Reflection of sheaves on a Calabi-Yau variety.
Operations on bundles preserving properties of their moduli can be found in one form or another in almost all papers on bundles. In this paper the operation of ``reflection'' is considered in the case of bundles on some Calabi-Yau varieties \(X\). Starting with the 2-dimensional case of a K3 surface \(X\), if \(E\) is a torsion free sheaf on \(X\) and \(\varphi:H^0(E) \otimes O_X \to E\) is the natural evalutation map then, depending on whether \(\varphi\) is injective or surjective, the reflection of \(E\) is by definition the cokernel or the kernel of \(\varphi\), respectively. Since the work of \textit{S. Mukai} [in: Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 341--413 (1987; Zbl 0674.14023)] and \textit{A. N. Tyurin} [Duke Math. J. 54, 1--26 (1987; Zbl 0631.14009)] it has been important to study the properties preserved by such transformations. The author here considers the case of sheaves on 3-dimensional Calabi-Yau varieties. Under suitable assumptions on the first Chern class, he shows that the stability condition is preserved under the reflection operation. Further a Brill-Noether locus of the moduli of sheaves on \(X\) is defined, and it is showed that the reflection induces isomorphisms between the Brill-Noether loci for different Mukai vectors.
scientific article; zbMATH DE number 1925772