Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces

Robin de Jong

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Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces

Robin de Jong

Algebraic Geometry · Submitted 10 Jul 2012

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

The authors' main claim In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.93 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces.

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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	Around 2008 N. Kawazumi and S.	background 1.00	0.03	0.01	unclear 0.75
s1	Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces.	background 1.00	0.03	0.02	asserted 1.00
s2	One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials.	background 0.90	0.07	0.03	asserted 0.79
s3	In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces.	contribution 0.76	0.96	0.93	asserted 1.00
s4	As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed.	result 0.94	0.83	0.12	asserted 1.00
s5	These formulas are a refinement of a result Hain and Reed prove in their paper.	interpretation 0.47	0.23	0.05	asserted 1.00
s6	We illustrate our results with some explicit calculations on degenerating genus two surfaces.	contribution 0.64	0.92	0.03	asserted 1.00

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

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The paper

Original abstract

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces.

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References

Forward citations

Cited by 2 Pith papers

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

Zhang-Kawazumi Invariants and Superstring Amplitudes
hep-th 2013-08 conditional novelty 7.0 of 10

The two-loop D^6 R^4 coefficient in Type IIB string theory is proportional to the Zhang-Kawazumi invariant, and S-duality then predicts the integral of that invariant over genus-two moduli space to be 2π^3/45.
Matching the $D^6 {\cal R}^4$ interaction at two-loops
hep-th 2014-05 accept novelty 6.0 of 10

The Zhang-Kawazumi invariant is an eigenfunction of the Laplace-Beltrami operator with eigenvalue 5 on genus-two moduli space, and its integral over that space equals 2π³/45, matching the S-duality prediction for the ...

How the literature uses it

6 arXiv papers cite this in 9 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 · 6 sentences without a confident label

Compares with it 1 paper

These limits reproduce the genus-two results of [ref] , where the degeneration limits of the Arakelov Green function and the Faltings invariant on a genus $h$ surface were considered, and the results of [ref] on the asymptotic limits of $φ$.

Zhang-Kawazumi Invariants and Superstring Amplitudes · Asymptotics of the $φ$ and $δ$ invariants in degeneration limits · 0.89

Cites it as support 1 paper

This property was shown in [ref] and [ref] .

Zhang-Kawazumi Invariants and Superstring Amplitudes · Holomorphy properties of $Φ$ · 0.98

Cites it as background 1 paper

Various properties of the amplitude have been analyzed in detail upto the $D^6ℛ^4$ term, and some properties of the integrand have also been studied at higher orders in the derivative expansion [ref] .

Eigenvalue equation for genus two modular graphs · Introduction · 0.81

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