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Arakelov–Milnor inequalities and maximal variations of Hodge structure.
- Source :
- Compositio Mathematica; May2023, Vol. 159 Issue 5, p1005-1041, 37p
- Publication Year :
- 2023
-
Abstract
- In this paper we study the $\mathbb {C}^*$ -fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. An important result of this paper is a bound on this invariant which generalizes the Milnor–Wood inequality for a Hodge bundle in the Hermitian case, and is analogous to the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case. [ABSTRACT FROM AUTHOR]
- Subjects :
- SEMISIMPLE Lie groups
RIEMANN surfaces
TOPOLOGICAL property
LIE groups
VECTOR spaces
Subjects
Details
- Language :
- English
- ISSN :
- 0010437X
- Volume :
- 159
- Issue :
- 5
- Database :
- Complementary Index
- Journal :
- Compositio Mathematica
- Publication Type :
- Academic Journal
- Accession number :
- 163518579
- Full Text :
- https://doi.org/10.1112/S0010437X23007157