Toward the noncommutative minimal model program for Fano varieties

TL;DR

The paper advances the noncommutative minimal model program (NMMP) for smooth Fano varieties by tying quantum cohomology to derived-category stability via lifted central charges $\mathcal{Z}_w^\tau$, constructing geometric stability conditions in new geometric settings, and establishing quasi-convergent stability paths that induce semiorthogonal decompositions. It develops an isomonodromic deformation framework for the quantum connection, derives detailed asymptotics for quantum cohomology central charges, and proves NMMP-related conjectures in several key examples (Grassmannians, quadrics, cubic threefolds, and cubic fourfolds in substantial cases). The work also provides gluing techniques for stability conditions along Kuznetsov-type decompositions, enabling the construction of geometric or almost-geometric stability conditions in complex settings without relying on BG-type inequalities. Overall, these results illuminate the deep connections between semisimple quantum cohomology, stability conditions, and categorical decompositions, with Gamma conjectures playing a guiding role in understanding canonical path choices and mutations. The findings pave the way for broader applications to NMMP in Fano geometry and offer concrete schemes to derive canonical SODs from quantum-differential equation data.

Abstract

We study the noncommutative minimal model program, as proposed by Halpern-Leistner, for Fano varieties. We construct lifts of Iritani's quantum cohomology central charge in the following examples: Grassmannians, smooth quadrics, and smooth cubic threefolds and fourfolds. Moreover, we verify that these lifted paths are quasi-convergent and give rise to the expected semiorthogonal decompositions of the bounded derived category. We also construct geometric stability conditions in the examples above and observe that, after suitable isomonodromic deformation of the quantum cohomology central charge, the quasi-convergent paths for Grassmannians and quadrics can be chosen to start in the geometric region.

Figures (3)

• Figure 1: The path of stability conditions constructed in this paper. $\sigma_t$ is defined for $t$ in an interval $(0,a)$. The arrow-head indicates the direction as $t\to 0$ and the horizontal line can be interpreted as boundary points in the partial compactification $\mathcal{A}\mathop{\mathrm{Stab}}\nolimits(X)$ constructed in augmented.
• Figure 2: The points $u_1,\ldots, u_N$ represent a configuration of points in $\bf{C}$. $(A_1,\ldots, A_N)$ is an asymptotically exponential basis of ${\rm H}^{\bullet}(X)$ such that $\mathcal{Z}_w^u(A_i)\sim (2\pi w)^{\dim X/2} e^{-u_i/w}\int_X \Psi_u(e_i)$ and $(E_1,\ldots, E_N)$ is its lift to a full exceptional collection of $\mathcal{D}^{b}(X)$. For any $i=1,\ldots, N$, we have $A_i = \widehat{\Gamma}_X\mathrm{Ch}(E_i)$.
• Figure 3: On the left side is the spectrum of $\tfrac{1}{4}\mathcal{E}_\tau$. The value $0$ appears with multiplicity two. On the right side is the spectrum of $\mathcal{E}_\tau/e^{\mathtt{i}\theta}$. After applying a small rotation, the eigenvalues are in general position so that $\Im(\lambda_3)<\Im(\lambda_2)<0<\Im(\lambda_1)<\Im(\lambda_0)$.

Theorems & Definitions (196)

• Conjecture A: = \ref{['conj:noncommutativeGamma']}\ref{['conj2SOD']}, simplified
• Conjecture B: = \ref{['conj:noncommutativeGamma']}\ref{['conj2independence']}, simplified
• Conjecture C: = \ref{['conj:noncommutativeGamma']}\ref{['conj2isomonodromic']}
• Theorem A
• Theorem B: = rest of \ref{['T:geomexistence']}
• Theorem C: = \ref{['thm_glued_geom_stab_cubic3']} + \ref{['thm_geomstab_cubic4_noplane']}
• Theorem D: = \ref{['C:NMMPFanoexamples']} + \ref{['T:conj2.1threefold']} + \ref{['T:conj2.1fourfold']}
• Theorem E: = \ref{['T:NCgammaforGrandQ']}
• Definition 2.1
• Definition 2.2