A construction of stable vector bundles on Calabi-Yau manifolds
Holomorphic vector bundles on Calabi-Yau manifolds have recently attracted some interest in the context of string theory. By mirror symmetry, stable ones should correspond to special Lagrangians in the mirror manifold. Unfortunately, the construction of vector bundles on higher dimensional varieties is a difficult problem. The often exploited tool of Serre's correspondence transposes this problem to the existence of special subvarieties; unfortunately, this question is usually intricate as well. In the paper under review, the author uses \textit{M. Maruyama}'s method of elementary transformations [in: Algebraic geometry, Proc. Int. Conf., La Rábida 1981, Lect. Notes Math. 961, 241-266 (1982; Zbl 0505.14009)] to construct special stable bundles on a polarized Calabi-Yau manifold \(X\). More precisely, these bundles are kernels of surjections \(H^0(D,L)\otimes \mathcal O_X\to i_* L\), where \(D\) is an effective divisor on \(X\) and \(L\) a globally generated line bundle on \(D\). It turns out that if \(L . H^{\dim X-1}\) is minimal for a polarization \(H\) of \(X\), then such a kernel is \(H\)-stable. This result is applied to the three-dimensional case. As a corollary, the author obtains the existence of stable bundles for certain triples of Chern classes, and studies the examples of the quintic threefold \(X \subset \mathbb{P}^4\) and of some other Calabi-Yau threefold hypersurfaces in weighted projective spaces in more detail.
scientific article; zbMATH DE number 2182215
scientific article