MathDB

Rational noncommutative Hodge conjecture

For every smooth proper differential graded category A\mathcal{A}, the rational Chern character is surjective onto the rational noncommutative Hodge classes: ch⁡A ⁣:K0(A)⊗ZQ↠Hdg⁡(A,Q)\operatorname{ch}_{\mathcal{A}}\colon K_0(\mathcal{A})\otimes_{\mathbb{Z}}\mathbb{Q}\twoheadrightarrow \operatorname{Hdg}(\mathcal{A},\mathbb{Q}).

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Classical Hodge conjecture for smooth projective hypersurfaces

    For a smooth projective hypersurface XX, the rational noncommutative Hodge conjecture for Perf⁡(X)\operatorname{Perf}(X) is equivalent to the classical rational Hodge conjecture for XX; equivalently, the Chern character from algebraic KK-theory is surjective onto the rational Hodge classes of XX.

    source: A note on the noncommutative Hodge conjecture for graded matrix factorizations

References

Primary source

arXiv

Additional references

algebraic-geometrymatrix-factorizationsnoncommutative-geometry
Posted

Progress summary

Refreshed
Claimed progress

A new paper claims the conjecture for a substantial but restricted family, while the general conjecture remains open.

The rational noncommutative Hodge conjecture asserts that the Chern character is surjective onto the rational Hodge classes; for categories of perfect complexes, it recovers the classical rational Hodge conjecture. No proposer is identified in the retrieved sources.

Known results

  • The conjecture is proved for smooth proper connective dg algebras and is compatible with semi-orthogonal decompositions (2021).
  • For a smooth projective hypersurface, the corresponding noncommutative Hodge condition is equivalent to the classical Hodge conjecture for the hypersurface (2024).

September 2026 restricted-family result

On September 3, 2026, A note on the noncommutative Hodge conjecture for graded matrix factorizations claimed the conjecture for homogeneous polynomials in 2m+22m+2 variables arising from very general squarefree binary forms of degree d≥7d \ge 7. It also computed the Hodge-class dimension and derived the associated hypersurface statement. This is a family result, not a solution of the general conjecture, and remains unverified.

Current status (as of September 2026): A restricted very-general family is claimed to satisfy the conjecture, but the general rational noncommutative Hodge conjecture remains open and the new claim is unverified.

Sources

Solutions 0

No solutions have been posted yet.