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Perverse sheaves with singularities along the curve \(y^ n=x^ m\). (English) Zbl 0638.32014

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.

MSC:

32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
32B15 Analytic subsets of affine space
14E20 Coverings in algebraic geometry
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