Deligne pairings and the Knudsen-Mumford expansion

D. H. Phong; Jacob Sturm; Julius Ross

RECORD

Deligne pairings and the Knudsen-Mumford expansion

D. H. Phong, Julius Ross, Jacob Sturm

Differential Geometry · Submitted 19 Dec 2006

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

The authors' main claim We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (\pi_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$.

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, 4,420 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.22 · compares with prior results 0.94 · 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 $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (\pi_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$. This generalizes a theorem of Deligne which holds for families of relative dimension one. As a corollary, we show that when $X$ is smooth (or, more generally, if $X$ fits in a smooth family), the line bundle associated to $X\to B$, which was introduced by the first and third authors, coincides with the CM bundle defined by Paul-Tian. In a second corollary, we establish asymptotics for the K-energy along Bergman rays, generalizing a formula obtained by Paul-Tian.

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 $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers.	background 1.00	0.09	0.02	asserted 0.69
s1	We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (\pi_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$.	result 0.57	0.96	0.89	asserted 1.00
s2	This generalizes a theorem of Deligne which holds for families of relative dimension one.	interpretation 0.54	0.24	0.08	asserted 1.00
s3	As a corollary, we show that when $X$ is smooth (or, more generally, if $X$ fits in a smooth family), the line bundle associated to $X\to B$, which was introduced by the first and third authors, coincides with the CM bundle defined by Paul-Tian.	result 0.97	0.87	0.14	asserted 1.00
s4	In a second corollary, we establish asymptotics for the K-energy along Bergman rays, generalizing a formula obtained by Paul-Tian.	result 0.93	0.92	0.07	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 $X\to B$ be a proper flat morphism between smooth quasi-projective varieties of relative dimension $n$, and $L\to X$ a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for $\det (\pi_* L^k)$ in terms of Deligne pairings of $L$ and the relative canonical bundle $K$. This generalizes a theorem of Deligne which holds for families of relative dimension one. As a corollary, we show that when $X$ is smooth (or, more generally, if $X$ fits in a smooth family), the line bundle associated to $X\to B$, which was introduced by the first and third authors, coincides with the CM bundle defined by Paul-Tian. In a second corollary, we establish asymptotics for the K-energy along Bergman rays, generalizing a formula obtained by Paul-Tian.

Human review

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

How the literature uses it

1 arXiv paper cite this in 1 sentence

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

Cites it as background 1 paper

It must be pointed out that expansions for the K-energy of the form [ref] are known to hold for the Fubini-Study metric along the fibres of a $^*$-action in projective space under some regularity assumptions, thanks to the work of Paul-Tian [ref] , Phong-Sturm-Ross [ref] and others.

Twisted cscK metrics and K\"ahler slope stability · Proof of stability condition · 0.83

Discussion

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