Perverse sheaves with singularities along the curve \(y^ n=x^ m\)
The authors prove that the category of perverse sheaves on the complex two-space \({\mathbb{C}}^ 2\), constructible with respect to the stratification \(\{0\}\subset \{y^ n=x^ m\}\subset {\mathbb{C}}^ 2\), \(n\leq m\), is equivalent to the category of \((n+2)\)-tuples of vector spaces \(A,B_ 1,...,B_ n,C\) together with maps \(A\rightleftarrows^{q_ k}_{p_ k}B_ k\rightleftarrows^{s_ k}_{r_ k}C\) and \(\theta_ k:B_ k\to B_{k+m}\) (all the indices are to be considered as integers modulo n), which satisfy some relations, explicitely given. The proof uses methods of the authors given in C. R. Acad. Sci., Paris, Ser. I. 299, 443-446 (1984; Zbl 0581.14013), and Invent. Math. 84, 403-435 (1986; Zbl 0597.18005). As an application one classifies perverse sheaves with no vanishing cycles at the origin for the case \(y^ 2=x^ 3\); in this case the nontrivial irreducible perverse sheaves are parameterized by one complex number.
scientific article; zbMATH DE number 4040291