Abstract: 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 the Milnor formula, 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.
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