Stability conditions on non-commutative curves

TL;DR

This paper establishes that non-commutative smooth projective varieties with certain low-dimensional stability conditions are necessarily curves, and provides a reconstruction method for higher genus curves based on stable object moduli spaces.

Contribution

It proves a dimension threshold for non-commutative varieties to be curves and introduces a reconstruction technique for higher genus curves using moduli spaces.

Findings

Non-commutative varieties with stability dimension < 6/5 are curves.

No such categories exist with dimension in (1, 6/5).

Reconstruction of higher genus curves via stable object moduli spaces.

Abstract

We prove that any non-commutative smooth projective variety with a Bridgeland stability condition of dimension less than $\frac{6}{5}$ must be a smooth projective curve. As a consequence, we deduce the non-existence of such categories with dimension in the interval $(1,\frac{6}{5})$. Moreover, we prove a sharp reconstruction result for smooth projective curves of higher genus using the moduli space of stable objects in a category of dimension $1$, and deduce a structural result for their semi-orthogonal decompositions.