arXiv subjects

Alexander N. Varchenko

Publications and source records attributed to Alexander N. Varchenko.

2 recordsLinked to original sources

arXiv2011

The contravariant form on singular vectors of a projective arrangement

We define the flag space and space of singular vectors for an arrangement A of hyperplanes in projective space equipped with a system of weights a: A --> C. We show that the contravariant bilinear form of the corresponding weighted central arrangement induces a well-defined form on the space of singular vectors of the projectivization. If the sum of the weights a(H), H in A, is zero, then this form is naturally isomorphic to the restriction to the space of singular vectors of the contravariant form of any affine arrangement obtained from A by dehomogenizing with respect to one of its hyperplanes.

math.CO↗

arXiv2002

Resonant local systems on complements of discriminantal arrangements and sl_2 representations

We calculate the skew-symmetric cohomology of the complement of a discriminantal hyperplane arrangement with coefficients in local systems arising in the context of the representation theory of the Lie algebra sl_2. For a discriminantal arrangement in C^k, the skew-symmetric cohomology is nontrivial in dimension k-1 precisely when the "master function" which defines the local system on the complement has nonisolated critical points. In symmetric coordinates, the critical set is a union of lines. Generically, the dimension of this nontrivial skew-symmetric cohomology group is equal to the number of critical lines.

math.AG↗