The metric geometry of singularity types

Chinh H. Lu; Eleonora Di Nezza; Tam\'as Darvas

REVIEW 1 major objections 5 minor 1 cited by

The metric geometry of singularity types

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper introduces a pseudometric on singularity types that becomes a complete metric on positive-mass subspaces, with atoms equal to relative full mass classes.

desk verdict A genuinely useful metric on singularity types, with solid positive-mass theorems, but the abstract overstates the zero-mass atom characterization by relying on the still-open equality P=C. read the letter →

arxiv 1909.00839 v1 pith:Y3EBVUTB submitted 2019-09-02 math.DG math.CV

Tamás Darvas , Eleonora Di Nezza , Chinh H. Lu This is my paper · ORCID

classification math.DGmath.CV MSC 32U0532Q1532W2053C55

keywords singularitytypestheta-plurisubharmonicfunctionsbigcohomologyclassesgeodesicraysmodelpotentialscomplexMonge-AmpèreequationsmultiplieridealsheavesKählermanifolds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper gives singularity types of plurisubharmonic potentials in a big cohomology class a natural pseudometric $d_{\mathcal S}$, built from finite-energy geodesic rays. It shows that two types are at zero distance exactly when they have the same relative full mass class, so the degenerate directions of the pseudometric are completely understood. On subspaces where the total Monge-Ampère mass is bounded below by $\delta>0$, the pseudometric is complete. This metric topology makes precise what it means for prescribed singularity types to vary, yielding convergence of Monge-Ampère solutions in capacity and a semicontinuity statement for multiplier ideal sheaves.

What carries the argument

The engine is the map $r[\,\cdot\,]:\mathcal S(X,\theta)\to\mathcal R(X,\theta)$ that attaches to each singularity type the geodesic ray with minimal singularity type, along with the ceiling operator $C$ whose fixed points are the model potentials. The metric is the chordal limit $d_{\mathcal S}([\psi],[\chi])=\lim_{t\to\infty} d_1(r[\psi]_t,r[\chi]_t)/t$, and the paper repeatedly uses the identity $P[\psi]=C(\psi)$ for positive mass, so model potentials are exactly envelope-fixed points. This machinery turns singularity-type questions into geodesic-ray questions, where completeness is already available.

What would settle it

Construct a $d_{\mathcal S}$-Cauchy sequence $[u_j]$ in $\mathcal S_\delta(X,\theta)$ whose decreasing envelope limit $v=\lim_j C(v_j)$ is not a fixed point of $C$ or has total mass below $\delta$; Theorem 1.1 predicts that no such sequence can exist. On the elliptic ruled surface example showing incompleteness for $\delta=0$, the analogous sequence loses mass precisely at the limit, so the comparison is explicit.

Watch

Extended reading notes

Core claim

The paper establishes that $\mathcal S(X,\theta)$ admits a pseudometric $d_{\mathcal S}$ whose zero classes are precisely the relative full mass classes: $d_{\mathcal S}([\psi],[\chi])=0$ iff $C(\psi)=C(\chi)$, equivalently the associated geodesic rays coincide. On $\mathcal S_\delta(X,\theta)$, the subspace of types with $\int_X\theta_u^n\ge\delta>0$, this pseudometric is complete. The paper further proves a volume diamond inequality $\int_X\theta_u^n+\int_X\theta_v^n\le \int_X\theta_{\max(u,v)}^n+\int_X\theta_{P(u,v)}^n$ when $P(u,v)\in\mathrm{PSH}(X,\theta)$, and derives from it two applications: $d_{\mathcal S}$-convergence forces eventual inclusion of multiplier ideal sheaves, and solutions of complex Monge-Ampère equations with prescribed singularity converge in capacity when the prescribed types converge.

Load-bearing premise

The completeness proof assumes an imported identity: for every positive-mass potential, the envelope $P[\psi]$ equals the ceiling $C(\psi)$; if that external theorem were false, the Cauchy-sequence limit argument in Theorem 4.9 would not go through.

Editorial extensions

If this is right

Increasing sequences of $\theta$-psh potentials satisfy $d_{\mathcal S}([u_j],[u])\to0$, and $d_{\mathcal S}$-convergence can be characterized by sandwiching a sequence between increasing and decreasing approximating sequences.
For any $\delta>0$, every $d_{\mathcal S}$-Cauchy sequence in $\mathcal S_\delta(X,\theta)$ converges to a singularity type whose total Monge-Ampère mass is at least $\delta$.
The volume diamond inequality $\int_X\theta_u^n+\int_X\theta_v^n\le\int_X\theta_{\max(u,v)}^n+\int_X\theta_{P(u,v)}^n$ holds whenever the rooftop $P(u,v)$ is $\theta$-psh, and it is an identity in complex dimension one.
If $d_{\mathcal S}([u_j],[u])\to0$, then the multiplier ideal sheaf inclusion $\mathcal J[u]\subseteq\mathcal J[u_j]$ holds for all sufficiently large $j$.
Prescribed singularity types that converge in $d_{\mathcal S}$ make the solutions of the corresponding complex Monge-Ampère equations converge in capacity and in $L^1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the $d_{\mathcal S}$-topology gives a quantitative notion of approximation for singularity types, so the effectiveness of kernel-type or envelope-type approximation schemes could be measured by rates of $d_{\mathcal S}$-convergence rather than only by stabilization of cohomological invariants.
Editorial inference: if the zero-mass identity $P[\psi]=C(\psi)$ stated as Conjecture 2.5 holds, the completeness and diamond arguments may extend below the positive-mass threshold, potentially making selected subspaces of $\mathcal S(X,\theta)$ complete even when the full space is not.
Editorial inference: a local analog of $d_{\mathcal S}$ on singularity germs of plurisubharmonic functions would turn the multiplier-ideal semicontinuity theorem into a quantitative strong-openness statement, though the paper only records this as a motivating question.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something real: it gives the first metric topology on S(X,θ), the space of singularity types, and the main completeness theorem for Sδ with δ>0 appears sound. The construction via chordal L1 geometry of geodesic rays is natural, and the applications—semicontinuity of multiplier ideals and stability of Monge–Ampère solutions—are valuable. The proofs are detailed and mostly self-contained; external results are published and clearly cited.

The soft spot the stress-test flags is legitimate. The abstract and introduction claim that dS-atoms are exactly the relative full mass classes. What Theorem 3.3 actually proves is dS([ψ],[χ])=0 iff C(ψ)=C(χ). The bridge from C-equivalence to relative full mass classes is the identity P[φ]=C(φ), which is imported for positive mass and left as Conjecture 2.5 for zero mass. So the advertised atom characterization is not established in full generality. The zero-mass case is not a corner: Section 4.2 exhibits singularities with zero mass, so the overstatement touches the paper's stated scope, not just a remoter application.

That said, the issue is fixable and does not sink the core. For positive mass—the setting of the completeness theorem, the diamond inequality, and the CMAE applications—the identity P=C is proven in the cited literature, and the paper's own arguments are coherent. The zero-mass gap is explicitly acknowledged in the body via Conjecture 2.5. The abstract and intro should be reworded to state the C-equivalence criterion plainly, present the relative-full-mass description for positive mass, and flag the zero-mass case as conjectural. Also worth asking the authors to double-check whether the zero-mass atom claim can be obtained directly, or whether it genuinely needs the conjecture.

The incompleteness of S(X,θ) and the Demailly–Peternell–Schneider example are good and handled honestly. The volume diamond inequality and its use in the sandwich theorem are thoughtful.

Bottom line: this is a strong paper from serious people, with one real overstatement in the presentation. I would send it to a good referee and expect acceptance after the abstract and introduction are aligned with what is actually proven.

Referee Report

1 major / 5 minor

Summary. The paper introduces a pseudometric d_S on the space S(X,θ) of singularity types of θ-psh potentials on a compact Kähler manifold with a big cohomology class. The construction embeds S(X,θ) into the chordal metric space of finite-energy geodesic rays and pulls back the metric d_1^c. The main results are: (i) d_S is a pseudometric whose zero-distance relation is characterized by equality of the ceiling operator C (Theorem 3.3); (ii) for δ>0 the subspace S_δ(X,θ) is complete (Theorem 1.1/4.9), while the full space is incomplete (Section 4.2); (iii) a volume diamond inequality for non-pluripolar masses (Theorem 1.2/5.4); and applications to semicontinuity of multiplier ideal sheaves (Theorem 1.3/6.1) and to stability of complex Monge–Ampère equations with prescribed singularity type (Theorem 1.4/7.1). The proofs are detailed and rely on prior work of the authors, especially the identity P[φ]=C(φ) for positive-mass potentials.

Significance. If the main results hold, this paper provides a natural metric topology on singularity types, filling a gap in the literature where convergence of singularity types was previously treated only in ad-hoc ways. The completeness theorem for positive-mass subspaces and the stability applications are substantial and will likely be useful in pluripotential theory and transcendental algebraic geometry. The paper is carefully written and contains numerous proved lemmas rather than black-box statements. However, the advertised atom characterization in the abstract is stronger than what is formally established, as detailed in the major comment; this does not undermine the positive-mass results but requires a correction of the claimed scope.

major comments (1)

[Abstract and §1; Theorem 3.3; Conjecture 2.5] The headline assertion that the atoms of d_S are 'exactly the relative full mass classes' (Abstract and §1, p. 2) is stronger than what is proved. Theorem 3.3 shows d_S([ψ],[χ])=0 iff C(ψ)=C(χ), and the identification of C-equivalence classes with relative full mass classes requires the identity P[φ]=C(φ). For positive mass this is imported from [DDL2, Remark 2.5, Theorem 3.12] and is used in Lemma 4.3, Proposition 4.6, Corollary 4.7, Proposition 4.8 and Theorem 4.9. For zero mass the identity is not proved and is explicitly left open as Conjecture 2.5; zero-mass singularity types do occur in S(X,θ) (see φ_2 in §4.2). Consequently the advertised atom characterization is not established in full generality. Please either prove the zero-mass case or reformulate the abstract/introduction to state that for positive mass the atoms are the relative full mass classes, while in general the atoms are exactly the fibers of C, with the identification to relative full mass classes being conjectural (Conjecture 2.5). This change is needed for the central claim to match the formal results.

minor comments (5)

[§1, p. 2] In the introduction, the definition of the envelope reads 'P [φ] := sup {v ∈ PSH(X,θ ) : [ v] ≤ [u],v ≤ 0}', using both φ and u; this is a mismatch and should be corrected (likely [v] ≤ [φ]).
[§4.2] In the computation of the mixed mass of φ_t, the inequality '≥ {θ}.{η} = ∫_C η >0' would be clearer if the authors explicitly noted that {θ}.{η} is an intersection number and that ∫_C η >0 because η is positive along the curve C.
[§2.1, Lemma 2.4] In the proof of Lemma 2.4, the sentence 'we used [DDL2, Proposition 2.1, Theorem 2.2]' compresses two different external facts into one citation; a more detailed pointer would improve readability.
[§4, Proposition 4.2] The geometric series bound in inequality (18), '∑_{k≥j} 1/C^{k+j−1} ≤ 1/C^{j−1} C/(C−1)', is correct but the intermediate step ∑_{k≥j} C^{-(k+j-1)} = C^{-(2j-1)}/(1-1/C) could be displayed to help the reader verify the estimate.
[§5, Theorem 5.6] The notation w_j := P(u_j,u_{j+1},...) in (30) denotes an infinite envelope; the accompanying text explains it as a decreasing limit of finite envelopes, but stating this explicitly in the theorem statement would prevent confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: d_S is genuinely constructed and its zero-distance classes are proved to equal C-equivalence; the only caveat is a zero-mass atom claim left to Conjecture 2.5, which is an unsupported assertion, not a circular reduction.

full rationale

dS is not defined to have the claimed atoms; it is pulled back from the chordal metric on geodesic rays (Section 2.2, (8)), and the equivalence dS([ψ],[χ])=0 iff C(ψ)=C(χ) is proved in Proposition 3.2 and Theorem 3.3 by showing r[ψ]∞=C(ψ) and using the metric completeness of the ray space. The ceilings C(ψ) are defined independently by a mixed-mass envelope family (4), not by dS. The completeness of Sδ(X,θ) (Theorem 4.9) is a proof from Lemma 4.3, Propositions 4.6 and 4.8, and the reduction dS([vj],[C(vj)])=0 follows from Theorem 3.3. The most delicate input is the positive-mass identity P[φ]=C(φ), imported from [DDL2, Remark 2.5, Theorem 3.12] and used in Lemmas 2.4 and 4.3; this is a load-bearing self-citation, but it is a published theorem with the needed hypotheses, not a parameter fitted in this paper, and the paper explicitly separates the unproved zero-mass case as Conjecture 2.5. Consequently the abstract's statement that atoms are exactly the relative full mass classes is not fully supported in the zero-mass case—if P[φ]=C(φ) fails there, the atom description based on C-equivalence would not match the P-based relative full mass classes. This is a correctness/incompleteness caveat, not circularity: no equation in the paper is equivalent to an input by construction, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

This is a pure mathematics paper with no fitted parameters and no invented physical entities. It relies on a network of previous theorems, many from the authors' own earlier papers; these are published results and do not beg the question. The metric d_S is a defined construction, not a postulated entity.

assumptions (6)

standard math Non-pluripolar Monge-Ampère products are well-defined for θ-psh potentials in big cohomology classes and satisfy mass monotonicity ([BEGZ10], [WN19]).
Pervasive in the paper (Sections 2, 3, 5, 7); underlies the definitions of mass, energy, and d_S.
standard math The space (E^1(X,θ), d_1) is a complete geodesic metric space and finite-energy geodesics exist ([DDL3, Theorem 1.1]).
Used to define the chordal metric d_c^1 on rays and to prove completeness of (R(X,θ), d_c^1) in Theorem 2.14.
standard math For positive mass potentials φ, the envelope P[φ] equals the ceiling C(φ) ([DDL2, Remark 2.5, Theorem 3.12]).
Used to identify model potentials in Lemma 4.3, Proposition 4.8 and elsewhere; the integrity of the completeness proof depends on it.
standard math Guan-Zhou strong openness: for increasing sequences of psh functions u_j ↑ u, the multiplier ideal sheaves stabilize ([GZh15], [GZh16]).
Core input in the proof of Theorem 6.1 for multiplier ideal sheaves.
standard math Domination principle and relative full mass characterization via envelopes ([DDL1, Proposition 2.4], [DDL2, Sections 2-3]).
Used to conclude equality of potentials from vanishing mass (Lemma 2.9, Corollary 4.7, Theorem 7.1).
standard math Skoda's uniform integrability theorem for negative psh functions with zero supremum ([Zer01], [GZ17, Theorem 2.50]).
Used in the proof of Theorem 7.1 to control exponentials of ψ_j.

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Pith. "Pith review of The metric geometry of singularity types." pith.science (2026). https://pith.science/paper/Y3EBVUTB

@misc{pith2026190900839,
  author       = {Pith},
  title        = {Pith review of: The metric geometry of singularity types},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3EBVUTB}},
  note         = {Machine review of arXiv:1909.00839}
}

abstract

Let $X$ be a compact K\"ahler manifold. Given a big cohomology class $\{\theta\}$, there is a natural equivalence relation on the space of $\theta$-psh functions giving rise to $\mathcal S(X,\theta)$, the space of singularity types of potentials. We introduce a natural pseudometric $d_{\mathcal {S}}$ on $\mathcal S(X,\theta)$ that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge-Amp\`ere equations with varying singularity type converge as governed by the $d_\mathcal S$-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds
math.CV 2019-09 conditional novelty 7.0 of 10

The total Hessian mass is monotone in singularity type, and Hessian equations H_m(u)=µ have unique solutions in prescribed singularity classes on compact Kähler manifolds.

Reference graph

Works this paper leans on

36 extracted references · 33 canonical work pages · cited by 1 Pith paper

[1]

Bedford, B.A

E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge-Amp\`ere equation, Invent. Math. 37 (1976), no. 1, 1--44

work page 1976
[2]

Bedford, B.A

E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions , Acta Math. 149 (1982), 1--40

work page 1982
[3]

Bedford, B.A

E. Bedford, B.A. Taylor, Fine topology, Silov boundary, and (i )^n , J. Funct. Anal. 72 (1987), no. 2, 225--251

work page 1987
[4]

Berman, From Monge-Amp\`ere equations to envelopes and geodesic rays in the zero temperature limit, Math

R.J. Berman, From Monge-Amp\`ere equations to envelopes and geodesic rays in the zero temperature limit, Math. Z., 2019, 291, 1--2, 365--394

work page 2019
[5]

Berman, S

R.J. Berman, S. Boucksom, M. Jonsson, A variational approach to the Yau--Tian--Donaldson conjecture, arXiv:1509.04561

arXiv
[6]

Berman, T

R.J. Berman, T. Darvas, C.H. Lu, Convexity of the extended K-energy and the long time behaviour of the Calabi flow, Geometry & Topology, 21 (2017) 2945--2988

work page 2017
[7]

Berndtsson, The openness conjecture and complex Brunn-Minkowski inequalities, Complex geometry and dynamics, 29--44, Abel Symp., 10, Springer, Cham, 2015

B. Berndtsson, The openness conjecture and complex Brunn-Minkowski inequalities, Complex geometry and dynamics, 29--44, Abel Symp., 10, Springer, Cham, 2015

work page 2015
[8]

Z. Blocki, The complex Monge-Amp\`ere operator in pluripotential theory, lecture notes, http://gamma.im.uj.edu.pl/ blocki/publ/ln/wykl.pdf http://gamma.im.uj.edu.pl/\ blocki/publ/ln/wykl.pdf

work page

Show all 36 references

[9]

Blocki, S

Z. Blocki, S. Ko odziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc., 2007, Vol 135, 7, 2089--2093

2007
[10]

Boucksom, On the volume of a line bundle

S. Boucksom, On the volume of a line bundle. Internat. J. Math. 13 (2002), no. 10, 1043--1063

2002
[11]

Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann

S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. \'Ecole Norm. Sup. (4) 37 (2004), no. 1, 45--76

2004
[12]

Boucksom, P

S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Amp\`ere equations in big cohomology classes, Acta Math. (2010), Volume 205, Issue 2, pp 199--262

2010
[13]

Cao, Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact K\"ahler manifolds

J. Cao, Numerical dimension and a Kawamata-Viehweg-Nadel-type vanishing theorem on compact K\"ahler manifolds. Compos. Math. 150 (2014), no. 11, 1869--1902

2014
[14]

Darvas, Weak geodesic rays in the space of K\"ahler potentials and the class E(X, ) , J

T. Darvas, Weak geodesic rays in the space of K\"ahler potentials and the class E(X, ) , J. Inst. Math. Jussieu 16 (2017), no. 4, 837--858

2017
[15]

Darvas, The Mabuchi geometry of finite energy classes, Adv

T. Darvas, The Mabuchi geometry of finite energy classes, Adv. Math. 285 (2015), 182--219

2015
[16]

Darvas, E

T. Darvas, E. Di Nezza, C.H. Lu, On the singularity type of full mass currents in big cohomology classes, Compos. Math. 154 (2018), no. 2, 380--409

2018
[17]

Darvas, E

T. Darvas, E. Di Nezza, C.H. Lu, Monotonicity of non-pluripolar products and complex Monge-Ampere equations with prescribed singularity, Analysis & PDE, Vol. 11 (2018), No. 8, 2049--2087

2018
[18]

Darvas, E

T. Darvas, E. Di Nezza, C.H. Lu, L^1 metric geometry of big cohomology classes, Annales de L'Institut Fourier, 68, no 7 (2018), p. 3053--3086

2018
[19]

Darvas, E

T. Darvas, E. Di Nezza, C.H. Lu, Log-concavity of volume and complex Monge-Amp\`ere equations with prescribed singularity, arXiv:1807.00276

arXiv
[20]

Darvas, C.H

T. Darvas, C.H. Lu, Geodesic stability, the space of rays, and uniform convexity in Mabuchi geometry, arXiv:1810.04661

arXiv
[21]

Demailly, J

J.-P. Demailly, J. Kollar, Semi-continuity of complex singularity exponents and K\"ahler-Einstein metrics on Fano orbifolds, Ann. Sci. \'Ecole Norm . Sup. 34 (2001), 525--556

2001
[22]

Demailly, Regularization of closed positive currents and intersection theory, J

J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992) 361--409

1992
[23]

Demailly, On the cohomology of pseudoeffective line bundles

J.-P. Demailly, On the cohomology of pseudoeffective line bundles. Complex geometry and dynamics, 51--99, Abel Symp., 10, Springer, Cham, 2015

2015
[24]

ahler cone of a compact K\

J.-P. Demailly, M. Paun, Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold. Ann. of Math. (2) 159 (2004), no. 3, 1247--1274

2004
[25]

Demailly, T

J.-P. Demailly, T. Peternell, M. Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., 3 (1994), 295--345

1994
[26]

Q. Guan, X. Zhou, A proof of Demailly's strong openness conjecture, Ann. of Math. (2) 182 (2015), no. 2, 605--616

2015
[27]

Q. Guan, X. Zhou, Effectiveness of Demailly's strong openness conjecture and related problems. Invent. Math. 202 (2015), no. 2, 635--676

2015
[28]

Guedj, A

V. Guedj, A. Zeriahi, Degenerate Complex Monge-Amp\`ere Equations, EMS Tracts in Mathematics, vol. 26, (2017)

2017
[29]

Guedj, C.H

V. Guedj, C.H. Lu, A. Zeriahi, Plurisubharmonic envelopes and supersolutions , arXiv:1703.05254

arXiv
[30]

Hartshorne, Algebraic Geometry, Springer Science and Business Media, (1977)

R. Hartshorne, Algebraic Geometry, Springer Science and Business Media, (1977)

1977
[31]

Hiep, The weighted log canonical threshold

P.H. Hiep, The weighted log canonical threshold. C. R. Math. Acad. Sci. Paris 352 (2014), no. 4, 283--288

2014
[32]

Lempert, Modules of square integrable holomorphic germs, Analysis meets geometry, 311--333, Trends Math., Birkhäuser/Springer, Cham, 2017

L. Lempert, Modules of square integrable holomorphic germs, Analysis meets geometry, 311--333, Trends Math., Birkhäuser/Springer, Cham, 2017

2017
[33]

J. Ross, D. Witt Nystr\"om, Analytic test configurations and geodesic rays, Journal of Symplectic Geometry Volume 12, Number 1 (2014), 125--169

2014
[34]

Xia, Mabuchi geometry of big cohomology classes with prescribed singularities, arXiv:1907.07234

M. Xia, Mabuchi geometry of big cohomology classes with prescribed singularities, arXiv:1907.07234

1907 arXiv
[35]

Witt-Nystr\"om, Monotonicity of non-pluripolar Monge-Amp\`ere measures, Indiana University Mathematics Journal, 2019, 68, 2, 579--591

D. Witt-Nystr\"om, Monotonicity of non-pluripolar Monge-Amp\`ere measures, Indiana University Mathematics Journal, 2019, 68, 2, 579--591

2019
[36]

Zeriahi, Volume and capacity of sublevel sets of a Lelong class of psh functions

A. Zeriahi, Volume and capacity of sublevel sets of a Lelong class of psh functions. Indiana Univ. Math. J. 50 (2001), no. 1, 671--703. footnotesize

2001

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bedford, B.A

E. Bedford, B.A. Taylor, The Dirichlet problem for a complex Monge-Amp\`ere equation, Invent. Math. 37 (1976), no. 1, 1--44

work page 1976

[2] [2]

Bedford, B.A

E. Bedford, B.A. Taylor, A new capacity for plurisubharmonic functions , Acta Math. 149 (1982), 1--40

work page 1982

[3] [3]

Bedford, B.A

E. Bedford, B.A. Taylor, Fine topology, Silov boundary, and (i )^n , J. Funct. Anal. 72 (1987), no. 2, 225--251

work page 1987

[4] [4]

Berman, From Monge-Amp\`ere equations to envelopes and geodesic rays in the zero temperature limit, Math

R.J. Berman, From Monge-Amp\`ere equations to envelopes and geodesic rays in the zero temperature limit, Math. Z., 2019, 291, 1--2, 365--394

work page 2019

[5] [5]

Berman, S

R.J. Berman, S. Boucksom, M. Jonsson, A variational approach to the Yau--Tian--Donaldson conjecture, arXiv:1509.04561

arXiv

[6] [6]

Berman, T

R.J. Berman, T. Darvas, C.H. Lu, Convexity of the extended K-energy and the long time behaviour of the Calabi flow, Geometry & Topology, 21 (2017) 2945--2988

work page 2017

[7] [7]

Berndtsson, The openness conjecture and complex Brunn-Minkowski inequalities, Complex geometry and dynamics, 29--44, Abel Symp., 10, Springer, Cham, 2015

B. Berndtsson, The openness conjecture and complex Brunn-Minkowski inequalities, Complex geometry and dynamics, 29--44, Abel Symp., 10, Springer, Cham, 2015

work page 2015

[8] [8]

Z. Blocki, The complex Monge-Amp\`ere operator in pluripotential theory, lecture notes, http://gamma.im.uj.edu.pl/ blocki/publ/ln/wykl.pdf http://gamma.im.uj.edu.pl/\ blocki/publ/ln/wykl.pdf

work page