Global boundedness for semistable decorated principal bundles with special regard to quiver sheaves (Q2871009)

The paper under review studies the wall-crossing problem for decorated principal bundles and quiver sheaves, leading to the result that there exists a finite number of chambers in the space of different stability parameters on which the stability notion does not change for the objects considered.NEWLINENEWLINEA decorated principal bundle consists on a \(G\)-principal bundle defined over a big open set of a complex projective variety \(X\), together with a representation \(\sigma: G\rightarrow \mathrm{GL}(V)\), a decomposition \(V=\bigoplus V_{i}\) in irreducible components, a choice of line bundles \(\mathcal{L}_{i}\) for each component and a homomorphism from the associated vector bundle by \(\sigma\) to each component to the corresponding line bundle.NEWLINENEWLINEQuiver sheaves are defined as representations of a finite quiver on the category of coherent sheaves, where the vertices correspond to torsion free \(\mathcal{O}_{X}\)-modules and the arrows are sheaf homomorphisms twisted by fixed vector bundles for each arrow. This way, quiver sheaves are particular cases of decorated principal bundles, where the group \(G=\mathrm{GL}_{\underline{n}}(\mathbb{C})\) (a product of general linear groups) acts on the finite dimensional vector space of representations which is the sum over all the arrows of the vector space of homomorphisms between the sheaves. The dimension vector \(\underline{n}\) is fixed by prescribing the ranks of the sheaves for each vertex of the quiver.NEWLINENEWLINEThe stability parameters for quiver sheaves appear to be of two kinds, ones denoted by \(\underline{\kappa}\) corresponding to the choice of a representation of the structure group \(G\), and others denoted by \(\underline{\alpha}\) related to the choice of a linearization of the action on the representation variety of the quiver.NEWLINENEWLINEThe article studies the stability chambers for the different parameters \(\underline{\alpha}\) . It proves the existence of bounds on the slopes of the subbundles appearing on the subobjects of quiver sheaves (c.f. Main Theorem 1.5), which leads to the existence of a stack of finite type over \(\mathbb{C}\) comprising all quiver sheaves of a fixed topological type, for which there exists at least a stability parameter \(\underline{\alpha}\) such that they are semistable with respect to it. Finally, Theorem 5.3 shows that there are a finite number of stability regions where the semistable objects do not change when varying the parameter \(\underline{\alpha}\). Besides, in the closure of such regions the natural implications between stable and semistable objects with respect to different parameters hold.