论文标题
对称的双线性形式和局部依次呈阳性特征中孤立的奇异性因子
Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic
论文作者
论文摘要
令$ f \ colon x \ to \ mathbb {a}^1_k $是从光滑品种到具有孤立单数点的仿射线的形态。对于这样的奇异性，我们有两个不变的人。一种是非脱位对称双线性形式（de rham），另一个是消失的周期复合物（étale）。 在本文中，我们给出了一个公式，该公式以双线性形式表示消失循环复合物的局部epsilon因子。特别是，局部epsilon因子的符号取决于双线性形式的判别。可以将此公式视为Milnor公式的改进，该公式比较了消失的周期的总维度和双线性形式的等级。 在特征性$ 2 $中，我们发现ARF不变性的概括，可以将其视为一般孤立的奇异性的非二次二次象征性的不变性。
Let $f\colon X\to\mathbb{A}^1_k$ be a morphism from a smooth variety to an affine line with an isolated singular point. For such a singularity, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (étale). In this article, we give a formula which expresses the local epsilon factor of the vanishing cycles complex in terms of the bilinear form. In particular, the sign of the local epsilon factor is determined by the discriminant of the bilinear form. This formula can be thought as a refinement of the Milnor formula, which compares the total dimension of the vanishing cycles and the rank of the bilinear form. In characteristic $2$, we find a generalization of the Arf invariant, which can be regarded as an invariant for non-degenerate quadratic singularities, to general isolated singularities.
