Homological Bondal-Orlov localization conjecture for rational singularities

TL;DR

This paper proves that for resolutions of rational singularities, the derived pushforward functor's image generates the entire derived category of the base, providing a weak form of the Bondal-Orlov localization conjecture using Hodge theory.

Contribution

It establishes a weak version of the Bondal-Orlov localization conjecture for rational singularities using Hodge-theoretic methods, extending to certain proper morphisms.

Findings

The derived pushforward generates the entire derived category for resolutions of rational singularities.

The result holds for more general proper morphisms with specific conditions.

Provides a positive answer to a question posed by Pavic and Shinder.

Abstract

Given a resolution of rational singularities $\pi\colon \tilde{X} \to X$ over a field of characteristic zero we use a Hodge-theoretic argument to prove that the image of the functor $\mathbf{R}\pi_*\colon \mathbf{D}(\tilde{X}) \to \mathbf{D}(X)$ between bounded derived categories of coherent sheaves generates $\mathbf{D}(X)$ as a triangulated category. This gives a weak version of the Bondal-Orlov localization conjecture, answering a question of Pavic and Shinder. The same result is established more generally for proper (non-necessarily birational) morphisms $\pi\colon \tilde{X} \to X$, with $\tilde{X}$ smooth, satisfying $\mathbf{R}\pi_*(\mathcal{O}_{\tilde{X}}) = \mathcal{O}_X$.