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| 開催情報 | 水曜日 17:00～18:00 数理科学研究科棟(駒場) 117号室 |
|---|---|
| 担当者 | 今井 直毅，ケリー シェーン |
2020年11月18日(水)
17:00-18:00 オンライン開催
竹内 大智 氏 (東京大学数理科学研究科)
Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic (Japanese)
竹内 大智 氏 (東京大学数理科学研究科)
Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic (Japanese)
[ 講演概要 ]
For a function on a smooth variety with an isolated singular point, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (\'etale). The latter is a Galois representation of a local field measuring a complexity of the singularity.
In this talk, I will 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 can be regarded as a refinement of Milnor formula in SGA 7, which compares the rank of the bilinear form and the total dimension of the vanishing cycles.
In characteristic 2, we find a generalization of Arf invariant, which can be regarded as an invariant for a non-degenerate quadratic singularity, to a general isolated singularity.
For a function on a smooth variety with an isolated singular point, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (\'etale). The latter is a Galois representation of a local field measuring a complexity of the singularity.
In this talk, I will 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 can be regarded as a refinement of Milnor formula in SGA 7, which compares the rank of the bilinear form and the total dimension of the vanishing cycles.
In characteristic 2, we find a generalization of Arf invariant, which can be regarded as an invariant for a non-degenerate quadratic singularity, to a general isolated singularity.
