Reflection of sheaves on a Calabi–Yau variety
Reflection of sheaves on a Calabi–Yau variety
复制标题
卡拉比-丘品种上滑轮的反映
DOI：
10.4310/ajm.2002.v6.n3.a8
复制
发表时间：
2002
影响因子：
0.6
通讯作者：
T. Nakashima
中科院分区：
文献类型：
--
作者：
T. Nakashima
本文研究了稳定层在Calabi-Yau簇上的反射及其对模空间的影响。它表明，反射定义的Brill-Noether轨迹之间的模空间的同构。导论.设E是光滑射影K3曲面X上的无挠层，tp：H（X，E）0 Ox -> E表示自然评价映射.如果（p）是内射或满射，则它的上核或核称为E的反射。反射函子首先由Mukai（[Mu]）引入，此后它被用于研究K3曲面（[Ma]，[N]，[Y]）上稳定层的模空间。考虑高维Calabi-Yau簇X上的反射函子似乎很有意义，因为Kontsevich的同调镜像猜想预言了X上相干层的导出范畴D（X）与其镜像的导出福谷范畴的等价性。受到猜想的启发，Seidel和托马斯最近引入了一个自动等价T？：D（X）-> D（X）称为关于球形对象f（[ST]）的扭曲函子。对于J ∈ D（X），Te（！F）定义为L是映射Hom（？）0 ε-> J，这与Mukai在ε = Ox的情况下的反射相一致然而，这个函子如何与层的稳定性相关的问题还没有得到解决。本文研究了反射函子对高维Calabi-Yau簇上稳定层模空间的影响。我们将证明，在适当的极小性假设的第一陈类，反射保持稳定的层上的任意光滑投影簇。进一步，我们定义了Calabi-Yau簇上层的模空间的Brill-Noether轨迹，并证明了反射诱导不同Mukai向量的Brill-Noether轨迹之间的同构.这是文献[Ma]，[Y]中关于K3曲面的结果的高维推广.我们还考虑了弦理论中出现的某个卡-丘三重性的反射的例子。最后，我们要感谢审稿人提出的宝贵意见和对原稿错误的纠正。1.束的反射。设X是定义在复数域C上的维数为d的光滑射影簇，H是X上的丰富线丛.对于线丛L E PicX，令degL = L · H~表示其度。最小H度dm[n（H）]被定义为以下正整数dmin（tf）= min{degM| M e Pic（X），deg M > 0}。X上的一个线丛C称为H-极小的，如果deg E = dm | n（H）.例如，在以下情况之一中，E是最小的：（1）PicX = Z[H} &nd E = H]（2）degC = 1。* 2001年8月30日接收; 2002年7月26日接受出版。t东京都立大学数学系，地址：Minami-Ohsawa 1-1，Hachiojishi，Tokyo 192-0397，Japan（nakasima@comp.metro-u.ac.jp）。
In this paper we study the reflection of stable sheaves on Calabi-Yau varieties and its effect on the moduli space. It is shown that the reflection defines isomorphisms between the Brill-Noether loci of moduli spaces. Introduction. Let E be a torsion-free sheaf on a smooth projective K3 surface X and let tp : H(X, E) 0 Ox —> E denote the natural evaluation map. If (p is either injective or surjective, then its cokernel or kernel is called the reflection of E. The reflection functor was first introduced by Mukai ([Mu]) and since then it has been exploited for the study of the moduli space of stable sheaves on K3 surfaces ([Ma], [N],[Y]). It seems significant to consider the reflection functor on higher dimensional CalabiYau variety X, in view of Kontsevich's homological mirror conjecture which predicts the existence of equivalence of the derived category D(X) of coherent sheaves on X and the derived Fukaya category of its mirror. Inspired by the conjecture, Seidel and Thomas recently introduced an auto equivalence T? : D(X) —> D(X) called the twist functor with respect to a spherical object f ([ST]). For J £ D(X), Te(!F) is defined to L be the cone of the map Hom^,.?) 0 £ —> J, which coincides with Mukai's reflection in case £ = OxHowever, the problem how this functor is related to the stability of sheaves has not been addressed. In this paper we study the effect of the reflection functor on the moduli space of stable sheaves on higher dimensional Calabi-Yau varieties instead of the derived category. We shall show that under suitable minimality assumption on the first Chern classes, the reflection preserves the stability of sheaves on arbitrary smooth projective varieties. Further we define the Brill-Noether locus of the moduli space of sheaves on Calabi-Yau varieties and prove that the reflection induces isomorphisms between the Brill-Noether loci for different Mukai vectors. This is a higher dimensional generalization of the results in [Ma],[Y] obtained for K3 surfaces. We also consider examples of reflections on a certain Calabi-Yau threefold which appears in string theory([COFKM]). Finally we would like to express our gratitude to the referee for giving valuable suggestions and correcting mistakes in the original manuscript. 1. Reflection of sheaves. Let X be a smooth projective variety of dimension d defined over the complex number field C and let H be an ample line bundle on X. For a line bundle L E PicX, let degL = L • H~ denote its degree. The minimal H-degree dm[n(H) is defined to be the following positive integer dmin(tf) = min{degM | M e Pic(X), deg M > 0}. A line bundle C on X is said to be H-minimal if deg£ = dm\n(H). For example, £ is iJ-minimal in one of the following cases: (1) PicX^Z[H} &nd£ = H] (2) degC = l. * Received August 30, 2001; accepted for publication July 26, 2002. t Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachiojishi, Tokyo 192-0397, Japan (nakasima@comp.metro-u.ac.jp).