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Hodge structures for orbifold cohomology

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Javier Fernandez

Algebraic Geometry · Submitted 3 Nov 2003

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

The authors' main claim We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''.

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arXiv IDmath/0311026 (version 2)
Pith IDDVOXWRRF, Pith's permanent record of this paper and its review
CitationJavier Fernandez. "Hodge structures for orbifold cohomology." arXiv:math/0311026 (2003). https://pith.science/paper/DVOXWRRF
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,856 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.49 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.88 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''. Furthermore, the total cohomology $H_{orb}^{*,*}(X)$ forms a mixed Hodge structure that is polarized by every element of the K\"ahler cone of $X$. Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified K\"ahler cone of $X$. This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of $X$, in the light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

background contribution method result interpretation main claim

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 construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''. contribution 0.75 0.95 0.73 asserted 1.00
s1 Furthermore, the total cohomology $H_{orb}^{*,*}(X)$ forms a mixed Hodge structure that is polarized by every element of the K\"ahler cone of $X$. result 0.97 0.85 0.25 asserted 1.00
s2 Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified K\"ahler cone of $X$. result 0.54 0.64 0.20 asserted 0.98
s3 This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of $X$, in the light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution. interpretation 1.00 0.34 0.15 hedged possibility 0.73

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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  • references and citing sentences
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  • P abstract
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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 construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology $H_{orb}^k(X)$ for projective $SL$-orbifolds $X$ satisfying a ``Hard Lefschetz Condition''. Furthermore, the total cohomology $H_{orb}^{*,*}(X)$ forms a mixed Hodge structure that is polarized by every element of the K\"ahler cone of $X$. Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified K\"ahler cone of $X$. This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of $X$, in the light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

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