{"paper":{"title":"A note on the noncommutative Hodge conjecture for graded matrix factorizations","arxiv_id":"2609.03784","primary_cat":"math.AG","submitted_at":"2026-09-03T12:58:41+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.87,"id":"s0","own":false,"role":"background","text":"Let $m\\geq 2$ and $d\\geq 7$.","hedge":{"p":0.56,"label":"unclear"},"own_p":0.09,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.0,"background":0.87,"limitation":0.08,"future_work":0.0,"contribution":0.03,"interpretation":0.0}},{"i":1,"p":0.44,"id":"s1","own":false,"role":"contribution","text":"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 $d$.","hedge":{"p":0.63,"label":"asserted"},"own_p":0.37,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.76,"role_probabilities":{"gap":0.0,"method":0.24,"result":0.0,"background":0.32,"limitation":0.0,"future_work":0.0,"contribution":0.44,"interpretation":0.0}},{"i":2,"p":0.68,"id":"s2","own":true,"role":"result","text":"We prove the rational noncommutative Hodge conjecture for the dg category $\\mathrm{MF}^{\\mathrm{gr}}(f)$ of graded matrix factorizations.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.86,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.68,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.32,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"result","text":"Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.65,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.93,"id":"s4","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.75,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":5,"p":0.63,"id":"s5","own":true,"role":"method","text":"A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.63,"result":0.3,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.01}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"result","text":"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 $\\bigl((d-1)^{2m+2}+d-1\\bigr)/d+d-1$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.26,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":7,"p":0.71,"id":"s7","own":true,"role":"result","text":"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.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.81,"main_claim_p":0.39,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.71,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.15}}]},"compiled":{"main_claim":{"p":0.58,"text":"We prove the rational noncommutative Hodge conjecture for the dg category $\\mathrm{MF}^{\\mathrm{gr}}(f)$ of graded matrix factorizations.","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per 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