The Michigan Mathematical Journal

Hyperplane Arrangements and Tensor Product Invariants

P. Belkale, P. Brosnan, and S. Mukhopadhyay

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Abstract

In the first part of this paper, we consider, in the context of an arbitrary weighted hyperplane arrangement, the map from compactly supported cohomology to the usual cohomology of a local system. We obtain a formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map.

In the second part, we apply these results to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels) the space of invariants acquires a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants and characterize the subspace of conformal blocks in Hodge theoretic terms.

Article information

Source
Michigan Math. J., Volume 68, Issue 4 (2019), 801-829.

Dates
Received: 17 November 2017
Revised: 7 December 2018
First available in Project Euclid: 8 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1565251217

Digital Object Identifier
doi:10.1307/mmj/1565251217

Mathematical Reviews number (MathSciNet)
MR4029630

Subjects
Primary: 32G34: Moduli and deformations for ordinary differential equations (e.g. Knizhnik-Zamolodchikov equation) [See also 34Mxx] 81T40: Two-dimensional field theories, conformal field theories, etc. 14D07: Variation of Hodge structures [See also 32G20]

Citation

Belkale, P.; Brosnan, P.; Mukhopadhyay, S. Hyperplane Arrangements and Tensor Product Invariants. Michigan Math. J. 68 (2019), no. 4, 801--829. doi:10.1307/mmj/1565251217. https://projecteuclid.org/euclid.mmj/1565251217


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