Moduli spaces of stable bundles on Calabi–Yau varieties and Donaldson–Thomas invariants

Let YY be a smooth Calabi–Yau hypersurface of P1×PP1×P where PP stands for a PdPd-bundle over P1P1. We will prove that for many ample line bundles LL and certain Chern characters cc, the moduli space M¯L(c) (resp.ML(c)ML(c)) of LL-Gieseker semistable (resp. LL-stable ) rank two torsion free sheaves (resp. vector bundles) on YY with Chern character cc are smooth and irreducible and we will compute its dimension. Moreover, we will prove that both moduli spaces coincide. As a byproduct of the geometrical description of these moduli spaces, we will compute the Donaldson–Thomas invariants of some Calabi–Yau 3-folds.

Let YY be some Calabi–Yau hypersurface.
► Prove that moduli spaces of semistable rank two torsion free sheaves on YY are smooth.
► We also prove that they are irreducible and compute its dimension.
► We compute the dimension of these moduli spaces.
► As a byproduct, we get the Donaldson–Thomas invariants of some Calabi–Yau 3-folds.