Gluing stability conditions.

*(English)*Zbl 1210.18011Let \(\mathcal D\) 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.

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.

Reviewer: Marcello Bernardara (Essen)

##### 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) |