A theoretical mathematics preprint presents a framework linking cyclic covers with singularity tests and differential-form formulas in complex algebraic geometry. It uses mixed Hodge-module decompositions and derived-category operations to obtain quasi-isomorphisms for differential-form complexes, including cases where the underlying variety is not smooth.
The units of analysis are abstract complex algebraic varieties, line bundles, sections, divisors and associated cyclic covers. The paper contains no sampled participants, animals, cells or empirical observations. The supplied document is an arXiv version 1 preprint dated 26 August 2026.
One decomposition, several consequences
At the center of the work is a theorem about the pushforward constant Hodge module of the cyclic cover. It splits into the constant Hodge module of the base and an extension-by-zero term from the complement of the divisor. In practical terms, the decomposition separates the contribution belonging to the base from the part associated with the region away from the divisor.
The same approach is used to compare complexes of differential forms. Derived-category operations produce quasi-isomorphisms, allowing the paper to treat different-looking complexes as equivalent within the mathematical setting of the construction.
Resolving smooth and singular spaces
For smooth irreducible bases, the relevant complex is analyzed after taking a strong log resolution of the pair formed by the base and the divisor. The resulting formulas give direct-sum descriptions of both the cover’s differential-form complex and its dual, using logarithmic forms twisted by two types of effective divisor.
For singular bases, the paper generalizes the approach using simplicial resolutions. For arbitrary algebraic varieties, its hyperresolution construction defines twisted logarithmic Du Bois complexes and produces an analogous direct-sum formula for the pushforward differential forms, with powers of the line bundle appearing in the decomposition.
Tests for the cover’s singularities
The two divisor twists lead to different criteria for two classes of singularities. In the smooth-base setting, Du Bois singularities of the cover are equivalent to quasi-isomorphism of the natural maps using the positive-twist divisors. Rational singularities are characterized instead by the corresponding maps using the nonnegative-twist divisors.
The paper also expresses these two quasi-isomorphism conditions through the log canonical threshold, a numerical threshold used to set the relevant boundary. The positive-twist condition is non-strict at the threshold, so equality is allowed, while the nonnegative-twist condition requires a strict inequality.
When the divisor has simple normal-crossing support, the constant Hodge module of the cover agrees with its intersection-complex Hodge module exactly when the covering degree is coprime to the multiplicity of every component of the divisor.
Vanishing results under added conditions
The framework also gives cohomology-vanishing results. For a projective base whose divisor complement is affine, the paper proves vanishing of the twisted logarithmic Du Bois cohomology below a bound set by the variety’s dimension minus its local cohomological defect.
A separate theorem applies to an irreducible projective variety with a big and nef line bundle. In that setting, the indicated degree-zero Du Bois cohomology vanishes below a bound adjusted by the defect measure.
The conclusions depend on explicit conditions that differ from one result to another. The formulas for smooth bases use a strong log resolution; the intersection-complex criterion requires simple normal-crossing support; and the vanishing results use either an affine divisor complement or a big-and-nef line bundle on a projective variety.
A theoretical result with a defined scope
Because the paper studies abstract varieties, line bundles, sections, divisors and cyclic covers, it contains no empirical sample or measured population. Its results are mathematical statements within the hypotheses attached to each construction, rather than estimates drawn from observations.
The supplied document is labeled an arXiv version 1 preprint dated 26 August 2026. Its acknowledgments thank named colleagues for help and discussions, but the supplied text does not report an external funding source.
Paper data and sources
Original title: Cyclic covers via mixed Hodge modules
Authors: Scott Hiatt
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text