论文标题
Fano品种的Lefschetz缺陷
The Lefschetz defect of Fano varieties
论文作者
论文摘要
该注释是对Lefschetz缺陷的简短调查，该缺陷是最近引入的光滑Fano品种不变的。它与x的picard rho（x）以及X中的PICARD数字有关。我们说明Lefschetz缺陷delta（x）及其起源的定义，已知结果，主要是在案例三角洲（x）> 1的情况下，以及几个示例，尤其是在3和4中允许在3和4中解释范围的范围。 Rho（X）> 4，由于Mori和Mukai。我们还以Rho（X）> 5的速度回顾了Fano 4倍的已知示例，这些例子很少：分开的产品和曲折的例子，只有6个已知的家庭，{6,7,8,9}的Rho（x）。
This note is a short survey on the Lefschetz defect, an invariant of smooth Fano varieties that has been recently introduced; it is related to the Picard number rho(X) of X, and to the Picard number of prime divisors in X. We explain the definition of the Lefschetz defect delta(X) and its origin, the known results - mainly on the case delta(X)>1, and several examples, especially in dimensions 3 and 4. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with rho(X)>4, due to Mori and Mukai. We also review the known examples of Fano 4-folds with rho(X)>5, that are remarkably few: apart products and toric examples, there are only 6 known families, with rho(X) in {6,7,8,9}.
