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Singularities in Hermitian pluripotential theory

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1.1 Conventions

We follow the normalizations of [SGPT]:

5.24 Maxima and the maximum principle

For a function $f$, we write $f^*(x)=\limsup _{y\to x}f(y)$ for its upper semicontinuous regularization. A subset of a complex manifold is pluripolar if each of its points has a neighborhood $U$ such that the part of the subset in $U$ is contained in $\{ p=-\infty \} $ for some $p\in \mathrm{PSH}(U)$. The local upper-envelope theorem recalled in [SGPT] states that if a nonempty family of plurisubharmonic functions is locally uniformly bounded above, then the regularization of its supremum is plurisubharmonic and differs from the supremum only on a pluripolar set.

5.25 Normalized compactness

Fix a smooth positive probability volume form $\mathrm{d}V$ on $X$, used for $L^1$ norms. Consider a sequence of plurisubharmonic functions on a connected domain that is locally uniformly bounded above. By the local compactness theorem recalled in [SGPT], it has a subsequence that either converges in $L^1_{\mathrm{loc}}$ or converges locally uniformly to $-\infty $. For normalized $\theta $-plurisubharmonic functions on $X$, a point at which the maximum is attained excludes the second alternative, and connectedness of $X$ propagates the first one across $X$.

5.27 Local products and capacity

Bedford–Taylor products of bounded plurisubharmonic functions, and the capacity defined through them, measure the exceptional sets of plurisubharmonic functions and control bounded approximation. We recall these classical facts of local pluripotential theory from [SGPT, GZ17]. The plurifine topology is the coarsest topology that makes all local plurisubharmonic functions continuous. It is finer than the ordinary topology; in particular, strict superlevel sets of quasi-plurisubharmonic functions are plurifine open.

5.27 Local products and capacity

Both are classical results of local pluripotential theory [Chapters 3 and 4, GZ17]; the conventions for bounded products and capacity in [Chapter 2, SGPT] agree with these up to fixed constants.

5.27 Local products and capacity

A function is quasi-continuous if, for every $\epsilon >0$, it is continuous outside an open set of capacity less than $\epsilon $. Plurisubharmonic functions have this property for the local Bedford–Taylor capacity, and a Borel set has zero capacity if and only if it is pluripolar. A locally uniformly bounded monotone sequence of plurisubharmonic functions converges in capacity: to its limit if it decreases, and to the upper semicontinuous regularization of its limit if it increases. If uniformly bounded plurisubharmonic functions converge in capacity, their local products converge weakly, and this remains true after the products are multiplied by uniformly bounded quasi-continuous coefficients converging in capacity. These local results, recalled in [SGPT, GZ17], also apply to quasi-plurisubharmonic functions after a common smooth correction.

5.27 Local products and capacity

The comparison principle of Bedford and Taylor also has mixed forms, recalled in [SGPT]. Let $a,b$ be locally bounded plurisubharmonic functions on $U$ such that $E=\{ a<b\} \Subset U$, and let $S$ be a positive closed $(n-1,n-1)$-current that is a product of Hessians of bounded plurisubharmonic functions and smooth closed positive forms. Then

5.27 Local products and capacity

The local envelope theorem for bounded obstacles, recalled in [SGPT, GZ17], shows where an envelope can carry Monge–Ampère mass. Let $B$ be a ball with a smooth strictly plurisubharmonic defining function $h<0$, and let $f$ be a bounded quasi-continuous obstacle with $Ah\leq f\leq 0$ for some $A>0$. Then the envelope

5.27 Local products and capacity

The gradient term, and its products with positive Bedford–Taylor factors of bounded potentials, are nonnegative. These identities belong to the bounded local calculus recalled in [SGPT, GZ17]; they follow by decreasing regularization, and bounded potentials have the local Sobolev regularity needed to define the gradient term. In particular, inequalities derived from this formula survive local smoothing with a lower curvature error tending to zero.

5.34 Analytic types and restriction

By the local growth theorem recalled in [SGPT], the limit exists and does not depend on the coordinates or on the correction $q$. With this normalization, $\log |z|^2$ has Lelong number one at the origin. For an irreducible analytic subset $Z$, the generic Lelong number of $u$ along $Z$ is

5.34 Analytic types and restriction

For analytic singularities we follow the convention of [SGPT]: we say that a quasi-plurisubharmonic function $u$ has analytic singularities if near every point there are holomorphic functions $f_1,\ldots ,f_ N$, not all identically zero, and a rational number $c>0$ such that

5.34.2 Lelong level sets

Siu's analyticity theorem states that, for every quasi-plurisubharmonic function $u$ and every $c>0$, the upper level set $\{ x:\nu (u,x)\geq c\} $ is an analytic subset. After a smooth local correction, this is the theorem for closed positive currents, which we use in the form recalled in [SGPT]; see section 25.3 for the original source.

5.36 The global curvature bound

This is the form of [Theorem 14.12, DemAM11] used in this book. The local generators in the construction are the weighted Bergman generators of $\mathcal I(mu)$; smooth local corrections do not change this ideal. On overlaps the local Bergman sums are comparable, so the glued logarithm keeps the analytic type $(1/m,\mathcal I(mu))$. The construction freezes the continuous lower bound $\gamma $ on sufficiently small coordinate balls, and its curvature error combines the modulus of continuity of $\gamma $ with an overlap term that tends to zero as $m$ increases. It is this global curvature estimate, and not merely the local convergence of proposition 5.35.1, that we take from the theorem.

5.36.3 Regularization of closed currents

We also use the regularization of closed currents in the form recalled in [SGPT]. Let $T=\eta +\mathrm{dd}^{\mathrm{c}}v$ be a closed real $(1,1)$-current, where $\eta $ is a smooth closed form and $v$ is quasi-plurisubharmonic, and set $\nu (T,x)=\nu (v,x)$. Suppose that $T\geq \gamma $ for a smooth real $(1,1)$-form $\gamma $. Smooth regularization gives smooth representatives $T_ j=\eta +\mathrm{dd}^{\mathrm{c}}v_ j$ with

5.37 Multiplier ideals and logarithmic poles

For a logarithmic pole along a submanifold, weighted integrability of a holomorphic germ is determined by its vanishing order in the normal directions. For general singularities, the valuative characterization of multiplier ideals and the strong openness theorem, both recalled in [SGPT], make the comparison of multiplier ideals compatible with maxima, sums and increasing limits.

5.37 Multiplier ideals and logarithmic poles

In the first item, the neighborhoods may be shrunk as necessary; we never need to extend a local modification to a modification of a compact manifold. The second item is the increasing-limit form of the strong openness theorem used in [SGPT]. Since adding a smooth function changes neither multiplier ideals nor Lelong numbers, both items apply to quasi-plurisubharmonic functions after the local correction of proposition 5.22.2.

5.37 Multiplier ideals and logarithmic poles

Let $v,w$ be quasi-plurisubharmonic functions on an open set, and let $Y$ be a complex submanifold on which $v|_ Y$ is not identically $-\infty $. The restriction and subadditivity theorems give the two inclusions recalled in [SGPT] and [Section 15, DemAM11]:

5.38 Lelong numbers and divisorial estimates

We use the following local consequence of Siu's decomposition theorem, recalled in [SGPT] and [DemAM11]. If $T$ is a closed positive $(1,1)$-current and $E$ is a prime divisor, the restriction of (the coefficient measures of) $T$ to $E$ equals the generic Lelong number of $T$ along $E$ times $[E]$. In particular, removing finitely many of these divisorial components from $T$ leaves a positive current. Distinct prime divisors do not contribute overlapping mass, since their intersections have codimension at least two. All these are local statements about closed currents.

Lemma 5.38.8

Every point of the smaller ball is the center of a Euclidean ball of a fixed radius contained in the intermediate ball. The local Lelong monotonicity formula, recalled in [SGPT], bounds the Lelong number at this center by a fixed multiple of the trace mass of the intermediate ball. The finite cover gives one bound on $Z$. Since smooth additions and additive constants do not change Lelong numbers, this proves the assertion for the original potentials. $\square$

5.41.4 Pointwise Legendre transforms

The minimum principle can also show that a supremum is already upper semicontinuous. For a concave family of potentials, the regularized Legendre transform is a potential on a product. The minimum principle identifies its inverse transforms with the original family everywhere, and applying the one-variable duality once more identifies the regularization with the pointwise supremum. We first state this duality precisely, including its behavior at the endpoints; the general Legendre–Fenchel theorem is recalled in [appendix on convex analysis, SGPT].

Lemma 5.46.4

Proof. On a small coordinate ball, take plurisubharmonic local potentials $a,b$ with $T=\mathrm{dd}^{\mathrm{c}}a$ and $S=\mathrm{dd}^{\mathrm{c}}b$, by [Chapter III, Proposition 1.19, p. 135, DemCADG]. Put $h=a-b\in L^1_{\mathrm{loc}}$. The trace identity gives $\mathrm{dd}^{\mathrm{c}}h\wedge \eta ^{n-1}=0$ in the sense of distributions. Dividing by the smooth positive volume form $\eta ^ n$ gives a scalar second-order operator with smooth coefficients whose principal matrix is a positive multiple of the inverse of the Hermitian matrix of $\eta $; it is uniformly elliptic on every smaller ball. Local elliptic regularity therefore makes $h$ smooth. Thus $T-S=\mathrm{dd}^{\mathrm{c}}h$ is smooth and carried by a Lebesgue null set, so its coefficient measures vanish. This proves equality locally, and hence globally. $\square$

11.37 Entropy and the closed case

Suppose now that $\theta $ is a Kähler form. By the closed-background theory of [SGPT], all potentials of the ordinary type of $\varphi $ have the same non-pluripolar mass $m_\varphi $, which is positive under the standing hypotheses. For $F\in L^\infty (\mathrm{d}V)$, integrating the equation gives

18.3 The Kähler characterization[Section 6.2, SGPT]
18.3 The Kähler characterization[Theorem 7.1.1, SGPT]
18.4.7 Analytic and divisorial singularities

For a potential $a$ with analytic singularities, the multiplier-ideal order controls the ordinary order: by lemma 5.38.11, every quasi-plurisubharmonic $v$ with $v\preceq _{\mathcal I}a$ satisfies $v\preceq a$. The ideal envelope of $a$ is then a competitor for its ordinary envelope, and the two envelopes coincide. This extends [Proposition 3.2.10, SGPT] to smooth backgrounds.

Theorem 18.6.4

Let $Y$ be a compact connected Riemann surface and $\theta $ a smooth positive form of total mass one. Fix a smooth positive probability volume form on $Y$, again denoted by $\mathrm{d}V$, and use $\mathcal K_\theta $, the envelopes and the ideal metric on $Y$ in place of $X$. Then the ideal models with finitely many logarithmic poles and zero non-pluripolar mass are dense in $\mathcal K_\theta $ for the $L^1$ topology. The logarithmic coefficients can be chosen rational, so these approximants have analytic singularities in the convention of [SGPT]. Consequently the completion in proposition 18.6.1 is all of $\mathcal K_\theta $, and the metric on ideal classes is not complete.

18.6.5 Mass and capacity convergence

For comparison, let $d_{S,\theta }$ denote the singularity pseudometric $d_ S$ recalled at the beginning of the chapter, computed in the background $\theta $. Since $r_ j\leq 0$, the ordered-distance formula of [Lemma 6.2.2, SGPT] applies and gives

18.7.4 Dyadic regularization[Theorem 7.1.1, proof of $(3)\Rightarrow (1)$, Step 1, SGPT]
17.17 Test curves and Legendre duality

The transforms are the pointwise Legendre transforms of subsection 5.41.4, and the proofs follow the envelope arguments for the Ross–Witt Nyström correspondence in [SGPT].

20.3.5 Comparison with ordinary restriction

which is locally integrable. Thus all multiplier ideals of $u$ are trivial and the trace data vanishes, whereas $u|_ Y=\log |w|^2$ has Lelong number one at the origin. This example is taken from [Example 8.1.2, SGPT].

Proposition 20.3.17

Finally, suppose that $X$ is Kähler. The Kähler trace is defined in [Definition 8.1.1 and Proposition 8.1.2, SGPT] through a quasi-equisingular approximation of $u$ with analytic types $(2^{-j},\mathcal I(2^ ju))$, whose restriction to $Y$ has the ordinary type of $w_{2^ j}$. The same distance calculation shows that these restrictions converge to $w$ for $d_{S,A\eta }$, so $w$ represents the Kähler trace up to $P$-equivalence. $\square$

20.3.16 Singularity distance on a curve[Definition 6.2.1, SGPT]
20.3.16 Singularity distance on a curve[Section 8.1, SGPT]
19.8.8 Bounded weighted sections[Section 10.3, SGPT]
19.9.5 The body of a graded section algebra

We use the following semigroup counting theorem, recalled in [Theorem C.2.1, SGPT]. Let $r\geq 0$, and let $\Sigma \subset \mathbb Z^{r+1}$ be an additive semigroup that generates $\mathbb Z^{r+1}$ as a group and lies in a closed convex cone in $\mathbb R^ r\times \mathbb R_{\geq 0}$ whose height-one slice is compact and whose only point at height zero is the origin. Then the number of points of $\Sigma $ at height $m$, divided by $m^ r$, converges to the Euclidean volume of the height-one slice of the closed convex cone generated by $\Sigma $.

20.4.9 Bott–Chern classes and closed currents

Bott–Chern cohomology records a closed $(1,1)$-current modulo $\mathrm{dd}^{\mathrm{c}}$ of global functions, and so separates the smooth background of the current from its potential. When $\theta $ is closed, subtracting the trace current from the pulled-back background gives a residual class on each model, and compatibility with pushforward organizes these classes into a b-divisor class in the sense of [section The notion of b-divisors, SGPT].

20.4.9 Bott–Chern classes and closed currents

When the models are Kähler, the $\partial \bar\partial $-lemma identifies this cohomology with the real $(1,1)$-cohomology used in [section The notion of b-divisors, SGPT], and the definition agrees with the one given there.

Theorem 20.4.15

On a Kähler manifold, with Kähler models, it agrees with the b-divisor class of the same current in [subsection The b-divisors of currents, SGPT].

16.29.12 Changes of representative and limiting mass

For closed backgrounds on a compact Kähler manifold, [chapter “Non-Archimedean pluripotential theory”, section “The definition of non-Archimedean metrics”, SGPT] indexes such systems by Kähler perturbations, as above, and defines their volume as a limit of the masses of the initial potentials. Its components are general test curves fixed by the ideal envelope, with initial potentials of positive mass, whereas our curves are constant on a prescribed left half-line and our system spaces impose no mass condition. For nonclosed positive backgrounds, the corresponding limit of masses is not invariant under the gauge bijections.

18.30 Selected spaces and partial Bergman kernels[Section 14.3, SGPT]
10.14 Null loci and numerical restricted volumes[Chapter 8, SGPT]
10.24 An incomplete metric in a big class

Finite energy refers to the classical closed theory of [SGPT]. For a smooth closed real $(1,1)$-form $\theta $ with big class on a compact Kähler surface $Y$, and for $u\in \mathrm{PSH}(Y,\theta )$ with minimal singularities, the Monge–Ampère energy relative to the envelope $V_\theta $ is

23.2.1 The Lee form and the Kähler covering

Thus $\chi $ is a homomorphism from the deck group to the multiplicative group of positive real numbers, which records how the Kähler metric is rescaled. Forms on $\widetilde X$ transforming by $\chi $ may equivalently be viewed as forms on $X$ with values in the flat real line bundle defined by $\chi $, the weight bundle. In this sense an LCK form is closed as a form with values in the weight bundle [Section 3.4, OV24LCK].

23.2.18 Ordinary and twisted potentials

This is Bott–Chern cohomology with values in the weight bundle [Section 18.5, OV24LCK]. It differs both from ordinary Bott–Chern cohomology and from the quotient of all backgrounds by ordinary $\mathrm{dd}^{\mathrm{c}}$-terms considered in section 10.30.

23.2.25 Volume equations in Gauduchon gauge

This is the uniqueness problem posed in [Question 47.56(3), OV24LCK], with the identification of classes made explicit. The question concerns metrics: uniqueness of a scalar potential $u$ would additionally require control of the kernel of $D_\lambda $. The counterexamples with degenerate densities in section 14.20 concern ordinary potentials and do not decide this question.

13.17 Comparison of singularity volumes

If $\theta $ is closed, both volumes equal $\int _ X\mathrm{MA}_\theta (u)$, because in the closed theory the non-pluripolar mass depends only on the singularity type [SGPT]. For a nonclosed background the two volumes can differ even at $u=0$, as example 11.1.12 shows.

9.11 Potentials and equations on unibranch spaces

We use three facts about unibranch spaces. Resolutions and common smooth refinements of two resolutions exist, and a proper bimeromorphic map to a unibranch space has connected fibers [Theorems B.1.1–B.1.3, SGPT]. The Fornaess–Narasimhan criterion identifies plurisubharmonic functions on $X$ as the upper semicontinuous functions that are subharmonic on holomorphic disks [Theorem B.2.1, SGPT]. The ramified disk lift used below is the argument in [proof of Theorem B.2.2, SGPT].

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