Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants

By D. Piyaratne, Yukinobu Toda. Journal für die Reine und Angewandte Mathematik, 2015

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In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/67194c43ec8df0c24e040106ddf07dd60afacff8

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In Geometry and Topology

In Mathematical Physics

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Stability conditions on Kuznetsov components (2017, methodology): “ our general method applies; see Section 6. In order to prove Theorem 1.2 with the same method, we would have to tilt a third time; this would require a conjectural Bogomolov-Gieseker type inequality [PT15] that is not currently known for any fourfold. Instead, we use the rational ﬁbration in conics”