Reflection of sheaves on a Calabi-Yau variety.
type: Item
instance of: scholarly article
title: Reflection of sheaves on a Calabi-Yau variety.
zbMATH Open document ID: 1039.14022
author: Tohru Nakashima
published in: The Asian Journal of Mathematics
publication date: 2003-10-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14H51
zbMATH DE Number: 1925772
zbMATH Keywords: stable sheaves
zbMATH Keywords: Calabi-Yau varieties
reviewed by: Carlo Giovanni Madonna
MaRDI profile type: Publication
DOI: 10.4310/AJM.2002.V6.N3.A8
description: scientific article; zbMATH DE number 1925772