Spinor bundles on quadrics
Spinor bundles on quadrics
复制标题
二次曲面上的旋量丛
DOI：
--
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
G. Ottaviani
中科院分区：
文献类型：
--
作者：
G. Ottaviani
定义了维数为n的复二次超曲面Qn上的一些稳定向量束，作为Q4 ~ Gr（1,3）上的泛束和商束的对偶的自然推广。我们称之为旋量束。当n = 2fc - 1时，存在一个秩为2k~1的旋量束。当n = 2k时，存在两个秩为2k~1的旋量束。当n = 0 （mod 4）或n = 2 （mod 4）时，它们的行为略有不同。作为应用，我们描述了Q5和Qe上的一些3阶向量束的模空间。设Qn为复射影空间Pn+1的光滑二次超曲面。本文用几何方法定义了二次型Qn上的一些向量束，它们是Q4 ~ Gr（1,3）上的泛束的自然推广和商束的对偶。我们称之为旋量束。在Q4上，这个定义等同于通常的定义。旋量束是齐次且稳定的（根据Mum- ford-Takemoto的定义）。我们利用给出的几何描述和（OSS）中可用的一些标准技术研究了它们的第一性质。我们还使用了关于不可约表示诱导的齐次神经束的稳定性的Ramanan定理（见（Um））。当n为奇数时，只有一个旋量束，而当n为偶数时，有两个非同构旋量束。当n为偶数时，根据n = 0 （mod4）或n = 2 (mod4)，旋量束的行为略有不同。在(2)中，我们给出了二次曲面上包含旋量束的向量束的上同调分裂准则。Qn ~ Spin(n + 2)/P(cty) （St）是齐次流形，P（ai）的李代数的半单元部分为o(n)。在李代数的水平上，从0 (n)的自旋和半自旋表示定义了自旋束。本文分为三个部分。在§1中我们给出了一些初步的结果并定义了旋量束。在§2中，我们研究了旋量束的第一性质。在§3中，作为一个应用，我们描述了Q5和Qq上的一些3阶向量束的模空间。
We define some stable vector bundles on the complex quadric hypersurface Qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. When n = 2fc — 1 there is one spinor bundle of rank 2k~1. When n = 2k there are two spinor bundles of rank 2k~1. Their behavior is slightly different according as n = 0 (mod 4) or n = 2 (mod 4). As an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qe- Introduction. Let Qn be the smooth quadric hypersurface of the complex pro- jective space Pn+1. In this paper we define in a geometrical way some vector bundles on the quadric Qn as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. On Q4 this definition is equivalent to the usual one. Spinor bundles are homogeneous and stable (according to the definition of Mum- ford-Takemoto). We study their first properties using the geometrical description given and some standard techniques available in (OSS). We also use a theorem of Ramanan (see (Um)) about the stability of homoge- neous bundles induced by irreducible representations. When n is odd there is only one spinor bundle, while when n is even there are two nonisomorphic spinor bun- dles. When n is even the behavior of spinor bundles is slightly different according as n = 0 (mod4) or n = 2 (mod4). In (Ot2) we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles. Qn ~ Spin(n + 2)/P(cty) (St) is a homogeneous manifold, and the semisimple part of the Lie algebra of -P(ai) is o(n). At the level of Lie algebras, spinor bundles are defined from the spin and half-spin representations of o(n). The paper is divided into three sections. In §1 we give some preliminary results and we define the spinor bundles. In §2 we study the first properties of spinor bundles. In §3, as an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qq.