Understand the higher Du Bois singularities in the case of cubics and see what are some examples of Hodge structures in the case of cubics.

The problem asks for higher Du Bois singularities and illustrative Hodge structures associated with cubic varieties. The retrieved material treats important cubic threefold and fourfold examples, but does not identify a complete classification or a definitive solution.

Known results

• For a flat proper family with $k$-Du Bois local complete intersection fibers, $R^q f_*\Omega^p_{Y/S}$ is locally free and compatible with base change for $0\le p\le k$; the corresponding Hodge numbers are preserved.
• A Brieskorn–Pham hypersurface is $k$-Du Bois exactly when $\sum_i 1/d_i\ge k+1$; ordinary double points give explicit examples.
• For cubic fourfolds, the primitive structure $H^4_{\mathrm{prim}}(X,\mathbb{Z})$ has middle Hodge number $21$, and special cases contain K3-related sub-Hodge structures.
• For a singular cubic threefold, Collino’s example identifies a limiting structure with the weight-one structure of a genus-five hyperelliptic curve.

December 2023 and July 2026 developments

A 2023 result determines Hodge–Du Bois numbers for cubic threefolds with isolated singularities using global and local invariants, and identifies a weight-three cohomology piece with a curve’s weight-one structure in the non-cone case. A July 2026 preprint adds limiting Hodge-structure examples for singular cubic fourfolds. Neither source claims to settle the full problem.

Current status (as of September 2026): Several substantial cases and Hodge-structure examples are known, but the general higher Du Bois and cubic Hodge-structure problem remains open.