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Gluing stability conditions. (English) Zbl 1210.18011
Let $$\mathcal D$$ be a linear triangulated category. A stability condition on $$\mathcal D$$ is given by the heart $$H$$ of a bounded $$t$$-structure and a function $$Z: K_0({\mathcal D}) \to {\mathbb C}$$ satisfying some axioms (in particular the Harder–Narasimhan property). In this paper the authors describe, given a semiorthogonal decomposition $${\mathcal D} = \langle {\mathcal D}_1, {\mathcal D}_2 \rangle$$, how to glue stability conditions on $${\mathcal D}_1$$ and $${\mathcal D}_2$$ to give a stability condition on $$\mathcal D$$.
If $$(H_1,Z_1)$$ and $$(H_2,Z_2)$$ are stability conditions on $${\mathcal D}_1$$ and $${\mathcal D}_2$$ respectively, satisfying $$\mathrm{Hom}^{\leq 0} (H_1,H_2) = 0$$, then the smallest full subcategory $$H$$ of $$\mathcal D$$ containing $$H_1$$ and $$H_2$$ and closed under extensions is the heart of a $$t$$-structure. Then $$Z_1$$ and $$Z_2$$ naturally define a function $$Z: K_0({\mathcal D}) \to \mathbb{C}$$ and it is easy to show that $$Z$$ always satisfies all the required axiom but the Harder–Narasimhan property. The authors provide, in the case $$(H_i,Z_i)$$ enjoy the nice property of being reasonable, two sufficient criteria for $$Z$$ to be a stability condition: the first one requires discreteness conditions on the stabilities $$(H_i,Z_i)$$ and the second one a stronger orthogonality between $$H_1$$ and $$H_2$$.
Finally, they provide some example of glued stabilities, in particular looking at double covering $$X \to Y$$ of smooth projective varieties and at the derived category $${\mathcal D}_{{\mathbb Z}_2}(X)$$ of $${\mathbb Z}_2$$-equivariant sheaves on $$X$$. The first example they describe is given by $$Y={\mathbb P}^3$$. The case of $$X$$ and $$Y$$ curves is also extensively treated: the stability space is indeed completely described when the genus of $$Y$$ is positive.

MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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