References & Citations
Mathematics > Algebraic Geometry
Title: The noncommutative minimal model program
(Submitted on 30 Jan 2023 (v1), last revised 27 Mar 2024 (this version, v2))
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 $\mathbb{P}^1$.
Submission history
From: Daniel Halpern-Leistner [view email][v1] Mon, 30 Jan 2023 18:38:17 GMT (37kb)
[v2] Wed, 27 Mar 2024 05:26:50 GMT (37kb,D)
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