Weighted cscK metric (I): a priori estimates

Abdellah Lahdili; Eleonora Di Nezza; Simon Jubert

RECORD

Weighted cscK metric (I): a priori estimates

Eleonora Di Nezza, Simon Jubert, Abdellah Lahdili

Differential Geometry · Submitted 13 Jul 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,935 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.23 · compares with prior results 0.93 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Let $X$ be a compact K\"ahler manifold. In this paper we study the existence of constant weighted scalar curvature K\"ahler (weighted cscK) metrics on $X$. More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $X$ be a compact K\"ahler manifold.	background 1.00	0.04	0.01	asserted 0.76
s1	In this paper we study the existence of constant weighted scalar curvature K\"ahler (weighted cscK) metrics on $X$.	contribution 0.86	0.55	0.12	asserted 0.73
s2	More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.	contribution 0.58	0.95	0.87	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Let $X$ be a compact K\"ahler manifold. In this paper we study the existence of constant weighted scalar curvature K\"ahler (weighted cscK) metrics on $X$. More precisely, we establish a priori $C^{k}$-estimates ($k\geq 0$) for the K\"ahler potential associated with these metrics, thereby extending a result due to Chen and Cheng in the classical cscK setting.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

1 arXiv paper cite this in 13 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 12 sentences without a confident label

Compares with it 1 paper

Hence, our definition of the weighted scalar curvature is related to the one use in [ref] by $^Lah_(̌ω_φ)= (̌μ_φ) _(̌ω_φ)$.

Numerical invariants for weighted cscK metrics · Weighted curvatures and weighted trace · 1.00

What it cites · 2 sentences without a confident label

Compares with 1 paper

We would like to acknowledge Jiyuan Han and Yaxiong Liu for bringing to our attention the overlap between our project and their work [ref] .

2406.10939 · Acknowledgements · 0.87

Cites as background 1 paper

Some works have already been carried on in some special cases: [ref] deals with the semisimple fibration case, [ref] the $g$-solitons case, [ref] the extremal metrics case, and [ref] treats the case of $σ$-extremal metrics.

Modified extremal K\"{a}hler metrics and multiplier Hermitian-Einstein metrics · Introduction · 0.81

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.