The Hodge structure on the singularity category of a complex hypersurface

TL;DR

This paper establishes a link between the noncommutative Hodge structure of a hypersurface's singularity category and the classical Hodge structure of its projective counterpart, revealing an equivalence of Hodge conjectures.

Contribution

It identifies the noncommutative Hodge structure on the singularity category with the classical Hodge structure, and relates the Hodge conjecture to a dg-categorical analogue.

Findings

Noncommutative Hodge structure matches classical Hodge structure

Hodge conjecture for hypersurface is equivalent to a dg-categorical version

Provides a categorical perspective on classical Hodge theory

Abstract

Given a complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge structure on the primitive cohomology of the associated projective hypersurface. As a consequence, we show that the Hodge conjecture for the projective hypersurface is equivalent to a dg-categorical analogue of the Hodge conjecture for the singularity category.