z|), where χ : R→R is a convex increasing function.The following remark is a direct consequence of Lemma 2.3.Remark 2.4. Let ϕ be a plurisubharmonic function on a ball B(0, r) ⊂ Bn that is U (n) invariant. Then co(ϕ) = nν(ϕ,o) .Proof. By Definition 1.1, it is clear thatν(ϕ,o) = limt→−∞χ(t)t,i.e. for any ε > 0, there exists δ > 0 such that for any |z| < δ,(ν(ϕ,o) + ε) log |z| ≤ ϕ(z) ≤ (ν(ϕ,o) − ε) log |z|,and the present remark is deduced by looking at the integrability of e−cϕ . 2.3. A holomorphic change of variables for Lelong numbersTake a surjective linear map  : Ck →Cn (k ≥ n), and define a holomorphic map pk from Cn ×Ck such thatpk(z, w) = z + (w).Let ϕ be a plurisubharmonic function on D ⊂ Cn . Then the pull-back function p∗kϕ = ϕ ◦ pk is well defined on the open set p−1k (D) ⊂ Cn ×Ck .Lemma 2.5. For any z0 ∈ D, the pull-back function p∗kϕ satisfies by construction the following properties:(1) p∗kϕ(z0, o) = ϕ(z0);(2) ν(p∗kϕ, (z0, o)) = ν(ϕ, z0) = ν(p∗kϕ|z=z0 , (z0, o)),where the notation ν(p∗kϕ|z=z0 , (z0, o)) indicates that the Lelong number is computed along the submanifold {z = z0} = {z0} ×Ck.418 Q. Guan, X. Zhou / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 415–419Proof. (1) is obvious by definition of pk . In order to prove (2), it suffices to consider the case z0 = (0, · · · , 0) ∈ D . It is clear that p∗k log |z| = log |z + (w)| ≤ log(|z| + |w|) + O (1). Therefore, for any c > 0 satisfying ϕ(z) ≤ c log |z| + O (1) when z → 0, we havep∗kϕ(z, w) ≤ c log(|z| + |w|) + O (1),which implies ν(p∗kϕ, (z0, o)) ≥ ν(ϕ, z0). Conversely, if H ⊂ Ck is taken to be a linear subspace on which  : H → Cn is bijective, Lemma 2.2 yields the sequence of inequalitiesν(p∗kϕ, (z0,o)) ≤ ν(p∗kϕ|{z0}×Ck , (z0,o)) ≤ ν(p∗kϕ|{z0}×H , (z0,o)) = ν(ϕ, z0).The last equality comes from the invariance of Lelong numbers by linear changes of variable, which is obvious from Defini-tion 1.1. This implies (2), and the lemma is proved. 2.4. An invariance property of Lelong numbersLet ϕ be a plurisubharmonic function on a product domain  ⊂ Cn ×Ck containing the origin, and let (z, w) denote the coordinates. One can defineϕ̃(z, w) := supg∈U (k)ϕ(z, g w)on a U (k) invariant neighborhood of {w = o} in Ck . It is a plurisubharmonic function. By Definition 1.1, we immediately getLemma 2.6. The equalitiesν(ϕ̃, (z0,o)) = ν(ϕ, (z0,o)) and ν(ϕ̃|z=z0 , (z0,o)) = ν(ϕ|z=z0 , (z0,o))hold for any (z0, o) ∈ ( ∩ {w = o}).3. Proof of Theorem 1.3Let ϕ be a plurisubharmonic function on an open set D ⊂⊂ Cn . With the same notation as above, let us consider the plurisubharmonic functionϕk(z, w) := supg∈U (k)p∗kϕ(z, g w),which is defined on a neighborhood of D × {o} in Cn ×Ck . By Lemmas 2.6 and 2.5 (2), second equality, we haveν(ϕk|z=z0 , (z0,o)) = ν(p∗kϕ|z=z0 , (z0,o)) = ν(ϕ, z0). (3.1)However, by Remark 2.4, since w → ϕk(z0, w) is U (k)-invariant, it follows thatν(ϕk|z=z0 , (z0,o)) =kc(z0,o)(ϕk|z=z0). (3.2)From Proposition 2.1 and equalities (3.2), (3.1) respectively, one getsc(z0,o)(ϕk) ≥ c(z0,o)(ϕk|z=z0) =kν(ϕk|z=z0 , (z0,o))= kν(ϕ, z0).This completes the proof of Theorem 1.3.Remark 3.1. By taking  :Ck → Cn to be the projection onto the first n coordinates, one simply getsϕk(z, w) = supζ∈B(z,|w|)ϕ(ζ ), w ∈Ck. (3.3)In fact with such a choice of ϕk , Theorem 1.3 even holds for any k ≥ 1, as one can easily see.Q. Guan, X. Zhou / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 415–419 419References[1] B. Berndtsson, The openness conjecture for plurisubharmonic functions, arXiv:1305.5781.[2] E. 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Kollár, et al., Flips and abundance for algebraic threefolds, Astérisque 211 (1992).[20] A. Nadel, Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (3) (1990) 549–596.[21] V. Shokurov, 3-fold log flips, Izv. Russ. Acad. Nauk Ser. Mat. 56 (1992) 105–203.[22] Y.T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math. 27 (1974) 53–156.[23] Y.T. Siu, Multiplier ideal sheaves in complex and algebraic geometry, Sci. China Ser. A 48 (Suppl.) (2005) 1–31.[24] H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans Cn , Bull. Soc. Math. France 100 (1972) 353–408.[25] G. Tian, On Kähler–Einstein metrics on certain Kähler manifolds with C1(M) > 0, Invent. Math. 89 (2) (1987) 225–246.

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Lelong numbers, complex singularity exponents, and Siu's semicontinuity theorem

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