Stability conditions on generic complex tori (Q2885369)

A Bridgeland stability condition on a triangulated category \(\mathcal{D}\) consists, roughly speaking, of a heart \(\mathcal{A}\) of a bounded t-structure on \(\mathcal{D}\) and a group homomorphism \(Z: K(\mathcal{A})=K(\mathcal{D})\rightarrow \mathbb{C}\) (where \(K(-)\) denotes the Grothendieck group) satisfying some axioms. There is also a version where one uses the numerical Grothendieck group. The set of all (numerical) stability conditions forms a complex manifold. In particular, one would like to understand stability conditions when \(\mathcal{D}\) is the bounded derived category of coherent sheaves on a variety. Some investigated cases include curves, \(K3\) and abelian surfaces, and projective spaces, but in general it is very difficult to construct stability conditions in the geometric setting.NEWLINENEWLINEIn the paper under review the author studies the numerical stability manifold of a generic complex torus of dimension \(\geq 3\). Recall that a complex torus \(X\) is generic if \(H^{p,p}(X)\cap H^{2p}(X,\mathbb{Z})=0\) for all \(0<p<\text{dim}(X)\). The main result states that the set \(U(X)\) of all numerical stability conditions which satisfy that all skyscraper sheaves and all degree 0 line bundles are semistable of phase \(\phi\) resp.\ \(\psi\) is a simply connected component of the numerical stability manifold. Furthermore, \(U(X)\) can be written as a disjoint union of \(\widetilde{GL}^+(2,\mathbb{R})\)-orbits of certain stability conditions \(\sigma_{(p)}\) and \(\sigma_{(p)}^\gamma\), where \(\widetilde{GL}^+(2,\mathbb{R})\) is the universal cover of \(GL^+(2,\mathbb{R})\) and always acts on the stability manifold.NEWLINENEWLINEThe paper is organised as follows. Section 2 establishes some results on sheaves on a generic complex torus. In particular, any reflexive sheaf is proven to be locally free. In the next section the author constructs, using the method of tilting, hearts of bounded t-structures and stability functions on these hearts. This gives the stability conditions \(\sigma_{(p)}\) and \(\sigma_{(p)}^\gamma\) on \(X\) entering in the main result. Lastly, the last section investigates the topology of \(U(X)\).