矩阵奇异性的消失圈
Vanishing cycles of matrix singularities
摘要：
本文是关于具有足够多参数的任意平方、对称和反对称矩阵的全纯族的奇异Milnor纤维的消失拓扑。我们在这类纤维上定义了消失圈，证明了Damon-Pike $μ=τ$猜想关于一类特殊类型的族的一个推广形式，并对矩阵奇异性的单值性的理解迈出了第一步.我们还证明了一个Lyashko Looijenga型定理简单的矩阵家庭，并指出了一个令人惊讶的关系，某些Shephard-Todd组，简单的奇函数和零星的一部分，布鲁斯-塔里简单的矩阵分类。
英文摘要：
The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $μ=τ$ conjecture about the families of a special type, and make first steps towards understading of the monodromy of matrix singularities. We also prove a Lyashko-Looijenga type theorem for simple matrix families, and point out a surprising relationship between certain Shephard-Todd groups, simple odd functions and a sporadic part of the Bruce-Tari simple matrix classification.
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