A note on the noncommutative Hodge conjecture for graded matrix factorizations

Abstract

Let $m\ge 2$ and $d\ge 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots +F_m(u_m,v_m)$, 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 ${\mathrm{MF}}^{\mathrm{gr}}(f)$ 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, ${\dim }_{\mathbb{Q}}\mathrm{Hdg}({\mathrm{MF}}^{\mathrm{gr}}(f),\mathbb{Q})=(d-1{)}^{m+1}+d-1$, 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.

arXiv comment35 pages, comments are welcome