Existence of stable reflexive sheaves on certain threefolds (Q2332127)

\textit{M. R. Douglas}, \textit{R. Reinbacher} and \textit{S.-T. Yau} [``Branes, bundles and attractors: Bogomolov and beyond'', Preprint, \url{arXiv:math/0604597}], in relation with string theory, have conjectured the existence of \(\mu\)-stable reflexive sheaves on Calabi-Yau threefolds. The paper under review builds on the existing results on this subject by by giving a few new criteria for the existence of \(\mu\)-stable reflexive sheaves on smooth projective threefolds. The first main result of the paper review is as follows. Let \(X\) be a smooth projective threefold with \(H^i(\mathcal{O}_X) =0\) for \(i =1, 2\). Let \(Z_1,\dots, Z_n\) be disjoint irreducible Cohen-Macaulay curves of arithmetic genus \(p_a(Z_i)\), which are generically locally complete intersections. Assume \(\omega_{Z_i} \cong \omega_X|_{Z_i}\) (resp. \(p_a(Z_i)-1-K_X\cdot[Z_i]>0\)) for each \(i\), and also that there exists an irreducible surface \(S\subset X\) such that \(Z_i\cap S=\emptyset\) for each \(i\) and \(H^1(\mathcal{O}_S) =0\). Then there exists a rank two locally free (resp. reflexive) sheaf \(E\) on X \(\mu\)-stable with respect to any polarization and has Chern classes \(c_1(E) =0\), \(c_2(E) =\sum_{i=1}^n[Z_i]\), \(c_3(E)=\sum_{i=1}^n(2p_a(Z_i)-2-K_X\cdot Z_i\)). This result is used to prove the existence of \(\mu\)-stable reflexive sheaves on certain fibered threefolds.