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Singularities in global pluripotential theory

Mingchen Xia

These contributions appear throughout the shared chapters.

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Original source commentary and bibliographical references accompany the corresponding Pome entries here.

1.1 Conventions

The function $\mathbb {R}\ni x \mapsto x^2$ is convex instead of concave. This is different from the convention in [CLS11].

5.7 Plurisubharmonic functions

The aim of this chapter is to recall the basic theory of plurisubharmonic functions and to fix the conventions used throughout the rest of the book. We assume only standard background in several complex variables and complex geometry, but we record a number of details whose normalization will matter later. General references for the material include [GZ17], [Dem12] and [DemBook].

5.7 Plurisubharmonic functions

We then discuss the plurifine topology. This topology is the natural one for many local arguments in pluripotential theory, especially those involving contact sets and non-pluripolar products. We include the Bedford–Taylor base theorem in a form suited to later applications, following the approach of [BT87] and [EMW06].

5.7 Plurisubharmonic functions

In this book, unlike in many references, we use the language of nets a lot. We remind the readers that subnets in the whole book mean Willard subnets. In particular, a subnet of a monotone net is still monotone. The readers can find more details about nets or Moore–Smith convergence in Willard's textbook [Wil70] or Kelley's [Kel75]. But one has to be careful that Willard's notion of subnets is stricter than Kelley's.

5.13.4 The higher dimensional case

See [Theorem 2.12, HK76] for a more general result.

Corollary 5.14.20

Let $\Omega \subseteq \mathbb {C}^ n$ be a domain and $\Omega '\subseteq \Omega $ be a subdomainNote. In [Proposition 1.30, GZ17], Guedj–Zeriahi assumed in addition that $\Omega '$ is relatively compact in $\Omega $. This assumption is not necessary for the proof, and the applications of this proposition in [GZ17] usually do not satisfy this condition.. Consider $\varphi \in \mathrm{PSH}(\Omega )$ and $\psi \in \mathrm{PSH}(\Omega ')$. Assume that

5.14 Properties of plurisubharmonic functionsNote. We remind the readers that the third displayed formula in the proof of [Theorem 1.48, GZ17] Step 2 is wrong. But this is irrelevant for our purpose.
5.14 Properties of plurisubharmonic functionsNote. In [Proposition 1.28, GZ17], the second part is only stated for sequences, the net version is obvious using the sub-mean value inequality and the monotone convergence theorem for nets as in [Proposition 7.12, Fol99].
5.14 Properties of plurisubharmonic functionsNote. The proof of [Proposition 1.40, GZ17] contains a gap: The measurability of $u$ is not established during the proof, so the claim $u \ast \chi _{\epsilon }$ converges to $u$ in $L^1_{\mathrm{loc}}$ is not valid. One needs to choose the sequence $(v_ j)_ j$ carefully so that $v\leq u$, and use the fact that $v \ast \chi _{\epsilon }$ converges to $v$ in $L^1_{\mathrm{loc}}$ instead.
5.15 Plurifine topology

In this section, we introduce the notion of plurifine topology. Unlike other sections in this chapter, this section contains full details. This is mainly due to the unfortunate omissions and numerous minor problems in the foundational paper [BT87]. In particular, the key theorem theorem 5.15.11 was claimed by Bedford–Taylor without proof. Our presentation largely follows [EMW06]. We also include enough details so that this section is readable for those who are not familiar with the classical potential theory.

5.15 Plurifine topology

Very recently, El Kadiri–Fuglede published a book [EKF] about classical fine potential theory, containing essentially all results in this section. The interested reader may consult their book for further details.

Definition 5.15.4

A subset $E\subseteq \Omega $ is thinNote. A more proper name would be plurithin. But since we will never need the classical notion of thin sets à la Cartan in this book, we prefer omitting the prefix pluri-. at $x\in \Omega $ if one of the following conditions holds:

Proposition 5.15.6 (Bedford–Taylor)Note. The original argument in [Proposition 10.2, BT82] was quite intriguing: Neither the auxiliary functions $\varphi _ j$'s nor the simple computations were correct. However, I believe that Bedford–Taylor had a correct proof in mind. Something more than a typo, but not yet a mistake, could be properly called a thinkpo, a terminology invented by R. Berman.
5.16 Lelong numbers and multiplier ideal sheaves

In this section, we briefly recall the notions of Lelong numbers and multiplier ideal sheaves. Our presentation is by no means intended to be complete. Readers are encouraged to read the textbooks [Section 2.3, GZ17] and [Dem12].

Proposition 5.16.22Note. In [Proposition 14.3, Dem12], Demailly used the highly non-standard notation $f^*\mathcal{I}(\varphi )$ to denote the image of $f^*\mathcal{I}(\varphi )\rightarrow \mathcal{O}_ X$, even when $f$ is not flat.
5.17 Quasi-plurisubharmonic functionsNote. In the proof of [Proposition 8.5, GZ17], there is a misleading sentence: This is a dcreasing sequence of $\omega $-psh functions, which should be removed.
5.20 Plurisubharmonic metrics on line bundles
Pour exprimer d'une manière frappante que le monument que j'élève sera placé sous l'invocation de la Science, j'ai décidé d'inscrire en lettres d'or sur la grande frise du premier étage et à la place d'honneur, les noms des plus grands savantsNote. Gaspard Monge, Comte de Péluse (1746–1818), known oddly by his family name instead of de Péluse, is one of the 72 names scribed on the Eiffel tower. Apart from being a mathematician, he was also a politician, active mainly after the French Revolution. qui ont honoré la France depuis 1789 jusqu'à nos jours. — Gustave Eiffel, 1889
13.1 Non-pluripolar products

The reader may consult [BEGZ10] and [DDNLsurv] for broader background. Our presentation is adapted to the later use of envelopes and singularity types, so some estimates are stated in a form slightly more specific than in the standard references.

13.10 Bedford–Taylor theoryNote. In [Proposition 3.3, GZ17], the authors claim that $\mathrm{d}u\wedge \mathrm{d}^{\mathrm{c}}u\wedge T$ is closed, which is wrong already in the smooth case.
Remark 13.11.12

for any closed positive current $T$. This is the relative non-pluripolar product introduced by Vu [Vu20]. Unlike the relative Bedford–Taylor products, the relative non-pluripolar products present some pathological behaviors. For example, they are not linear in general.

Definition 13.12.6

Note. It is very tempting to write $\lim _{i\in I} \operatorname{Cap}_{\theta }\left( \{ |f_ i-f|>\delta \} \right)=0$, as in [Definition 4.23, GZ17] for example. But the set where $f_ i-f$ is not defined is not a pluripolar set in general. Hence this abuse of notation is not acceptable. We sometimes write $f_ i\xrightarrow {\mathrm{C}} f$.

13.13 Properties of non-pluripolar products
Politiques et scientifiques ont le sens des réalités, mais ce ne sont pas les mêmes. Il en résulte — et ce sera là un principe que le général de Gaulle fera sien que l'activité de recherche ne peut être évaluée, quant à sa qualité propre, que par des hommes qui la pratiquent eux-mêmes. — Pierre LelongNote. Pierre Lelong (1912–2011) was the husband of another famous mathematician Jacqueline Ferrand. During their marriage (1947–1977), the latter published under the name of Jacqueline Lelong-Ferrand., 1999
Definition 13.13.18

Note. In the original article [DDNLsurv], the authors required that $\varphi \leq f$ everywhere. But in the proof of their Theorem 2.7 (theorem 13.13.24 below), they actually relied on the current definition. In the most relevant case, if $f$ is $\mathcal{F}$-upper semicontinuous (for example, if $f$ is upper semicontinuous), then these two definitions coincide. This is due to 5.15.20 and proposition 5.15.18.

13.13 Properties of non-pluripolar products

We shall say that $\theta _{P_{\theta }(f)}^ n$ is carried by the contact set $\{ P_{\theta }(f)=f\} $. Note. It is tempting to say that $\theta _{P_{\theta }(f)}^ n$ is supported on the contact set $\{ P_{\theta }(f)=f\} $, as people are already doing in the literature. It should be mentioned that this is an abuse of the language, since the support of $\theta _{P_{\theta }(f)}^ n$ (as a closed subset of $X$) could be much larger in general. One could probably introduce a notion of plurifine support, similar to what Fuglede did in the classical potential theory [Fug72].

16.1 The envelope operators

Several results about $P$-envelopes are developed by Darvas–Di Nezza–Lu in various papers and summarized in their survey [DDNLsurv]. The reader is invited to read their survey for more details and further results.

16.6.26 Properties of the $\mathcal{I}$-envelope
In den Dreißiger Jahren besuchte ich regelmäßig die Schweiz, teils um mich auch auf den Viertausendern zu tummeln, zum großen Teil aber auch, um Emigrantenblätter zu lesen und mich mit Kollegen über Naziverbrechen zu unterhalten. Aber auch die Schweizer schauten sich, wenn sie offen reden wollten, ebenso ängstlich um wie das bei uns üblich war.Note. The recent policy of ETH against Chinese students makes me feel that nothing has changed in Switzerland after the collapsing of Nazi for almost 80 years. — Oskar PerronNote. Oskar Perron (1880–1975), after earning an Eisernes Kreuz during WWI, obtained a position in München in 1922, initiating the glorious period there. Among his colleagues were Carathéodory, Tietze and Sommerfield.
17.10.40 The metrics on the spaces of potentials and rays
There are two principal ways to formulate mathematical assertions (problems, conjectures, theorems, …): Russian and French. The Russian way is to choose the most simple and specific case (so that nobody could simplify the formulation preserving the main point). The French way is to generalize the statement as far as nobody could generalize it further. — Vladimir ArnoldNote. Vladimir Igorevich Arnold (1937–2010), who became a professor at l'Université Paris IX after the dissolution of USSR, was always sick of France (so am I!). In the public lecture entitled “Sur l'éducation mathématique” in 1997, he invented the famous joke “Combien font $2+3$?” to question the French education system.
17.10.40 The metrics on the spaces of potentials and rays

See [Definition 5.9.1, EKF].

21.12 Toric pluripotential theory on ample line bundles

We assume familiarity with basic toric geometry at the level of [CLS11]. Some convex-analytic facts used below are recalled in section 21.11. The chapter can be skipped on a first reading if the reader is only interested in the non-toric theory, but it provides useful examples and a guiding model for several later constructions.

21.12 Toric pluripotential theory on ample line bundles

Finally, throughout the book convex functions are understood in the usual sense: the function $\mathbb R\ni x\mapsto x^2$ is convex. This convention is opposite to the convention often used in toric geometry, including parts of [CLS11]; care is therefore needed when comparing formulas.

21.12.1 Toric setup

Let $P\subseteq M_{\mathbb {R}}$ be a full-dimensional smoothNote. Recall that smooth means that for every vertex $v\in P$, if $w_ E$ denotes the first lattice point after $v$ as one traverses each edge $E$ of $P$ containing $v$, then $\{ w_ E-v\} _ E$ forms a basis of $M$. See [Definition 2.4.2, CLS11]. We also say $P$ is a Delzant polytope in this case. lattice polytopeNote. A lattice polytope in $M_{\mathbb {R}}$ is the convex hull of finitely many points in $M$..

21.12.1 Toric setup

Note that our support function differs from [Proposition 4.2.14, CLS11], where instead of a maximum, they took the minimum.

21.12.1 Toric setup

where $D_{\rho }$ is the toric prime divisor defined by $\rho $ under the orbit-cone correspondence [Theorem 3.2.6, CLS11].

21.12.1 Toric setup

Note. Be careful when you compare with other references, some people prefer $\log |z|$, $-\log |z|$ or $-\log |z|^2$ instead. This choice is fixed throughout the book. We have a canonical identification

Example 21.12.12

The Legendre transform is givenNote. While reading an advanced mathematical textbook/paper, I usually tend to trust the authors for their elementary computations. A few years ago, I was asked to present the result of a landmark paper written by two respected mathematicians on a conference. After spending a few days on the elementary integrals, I found out that all non-trivial constants in that paper were wrong. So I ask the readers to really verify this expression, if it is not obvious to you. by

21.12.6 Toric plurisubharmonic functions

Recall that $\mathrm{H}^0(X,L)$ can be identified with the vector space generated by $\chi ^ m$ for all $m\in P\cap M$, see [Proposition 4.3.3, CLS11]. In other words, a character $\chi ^ m$ of $T$ gives a global section of $L$ if and only if $m\in P$. This gives a beautiful characterization of the lattice points in $P$. The following theorem of Yi Yao gives an analogous characterization of the lattice points in the Newton body.

16.9.54 Continuity of invariants
Le but de cette thèse est de munir son auteur du titre de Docteur.Note. Similarly, the purpose of the current book is to make my complaints about France in the acknowledgments published. — Adrien DouadyNote. Adrien Douady (1935–2006) was a French mathematician known for his pioneering work in complex dynamics and fractal geometry. Along with John H. Hubbard, he proved important results about the Mandelbrot set and developed renormalization theory for polynomial mappings. He discovered the Douady Rabbit, a famous fractal Julia set. Douady studied at École Normale Supérieure (the place where I began to hate France, thanks to Claude Viterbo) and taught at several French universities. He was also a member of the Bourbaki group. Tragically, he died in a swimming accident in 2006. , at the beginning of his thesis
18.1.44 The volumes of Hermitian pseudo-effective line bundles
Some of our philosophers made a serious mistake: without having grasped the substance of the matter, they began to deny the significance of a new direction in science. — S. L. SobolevNote. Sergei Lvovich Sobolev (1908–1989) was a Soviet mathematician whose work on weak derivatives, generalized functions, and embedding theorems became foundational for modern analysis and partial differential equations. He also contributed to mathematical physics, numerical analysis, cubature formulas, and cybernetics, and was later central to the mathematical life of Akademgorodok in Novosibirsk., A. I. Kitov and A. A. Lyapunov in

Вопросы философии

20.2.29 Relation to the classical restricted volumes

These definitions extend the more classical definition in the algebraic setting due to [ELMNP09].

20.2.44 Analytic Bertini theorems
Comment se fait-il que M. Gauss ait osé vous faire dire que la plupart de vos théorèmes lui étaient connus et qu'il en avait fait la découverte dès 1808. Cet excès d'impudence n'est pas croyable de la part d'un homme qui a assez de mérite personnel pour n'avoir pas besoin de s'approprier les découvertes des autres. — Adrien-Marie LegendreNote. Adrien-Marie Legendre (1752–1833) was a French mathematician known for his foundational contributions to number theory, statistics, and mathematical analysis. Apart from his mathematical contributions, he also helped formalize the metric system during the French Revolution., in a letter to Jacobi in 1827
17.15 Operations on test curves
It is very fortunate that, unlike people who dig for gold, mathematicians can freely share their precious treasures with everybody. Once you understand something really well, it feels great to explain it to all. — Andrei OkounkovNote. Andrei Yuryevich Okounkov (1969–) is a Russian-American mathematician renowned for his contributions to representation theory. He was one of the key organizers of the ICM 2022 in St. Petersburg, which was unfortunately canceled under the indiscriminate discrimination against Russian citizens by the virtue signalers all over the western world after the war waged by the ruling class.
19.1 The theory of Okounkov bodies

We assume some familiarity with the classical Okounkov bodies. Excellent references are the original papers of Lazarsfeld–Mustaţă [LM09] and Kaveh–Khovanskii [KK12]; we give a rapid review here.

19.1 The theory of Okounkov bodies

Let $X$ be an irreducible smooth projective variety of dimension $n$ and $L$ a big holomorphic line bundle on $X$. Given an admissible flag $X=Y_0\supseteq Y_1\supseteq \cdots \supseteq Y_ n$ on $X$ (definition 19.3.2), one attaches to $L$ a natural $n$-dimensional convex body $\Delta (L)\subseteq \mathbb {R}^ n$, generalizing the Newton polytope construction in toric geometry (definition 21.12.27). This construction was first considered by Okounkov [Oko96, Oko03] and extended by Lazarsfeld–Mustaţă [LM09] and Kaveh–Khovanskii [KK12]. The convex body $\Delta (L)$ is the Okounkov body of $L$ with respect to the chosen flag. Concretely, the successive vanishing order along the flag defines a valuation $\nu \colon \mathbb {C}(X)^{\times }\to \mathbb {Z}^ n$, and

19.1 The theory of Okounkov bodies

is a semigroup whose closed convex cone, intersected with the slice $\{ (x,1)\} $, is $\Delta (L)$. Lazarsfeld–Mustaţă [LM09] showed that $\Delta (L)$ depends only on the numerical class of $L$, and Jow [Jow10] showed that the Okounkov bodies for all flags determine this class, so $\Delta (L)$ may be regarded as a universal numerical invariant of $L$.

Remark 19.5.36

Also note that this result extends [Theorem 3.4, Jow10] and hence gives simpler proofs of [Theorem A, Theorem B, Jow10].

19.6 Okounkov test curves
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: There is no permanent place in this world for ugly mathematics. — Godfrey Harold HardyNote. Godfrey Harold Hardy (1877–1947) was a British mathematician famous for his work in number theory and mathematical analysis. Apart from his research, Hardy was a strong advocate for pure mathematics and believed that mathematics should be pursued for its own beauty, not just for practical use. He remained lifelong unmarried and dedicated much of his life entirely to mathematics, fitting into the common stereotype of a mathematician.
Remark 20.1.50

The algebraic nef b-divisors are introduced by Dang–Favre [DF20], as their intersection theory. It is a straightforward application of the results proved in this section that our transcendental intersection theory coincides with theirs in the algebraic setting.

21.13.1 Toric setup

be a $T$-invariant integral big divisor on $X$. Let $P_ D\subseteq M_{\mathbb {R}}$ be the following polytopeNote. Note that $P_ D$ is not necessarily a lattice polytope, see [Example 10.5.4, CLS11]. In fact, any rational polytope with positive volume can be realized in this manner.

Example 21.13.36

Note that it differs from the polytope in example 21.13.35.Note. Although these examples are almost trivial, they did confuse me a lot at the beginning of 2023, when Kewei Zhang, Tamás Darvas and I were collaborating on [DRWNXZ]. At that time, Kewei himself already proved the main theorem for a generic flag. I realized that some simple birational geometry would suffice to prove the same result for general flags. I persuaded myself and Kewei that the Okounkov bodies are always birationally invariant, and deduced some apparently wrong conclusions. I got no clue for a couple of weeks, then one day, on the noisy metro line 7 of Paris, I got nothing to do, so I said to myself: Why not compute the simplest toric examples? Then after a few minutes, all of a sudden, the whole picture became completely clear.

21.13.31 Partial Okounkov bodies

As an interesting consequence of (21.13.51), we have a geometric description of the divisorial Zariski decomposition of [Bou02] in the toric setting: The negative part of $D$ is given by

21.13.52 The pluripotential theory
A good theorem lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect. — John TateNote. John Torrence Tate Jr. (1925–2019), the grandfather of Dustin Clausen, was one of the greatest minds in the whole history of America. However, his aversion to publishing papers arguably impeded the progress of the development of mathematics to some extent. For example, his foundational work on rigid non-Archimedean geometry was written in 1962, but was not available to the public until 1971.
Theorem 21.13.53

There are canonical bijectionsNote. In the earlier version of this book, I required additional conditions in 2 and 3, namely $F\preceq F_{V_{\theta }}$ and $G\succeq G_{V_{\theta }}$ respectively. These conditions can be removed, since $G_{V_{\theta }}$ is always bounded, as suggested by Robert Berman. See theorem 21.13.45. between the following sets:

22.1 Non-Archimedean pluripotential theory

A related approach has been developed independently by Mesquita-Piccione [MP24], where a Berkovich-like analytification of a compact Kähler manifold is constructed. The present chapter is logically based on the test-curve formalism of section 17.11, especially the $\mathcal I$-model theory of section 17.14 and the operations of section 17.15. It is largely independent of section 21.13, section 18.28, while the comparison with Boucksom–Jonsson uses the non-Archimedean theory recalled in subsection 22.1.64.

22.1.65 A brief recap of Boucksom–Jonsson's theory

In this section, we briefly recall the non-Archimedean global pluripotential theory à la Boucksom–Jonsson [BJ22]. Our presentation is selective, so the original paper remains the best companion for this section.

22.1.66 Valuations

See [Berk93] for more details.

22.1.70 Non-Archimedean plurisubharmonic functions

Let $X$ be an irreducible complex projective variety of dimension $n$ and $L$ be a holomorphic pseudo-effective $\mathbb {Q}$-line bundle on $X$. Through the GAGA morphism $X^{\mathrm{An}}\rightarrow X$ of ringed spaces, $L$ can be pulled back to an analytic line bundle $L^{\mathrm{An}}$ on $X$. The purpose of this section is to study the psh metrics on $L^{\mathrm{An}}$. We will follow the approach of [BJ22], which avoids the direct treatment of $L^{\mathrm{An}}$ itself.

22.1.70 Non-Archimedean plurisubharmonic functions

Following [Definition 2.18, BJ22], we define $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}(L^{\mathrm{An}})$, the set of (rational) generically finite Fubini–Study functions $\phi \colon X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ that are of the following form:

Definition 22.1.71

A plurisubharmonic metric (or psh metric for short) on $L^{\mathrm{An}}$ is a function $\phi \colon X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ that is not identically $-\infty $, and is the pointwise limit of a decreasing net $(\phi _ i)_{i\in I}$, where $\phi _ i\in \mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}(L_ i^{\mathrm{An}})$ for some $\mathbb {Q}$-line bundles $L_ i$ on $X$ satisfying $c_1(L_ i)\to c_1(L)$ in $\mathrm{NS}^1(X)_{\mathbb {R}}$.

Conjecture 22.1.76

Assume that $X$ is unibranch. Then every bounded from above increasing net of elements in $\mathrm{PSH}(L^{\mathrm{An}})$ converges in $\mathrm{PSH}(L^{\mathrm{An}})$.

22.1.91 The algebraic setting

Next we point out a version of Siu's uniform global generation lemma [Siu88] that we will need in the proof of our next theorem:

22.1.101 The comparison theorem
I speak twelve languages: English is the bestest. — Stefan BergmanNote. Stefan Bergman (1895–1977), bearing a very Scandinavian name, was a Polish-American mathematician best known for his work in complex analysis, especially in several complex variables. He introduced the Bergman kernel, a fundamental concept in complex analysis that has influenced many areas of mathematics and theoretical physics. Bergman was born in Poland (then part of the Russian Empire), and studied in Berlin. He fled Europe during World War II and eventually settled in the United States.
18.28 Partial Bergman kernels

The main result is the convergence theorem theorem 18.28.49. Given a $\theta $-psh potential $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, a closed non-pluripolar weighted subset $(K,v)$, and a Bernstein–Markov measure $\mu $, the normalized partial Bergman measures on $X$ converge weakly to the partial equilibrium measure of $(K,v,\varphi )$ as $k\to \infty $. This is the relative version of the theorem of Berman–Boucksom–Witt Nyström [BBWN11], corresponding to the case $\varphi =V_\theta $. The assumptions are essentially the mildest ones under which the statement makes sense.

21.11 Convex functions and convex bodies

This appendix collects the basic facts about convex functions and convex bodies that are used throughout the book. Our standard reference is [Roc70]. All results stated for convex functions apply equally to concave functions, after replacing the function by its negative.

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