Duke Mathematical Journal

Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration

Dima Arinkin and Roman Fedorov

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Abstract

Let X be an abelian scheme over a scheme B. The Fourier–Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular.

Our main result provides a fully faithful functor from a twist of the derived category of Picτ(X/B) to the derived category of X. Here Picτ(X/B) is the algebraic space classifying fiberwise numerically trivial line bundles.

Next, we show that every algebraically integrable system gives rise to a degenerate abelian scheme and discuss applications to Hitchin systems.

Article information

Source
Duke Math. J., Volume 165, Number 15 (2016), 2991-3042.

Dates
Received: 10 November 2014
Revised: 19 October 2015
First available in Project Euclid: 16 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1471368610

Digital Object Identifier
doi:10.1215/00127094-3645223

Mathematical Reviews number (MathSciNet)
MR3557277

Zentralblatt MATH identifier
06656239

Subjects
Primary: 14D24: Geometric Langlands program: algebro-geometric aspects [See also 22E57]
Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14K30: Picard schemes, higher Jacobians [See also 14H40, 32G20] 14L15: Group schemes

Keywords
Fourier–Mukai transform abelian scheme Picard space integrable system Hitchin system Langlands duality

Citation

Arinkin, Dima; Fedorov, Roman. Partial Fourier–Mukai transform for integrable systems with applications to Hitchin fibration. Duke Math. J. 165 (2016), no. 15, 2991--3042. doi:10.1215/00127094-3645223. https://projecteuclid.org/euclid.dmj/1471368610


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