A note on the noncommutative Hodge conjecture for graded matrix factorizations
Abstract
Let m≥ 2 and d≥ 7. We consider homogeneous polynomials in 2m+2 variables of the form f=F0(u0,v0)+⋯+Fm(um,vm), where the Fi are independently very general squarefree binary forms of degree d. We prove the rational noncommutative Hodge conjecture for the dg category MFgr(f) of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and d−1 one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank (d−1)m+1 in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, dimℚHdg(MFgr(f),ℚ)=(d−1)m+1+d−1, and the rational topological K-rank is ((d−1)2m+2+d−1)/d+d−1. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.
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BibTeX
@misc{lin2026note,
title = {{A note on the noncommutative Hodge conjecture for graded matrix factorizations}},
author = {Xun Lin and Shizhuo Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03784},
eprint = {2609.03784},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03784},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A note on the noncommutative Hodge conjecture for graded matrix factorizations
AU - Xun Lin
AU - Shizhuo Zhang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03784
UR - https://arxiv.org/abs/2609.03784
AB - Let m≥ 2 and d≥ 7. We consider homogeneous polynomials in 2m+2 variables of the form f=F_0(u_0,v_0)+⋯+F_m(u_m,v_m), where the F_i are independently very general squarefree binary forms of degree d. We prove the rational noncommutative Hodge conjecture for the dg category MF^gr(f) of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and d−1 one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank (d−1)^m+1 in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, dim_ℚHdg(MF^gr(f),ℚ)=(d−1)^m+1+d−1, and the rational topological K-rank is ((d−1)^2m+2+d−1)/d+d−1. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.
ER -