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The Chern character of a coherent sheaf on a smooth projective hypersurface
Abstract
Given a coherent sheaf on a smooth projective hypersurface X, we prove an explicit formula for its Chern character as a Cech cocycle in terms of the free resolution of the associated module and calculate its image in the Jacobian ring under the Griffiths residue map. The formula is a geometric analogue of the Kapustin-Li formula for Landau-Ginzburg models, but proven directly using Hodge-theoretic techniques. This yields an effective method to compute the primitive part of the Chern character of any coherent sheaf using commutative algebra. We finish by proving the Hodge conjecture for the degree 33 Fermat fourfold.
Related
- Character variety of the five-punctured sphere and the determinantal quintic hypersurface
- Projective compactifications of hypersurface complements
- Odd mapping cone Chern character
- The canonical class and Chern character operators on Hilbert schemes of points
- Extremal character spectra of finite projective systems
BibTeX
@misc{favero2026chern,
title = {{The Chern character of a coherent sheaf on a smooth projective hypersurface}},
author = {David Favero and Tyler L. Kelly},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12759},
eprint = {2609.12759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12759},
}
RIS
TY - UNPB
TI - The Chern character of a coherent sheaf on a smooth projective hypersurface
AU - David Favero
AU - Tyler L. Kelly
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.12759
UR - https://arxiv.org/abs/2609.12759
AB - Given a coherent sheaf on a smooth projective hypersurface X, we prove an explicit formula for its Chern character as a Cech cocycle in terms of the free resolution of the associated module and calculate its image in the Jacobian ring under the Griffiths residue map. The formula is a geometric analogue of the Kapustin-Li formula for Landau-Ginzburg models, but proven directly using Hodge-theoretic techniques. This yields an effective method to compute the primitive part of the Chern character of any coherent sheaf using commutative algebra. We finish by proving the Hodge conjecture for the degree 33 Fermat fourfold.
ER -