Moduli spaces of stable bundles on Calabi-Yau varieties and Donaldson-Thomas invariants
Given a smooth projective variety \(X\) with an ample line bundle, moduli spaces of Gieseker semistable sheaves on \(X\) are very interesting objects of study. If \(X\) is a surface, then quite a lot of results are present in the literature, but the investigation of fairly basic questions concerning, for example, the smoothness or the topological invariants of these moduli spaces on higher-dimensional varieties are, in general, difficult. The first goal of the paper under review is to investigate moduli spaces of sheaves on certain Calabi--Yau varieties, namely hypersurfaces of \(\mathbb{P}^1\times \mathbb{P}\), where \(\mathbb{P}\) is a projective bundle over \(\mathbb{P}^1\). The author proves that for certain choices of the ample line bundle \(L\) and Chern character \(c\), the moduli space of sheaves which are semistable with respect to \(L\) and have Chern character \(c\) is smooth and irreducible, in fact, isomorphic to a projective space resp.\ to some Quot-scheme. In any case, strictly semistable sheaves do not exist. The second goal of the paper is to present some explicit computations of Donaldson--Thomas (DT) invariants. Recall that DT-invariants roughly count stable sheaves on Calabi--Yau threefolds. The author uses the explicit description given in the first part, applies it to the three-dimensional case and derives the generating series of DT-invariants for these Calabi--Yau threefolds.
scientific article; zbMATH DE number 5956427
scientific article