Toward the noncommutative minimal model program for Fano varieties

TL;DR

This paper advances the noncommutative minimal model program for Fano varieties by constructing quantum cohomology lifts, verifying their properties, and establishing geometric stability conditions in key examples.

Contribution

It constructs lifts of quantum cohomology central charges for specific Fano varieties and verifies their quasi-convergence and semiorthogonal decompositions.

Findings

Lifts of quantum cohomology central charge constructed for Grassmannians, quadrics, and cubic varieties.

Verified quasi-convergence and semiorthogonal decompositions of derived categories.

Established geometric stability conditions and deformation properties.

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.