Perverse sheaf

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In mathematics, a perverse sheaf is an element of the derived category of sheaves with certain properties. This means that it is not a sheaf as such (it is an equivalence class of chain complexes of sheaves). The name "perverse sheaf" is misleading, in other words: a perverse sheaf is not a sheaf, but can be represented by a complex of sheaves. The name comes from perversity in intersection homology.

The concept was introduced by Joseph Bernstein, Alexander Beilinson, Pierre Deligne, and Ofer Gabber (1982). In the Riemann-Hilbert correspondence, perverse sheaves correspond to holonomic D-modules. This application established the notion of perverse sheaf as occurring 'in nature'.

[edit] Definition

A perverse sheaf is an element C of the bounded derived category of sheaves with constructible cohomology on a space X such that the set of points x with

$H^{-i}(j_x^*C)\ne 0$ or $H^{i}(j_x^!C)\ne 0$

has dimension at most 2i, for all i. Here j_x is the inclusion map of the point x.

The category of perverse sheaves is an abelian subcategory of the (non-abelian) derived category of sheaves, equal to the core of a suitable t-structure, and is preserved by Verdier duality.