Chern bounds and tangent geometry of polarized Calabi-Yau threefolds
Abstract
We study the numerical geography and tangent geometry of very amply polarized Calabi--Yau threefolds (X,H) through the positivity of the first jet bundle J1(H). Writing d=∫X H3, c=∫Xc2(X) H, and e=∫X c3(X), we exploit two different positivity properties of this single bundle. Mixed intersections on ℙ(J1(H)*) give e≥−5d−c−c2/(4d), while a volume estimate for a perturbed tautological class gives e≥40d[(1−c/10d)3/2−1]; in particular e+6c≥0, improving Sun's inequality e+10c≥0. As consequences, we obtain the uniform Hodge bounds −4d−80≤ h1,1(X)−h2,1(X)≤173d/66, the lower bound degX∨≥78 for the dual hypersurface, and, in the critical case X⊂ℙ6, the upper bound d≤34. We also prove that, for every m≥2, the tangent-incidence morphism associated with |mH| is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings X⊂ℙN with N≥7, and verify it for several families.
Related
- Rigid and obstructed tangent bundles on Calabi--Yau threefolds
- Scalar potential bounds on one-parameter Calabi-Yau threefolds with fluxes
- The Tate Conjecture for Ordinary Calabi-Yau Threefolds
- Integral Bott--Chern cohomology of Generalized Calabi--Eckmann threefolds
- Degeneration of Calabi-Yau threefolds with Galois action induced by modular form of weight 3
BibTeX
@misc{kanazawa2026chern,
title = {{Chern bounds and tangent geometry of polarized Calabi-Yau threefolds}},
author = {Atsushi Kanazawa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17513},
eprint = {2609.17513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17513},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Chern bounds and tangent geometry of polarized Calabi-Yau threefolds
AU - Atsushi Kanazawa
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.17513
UR - https://arxiv.org/abs/2609.17513
AB - We study the numerical geography and tangent geometry of very amply polarized Calabi--Yau threefolds (X,H) through the positivity of the first jet bundle J^1(H). Writing d=∫_X H^3, c=∫_Xc_2(X) H, and e=∫_X c_3(X), we exploit two different positivity properties of this single bundle. Mixed intersections on ℙ(J^1(H)^*) give e≥−5d−c−c^2/(4d), while a volume estimate for a perturbed tautological class gives e≥40d[(1−c/10d)^3/2−1]; in particular e+6c≥0, improving Sun's inequality e+10c≥0. As consequences, we obtain the uniform Hodge bounds −4d−80≤ h^1,1(X)−h^2,1(X)≤173d/66, the lower bound degX^∨≥78 for the dual hypersurface, and, in the critical case X⊂ℙ^6, the upper bound d≤34. We also prove that, for every m≥2, the tangent-incidence morphism associated with |mH| is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings X⊂ℙ^N with N≥7, and verify it for several families.
ER -