Quantum Cohomology of Complete Intersections
Quantum Cohomology of Complete Intersections
复制标题
完全交集的量子上同调
DOI：
--
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
A. Beauville
中科院分区：
文献类型：
--
作者：
A. Beauville
引言射影流形X的量子上同调代数是赋予X不同代数结构的上同调，它考虑了X中有理曲线的几何形状。这种结构首先由数学物理学家[V，W]定义;最近阮和田[R-T]已经实现了一个严格的结构（以及结合性的证明，这是非常重要的）。当计算例如表面时，量子上同调看起来相当复杂[C-M]。本说明的目的是表明，当维数相对于次数变高时，情况会大大改善。我们的主要结果是：定理.−设X ∈ P是一个光滑的完全交的次数（d1，. . .，dr）且维数n ≥ 3，其中n ≥ 2 ∑（di − 1）− 1。设d = d1。. . dr和δ = ∑（di − 1）。量子上同调代数H <$（X，Q）是由超平面类H和本原上同调H（X，Q）o生成的代数，关系式为：
Introduction The quantum cohomology algebra of a projective manifold X is the cohomology of X endowed with a different algebra structure, which takes into account the geometry of rational curves in X . This structure has been first defined heuristically by the mathematical physicists [V,W]; a rigorous construction (and proof of the associativity, which is highly non trivial) has been achieved recently by Ruan and Tian [R-T]. When computed e.g. for surfaces, the quantum cohomology looks rather complicated [C-M]. The aim of this note is to show that the situation improves considerably when the dimension becomes high with respect to the degree. Our main result is: Theorem .− Let X ⊂ P be a smooth complete intersection of degree (d1, . . . , dr) and dimension n ≥ 3 , with n ≥ 2 ∑ (di − 1)− 1 . Let d = d1 . . . dr and δ = ∑ (di − 1) . The quantum cohomology algebra H∗(X,Q) is the algebra generated by the hyperplane class H and the primitive cohomology H(X,Q)o , with the relations: