Luận án Tiến sĩ Toán học về ngưỡng chính tắc log và ngưỡng chính tắc log có trọng số

MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION TRINH TUNG THE LOG CANONICAL THRESHOLD AND THE WEIGHTED LOG CANONICAL THRESHOLD DOCTORAL DISSERTATION OF MATHEMATICS HANOI - 2024 MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION THE LOG CANONICAL THRESHOLD AND THE WEIGHTED LOG CANONICAL THRESHOLD DOCTORAL DISSERTATION OF MATHEMATICS Speciality: Mathematical Analysis Speciality Code: 9. LE MAU HAI HANOI - 2024 COMMITTAL IN THE DISSERTATION I assure that my scientific results are complete under the guidance of Prof. Le Mau Hai. All the presented results in the dissertation are completely honest and have never been published by others.

Publications with other au- thors have been approved by them to be included in the dissertation. I take full responsibility for my research results in the dissertation. Hanoi, August, 2024 The author Trinh Tung 1 ACKNOWLEDGMENTS This dissertation has been carried out at the Department of Mathematics, Hanoi National University of Education, Vietnam, under the enthusiasm of my supervisor, Prof. Le Mau Hai.

Firstly, I would like to express my very sincere thanks to Prof. Le Mau Hai for agreeing to direct my research. During these three years of the PhD process at Hanoi National University of Education, Prof. Le Mau Hai has inspired me with a passion for mathematical research.

I am very appreciative and grateful for the time that he devoted to me without counting, mathematics or not, during which he taught me a lot of things, even though I couldn’t understand it all. He spent a lot of time encouraging me and sharing all his ideas with me on the research topic. He advised and transmitted to me not only specialized knowledge but also academic culture. Learning and then doing mathematics alongside him was, for me, a real honor and a great pleasure! I am deeply grateful to Prof.

Nguyen Quang Dieu, Prof. Nguyen Xuan Hong, Prof. Phung Van Manh, and all the members of weekly seminare of Labora- tory of Functional Theory, Hanoi National University of Education, as well as colleagues, for their suggestions and for giving numerous valuable comments that improved the presentation of the dissertation. I would also like to thank Prof.

Pham Hoang Hiep for his encouragement and many useful discussions on my research results. A special gratitude goes to the EPU (235 Hoang Quoc Viet Rd., Bac Tu Liem Dist., Hanoi), where I am currently working and teaching, for creating favorable conditions throughout the period studied in the doctoral program. I am also sincerely grateful to IMU and TWAS for supporting my PhD studies through the IMU Graduate Breakthrough Fellowship. Last but not least, I would like to express my gratitude to my parents, my family, and my close friends for their constant love, unconditional support, and their belief in me.

No text or wording can give a true value for everything they did for me. Sincerely grateful to all of you! Trinh Tung 2 DOCTORAL DISSERTATION OF MATHEMATICS TRINH TUNG Contents Table of Notations 4 Introduction 5 Motivation and overview of researching issues. 5 Objectives and scientific significance of dissertation. 10 Structure the dissertation.

10 Chapter 1: Comparison principle for the weighted log canonical thresholds 12 1.1 Introduction, main results, and layout of chapter .2 Some auxiliary results .3 Proof of Theorem 1.4 Proof of Theorem 1. 22 Chapter 2: Estimates of level sets of holomorphic functions and applications to the weighted log canonical thresholds 23 2.1 Introduction, main results, and layout of chapter .2 Some properties of weighted log canonical thresholds .3 Level sets of holomorphic functions in one variable .4 Estimates on solutions of polynomials in one variable .5 Some computations for weighted log canonical thresholds .6 Continuity of weighted log canonical thresholds .7 Analyticity of sublevel sets of the log canonical thresholds. 50 Chapter 3: Relation between the Lelong number, log canonical thresholds, and intersection numbers of subextension and maximal subextensions in Cegrell’s class E(Ω).1 Introduction, main results, and layout of chapter .2 Some auxiliary results .3 Results on maximal subextension of psh functions .4 Proof of Theorem 3. 65 Conclusion and Future Work 67 List of Publications 68 References 73 3 Table of Notations N: the set of natural numbers Z: the set of integral numbers Q: the set of rational numbers R: the set of real numbers Rn : real vector space of dimension n C: the set of complex numbers Cn : complex vector space of dimension n B(z, r) := {z ∈ Cn : |z − a| < r}: the open ball of center z and radius r > 0 ∆n (0, δ): polydisk of center 0 and radius δ > 0 Ω: a domain (an open connected set) in complex vecter space V2n : the Lebesgue measure on Cn psh functions: abbreviated plurisubharmonic functions P SH(Ω): the set of psh functions on Ω P SH − (Ω): the set of negative psh functions on Ω M P SH(Ω): the set of maximal psh functions on Ω OX : the space of germs of holomorphic functions at a point in X cϕ (z): the log canonical threshold of a psh function ϕ at a point z cϕ,µ (z): weighted log canonical threshold of a psh function ϕ with respect weight µ at a point z νu (z): the Lelong number of a psh function u at a point z ej (u): the intersection numbers of a psh function u Lp (Ω) the set of p-th power integrable functions on Ω Lploc (Ω): the set of locally p-th power integrable functions on Ω L∞loc (Ω): the set of locally bounded functions on Ω ∂ − ∂¯ i d = ∂ + ∂¯; dc = ; and ddc = ∂ ∂¯ 4i 2 (ddc u)n = |ddc u ∧ ·{z · · ∧ ddc u}: the complex Monge-Ampère operator in Cn n−times ACC conjecture: ascending chain condition conjecture A ≈ B if and only if there exists a constant C > 0 such that A = CB 4 Introduction 1.

Motivation and overview of researching issues Plurisubharmonic (often abbreviated as psh) functions appear naturally and have many applications in several complex analysis and complex algebraic geometry or complex dynamics and introduced independently in 1942 by Le- long in France and Oka in Japan. Oka used them to define pseudoconvex sets and solved the Levi problem and ∂¯-equations in a two-dimensional complex domain. Meanwhile, Lelong established their first properties and asked influ- ential questions eg. potential; analytical sets; and positive closed currents., some of which remained open for decades.

These problems were eventually solved by Bedford and Taylor in 1976 and 1982, and nowadays they are the foundations of the pluripotential theory. One of the main subjects of interest in pluripotential theory is to investigate plurisubharmonic functions and prob- lems related to these functions. From the 70s of the last century, Bedford and Taylor constructed the Monge-Ampère operator (ddc ·)n on the class of locally bounded plurisubharmonic functions and studied the existence of solutions of the Dirichlet problem on bounded strictly pseudoconvex domains (refer [6]). After that, in the seminar work (see [8]) they established deep properties of plurisubharmonic functions on open sets in Cn.

Next, many essential results for these functions have been done by other authors, for example, Caffarelli-Kohn- Nirenberg-Spruck [13], Cegrell [14], [15], [16], Kołodziej [62], [64], Błocki [11], Guedj-Zeriahi [37], etc. Besides the above-mentioned important research direction, what also gains attention is investigating singularities of plurisubharmonic functions. The psh function ϕ is said to have a singularity at z if ϕ(z) = −∞. A natural ques- tion is: How to measure the magnitude (or “heaviness”) of the singularities of a plurisubharmonic function? To measure singularities of a psh function at a point a ∈ Ω, where Ω is an open subset in Cn , Lelong and some other mathematicians as Skoda [72], Siu [71], Kiselman [56] have introduced and studied the Lelong number νϕ (a) of the functions ϕ.

One of the outstand- 5 DOCTORAL DISSERTATION OF MATHEMATICS TRINH TUNG ing results concerning this quantity is due to Siu. In [71], Siu proved that for all α > 0, the upper-level sets of Lelong numbers of a psh function ϕ: Λ(ϕ, α) = {z ∈ Ω : νϕ (z) ≥ α} is an analytic set in an open subset Ω of Cn. An- other approach to measuring singularities of a plurisubharmonic function was introduced and investigated by Demailly and Kollár [29]. This is to study the complex singularity exponents (also known by other names as log canonical thresholds).

The first informal definitions of the notion for the log canonical threshold of psh functions were established long ago through the languages of commu- tative algebra and also in algebraic geometry, such as coherent ideal sheaves, divisors, Arnold multiplicity, etc [3], [4], [58], [60]. It was not until the 2000s of the last century that, this concept was further elucidated by mathematicians Demailly and Kollár through analytical methods [29]. Their approach was to look at the L1 - integrability of e−2cϕ in terms of the Lebesgue measure in some local coordinates and it has been stated as follows Definition (Demailly–Kollár’s original definition) Let X be a complex manifold and ϕ be a plurisubharmonic (psh) function on X. For any compact set K ⊂ X , the “complex singularity exponent” of ϕ on K to be the nonnegative number cK (ϕ) = sup c > 0 : exp(−2cϕ) is L1 on a neighborhood of K ,  and the “Arnold’s multiplicity” is defined accordingly λK (ϕ) = cK (ϕ)−1 : λK (ϕ) = inf λ > 0 : exp −2λ−1 ϕ is L1 on a neighborhood of K.


If ϕ ≡ −∞ near some connected component of K , we put cK (ϕ) = 0, and λK (ϕ) = +∞. The singularity exponent cK (ϕ) only depends on the singularities of ϕ, namely on the behavior of ϕ near its −∞ poles. Revolve around this con- cept, there have been many important conjectures established, one of which is the “Openness conjecture” (Remark 5.3) in [29] which implies the inclusion of a stronger statement (“Strong openness conjecture 5. Strong openness conjecture: “Let U 0 b U b X be relatively compact open sets in a complex manifold X.

Let ϕ be a psh function on X such that e−ϕ dV < +∞. R U Then there exists ε = ε (ϕ, U, U 0 ) such that for every ψ psh on X Z kψ − ϕkL1 (U ) < ε ⇒ e−ψ dV2n < +∞. U0 In other words, the integrability of e−ϕ near a given compact set K should be an open property for the L1loc (= weak) topology on P(X). Until now one only 6 DOCTORAL DISSERTATION OF MATHEMATICS TRINH TUNG establish a result of the weaker form to this conjecture.

This is there exists δ > 0 such that e−(1−δ)ψ dV2n < +∞ for kψ − ϕkL1 (U ) < ε = ε (ϕ, U, U 0 , δ)”. R U0 In 2005, this conjecture was proved in case 2-dimension by Favre-Jonsson [32]. For arbitrary dimension, an algebraic approach has been proposed by Jonsson and Mustaţă [54] to reduce the “openness conjecture” of Demailly- Kollár to a purely algebraic statement and then it was completely proved by Bo Berndtsson [9] in 2013. An alternative approach to this conjecture was given by Guan-Zhou [33], [34], [35] using a variant version of the L2 -extension the- orem in combination with curve selection lemma of ∂¯-equations.

Notable in 2014, Hiep proposed the concept of weighted log canonical threshold [49], to- gether with applying the original version of Ohsawa-Takegoshi’s theorem [68] to the members of a standard basis for multiplier ideal sheaves of holomorphic functions associated with a psh function to prove “strong openness conjecture” of Demailly–Kollár and proved the continuity of the weighted log canonical thresholds for a certain measure is µ = kzk2t dV2n with −n < t < +∞ in [50].