Hitchin and Calabi–Yau integrable systems via variations of Hodge structures. (24th November 2020)

• Record Type: Journal Article
• Authors: Beck, Florian
• Is Part Of: Quarterly journal of mathematics. Volume 71:Part 4(2020)
• Page Start: 1345
• Page End: 1375
• Publication Date: 2020-11-24
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/qmath/haaa037
• Languages: English
• ISSNs: 0033-5606

Abstract

Abstract: Since its discovery by Hitchin in 1987, G -Hitchin systems for a reductive complex Lie group G have extensively been studied. For example, the generic fibers are nowadays well-understood. In this paper, we show that the smooth parts of G -Hitchin systems for a simple adjoint complex Lie group G are isomorphic to non-compact Calabi–Yau integrable systems extending results by Diaconescu–Donagi–Pantev. Moreover, we explain how Langlands duality for Hitchin systems is related to Poincaré–Verdier duality of the corresponding families of quasi-projective Calabi–Yau threefolds. Even though the statement is holomorphic-symplectic, our proof is Hodge-theoretic. It is based on polarizable variations of Hodge structures that admit so-called abstract Seiberg–Witten differentials. These ensure that the associated Jacobian fibration is an algebraic integrable system.