The noncommutative minimal model program

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 $\mathbb{P}^1$.

Figures (1)

• Figure 1: A visualization of $\mathop{\mathrm{Stab}}\nolimits(\bP^1)/\bG_a \cong \bC$. The red region is the $\widetilde{\mathop{\mathrm{GL}}\nolimits}^+(2,\bR)$-orbit of slope stability. The blue regions are stability conditions that are glued from the full exceptional collections shown, which correspond to the regions with imaginary part $> \pi$ in each of the coordinate charts $X_k$. The black path is determined by a particular solution to the quantum differential equation. The green vertical line represents the line added at infinity in the partial compactification of $\mathop{\mathrm{Stab}}\nolimits(\bP^1)/\bG_a$. The dotted horizontal lines differ by integer multiples of $\pi i$.

Theorems & Definitions (46)

• Lemma 1
• proof : Proof idea
• Example 2
• Conjecture 1
• Conjecture 2
• Example 3
• Example 4
• Definition 5
• Lemma 6
• proof