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A generalized Poincar\'e-Lelong formula

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Mats Andersson

Complex Variables · Submitted 22 Dec 2004

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

The authors' main claim We prove a generalization of the classical Poincar\'e-Lelong formula.

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arXiv IDmath/0412446 (version 2)
Pith IDI647SWO7, Pith's permanent record of this paper and its review
CitationMats Andersson. "A generalized Poincar\'e-Lelong formula." arXiv:math/0412446 (2004). https://pith.science/paper/I647SWO7
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,375 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.59 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We prove a generalization of the classical Poincar\'e-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that $dd^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. In particular, the top Bott-Chern class is represented by a current with support on $Z$. We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappi\`e-Leray type.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove a generalization of the classical Poincar\'e-Lelong formula. contribution 0.95 0.96 0.80 asserted 1.00
s1 Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. background 0.69 0.19 0.03 asserted 0.65
s2 Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that $dd^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. result 0.99 0.89 0.72 asserted 1.00
s3 In particular, the top Bott-Chern class is represented by a current with support on $Z$. result 0.64 0.86 0.22 asserted 1.00
s4 We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappi\`e-Leray type. contribution 0.85 0.89 0.06 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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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

We prove a generalization of the classical Poincar\'e-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that $dd^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. In particular, the top Bott-Chern class is represented by a current with support on $Z$. We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappi\`e-Leray type.

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