2024/12/11 by Tom Bridgeland, Bridgeland, Tom, Fabrizio Del Monte +3 · 2 citations
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2412.08531
We apply results on inducing stability conditions to local Calabi-Yau threefolds and obtain applications to Donaldson-Thomas (DT) theory. A basic example is the total space of the canonical bundle of Z=ℙ1× ℙ1. We use a result of Dell to construct stability conditions on the derived category of X for which all stable objects can be explicitly described. We relate them to stability conditions on the resolved conifold Y=\mathscrOℙ1(-1)⊕ 2 in two ways: geometrically via the McKay correspondence, and algebraically via a quotienting operation on quivers with potential. These stability conditions were first discussed in the physics literature by Closset and del Zotto, and were constructed mathematically by Xiong by a different method. We obtain a complete description of the corresponding DT invariants, from which we can conclude that they define analytic wall-crossing structures in the sense of Kontsevich and Soibelman. In the last section we discuss several other examples of a similar flavour.