Quiver $D$-Modules and Homology of Local Systems over an Arrangement of Hyperplanes

Details Quiver $D$-Modules and Homology of Local Systems over an Arrangement of Hyperplanes Quiver $D$-Modules and Homology of Local Systems over an Arrangement of Hyperplanes

2005-10-21

arxiv.org/abs/math/0510451v2

Mathematics

Quantum Algebra

Revised version, 92 pages

Scientific paper

Let $C$ be an arrangement of affine hyperplanes in a complex affine space $X$, $D$ the ring of algebraic differential operators on $X$. We define a category of quivers associated with $C$. A quiver is a collection of vector spaces, attached to strata of the arrangement, and suitable linear maps between the vector spaces . To a quiver we assign a $D$-module on $X$, called the quiver $D$-module. We describe basic operations for $D$-modules in terms of linear algebra of quivers. We give an explicit construction of a free resolution of a quive $D$-module and use the construction to describe the associated perverse sheaf. As an application, we calculate the cohomology of $X$ with coefficients in the quiver perverse sheaf (under certain assumptions).

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