关于分次矩阵分解的非交换霍奇猜想的注记
A note on the noncommutative Hodge conjecture for graded matrix factorizations
浏览论文内容
中文总结 AI 辅助
该论文证明分次矩阵分解的dg范畴MFᵍʳ(f)的有理非交换霍奇猜想成立，进而结合相关分解证明对应光滑射影超曲面的有理霍奇猜想成立。
中文摘要 AI 辅助
设m≥2且d≥7，我们考虑2m+2个变量中的齐次多项式，形式为f=F₀(u₀,v₀)+⋯+Fₘ(uₘ,vₘ)，其中Fᵢ是独立的非常一般的无平方因子二元d次型。我们证明了分次矩阵分解的dg范畴MFᵍʳ(f)的有理非交换霍奇猜想成立。其霍奇同调是标量不变雅可比部分与d-1个一维点部分的直和。显式秩1分解的边界- bulk像在单位部分生成秩为(d-1)ᵐ⁺¹的格，而稳定化剩余域的分次平移生成所有点部分。约化- Burau计算表明，在非常一般的参数下，单位部分的格穷尽了有理霍奇类，因此dim_Q Hdg(MFᵍʳ(f),Q)=(d-1)ᵐ⁺¹+d-1，有理拓扑K-秩为((d-1)²ᵐ⁺²+d-1)/d+d-1。最后，利用半正交分解的非交换霍奇猜想的可加性，结合法诺、卡拉比-丘及一般型超曲面的Orlov分解，证明了对应光滑射影超曲面的有理霍奇猜想成立。
英文摘要
Let $m\geq 2$ and $d\geq 7$. 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, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$. 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.
发表机构
- The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
- Sun Yat-sen University（中山大学）
机构由 AI 辅助整理，请以论文原文为准。
