The Michigan Mathematical Journal

Hyperplane Arrangements and Tensor Product Invariants

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

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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

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

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

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Mathematical Reviews number (MathSciNet)

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]


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.

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