The noncommutative minimal model program

TL;DR

This paper explores a noncommutative analog of the minimal model program by linking birational geometry with semiorthogonal decompositions of derived categories, proposing conjectures and mechanisms involving stability conditions and quantum differential equations.

Contribution

It introduces a conjecture on canonical semiorthogonal decompositions, connecting birational modifications with noncommutative geometry and stability conditions, and verifies these for smooth projective curves.

Findings

Implications for Dubrovin's conjecture and D-equivalence

Proposal of categorical birational invariants

Verification for smooth projective curves

Abstract

This note aims to clarify the deep relationship between birational modifications of a variety and semiorthogonal decompositions of its derived category of coherent sheaves. The result is a conjecture on the existence and properties of canonical semiorthogonal decompositions, which is a noncommutative analog of the minimal model program. We identify a mechanism for constructing semiorthogonal decompositions using Bridgeland stability conditions, and we propose that through this mechanism the quantum differential equation of the variety controls the conjectured semiorthogonal decompositions. We establish several implications of the conjectures: one direction of Dubrovin's conjecture on the existence of full exceptional collections; the $D$-equivalence conjecture; the existence of new categorical birational invariants for varieties of positive genus; and the existence of minimal noncommutative resolutions of singular varieties. Finally, we verify the conjectures for smooth projective curves by establishing a previously conjectured description of the stability manifold of $\mathbb{P}^1$.