# Dg categories of cubic fourfolds

## TL;DR

This paper establishes a criterion linking the geometricity of the Kuznetsov category of a cubic fourfold to the existence of a moduli space of points that forms a (possibly twisted) K3 surface, using dg category reconstruction techniques.

## Contribution

It proves a reconstruction theorem for moduli spaces of skyscraper sheaves in dg categories and applies it to characterize when the Kuznetsov category of a cubic fourfold is geometric.

## Key findings

- The Kuznetsov category is geometric iff it admits a system of points with a moduli space as a K3 surface.
- Reconstruction theorem for skyscraper sheaves in dg categories.
- Connection between dg enhancements and geometricity of categories.

## Abstract

We prove a reconstruction theorem \`a la Calabrese-Groechenig for the moduli space parametrizing skyscraper sheaves on a smooth projective variety when these are considered as a system of points in the dg category of perfect complexes on the variety, as axiomatized by To\"en and Vaqui\'e. This result is then used to show that, for a cubic fourfold $Y\subset \mathbb{P}^5_\mathbb{C}$, the Kuznetsov category $\mathcal{A}_Y$ is geometric (possibly twisted) if and only if a dg enhancement $T_Y$ of $\mathcal{A}_Y$ admits a system of points whose associated moduli space is a (possibly twisted) K3 surface.