Oscar García-Prada (ICMAT-CSIC) is co-author of the paper “Arakelov-Milnor inequalities and maximal variations of Hodge structure”, published in Compositio Mathematica in 2023.

Since their introduction more than 35 years ago, Hitchin moduli spaces of Higgs bundles over compact Riemann surfaces have been of tremendous interest in geometry, topology and theoretical physics. These spaces have an extremely rich geometry coming from the fact that they are hyper-Kähler, they define completely integrable systems, and by the non-abelian Hodge correspondence, they are identified with character varieties of surface group representations. These moduli spaces also play a central role in mirror symmetry and Langlands duality.

Within the moduli space of Higgs bundles there is a special subvariety determined by the fixed points of the C*-action obtained by scaling the Higgs field. These fixed points are called Hodge bundles and correspond to holonomies of complex variations of Hodge structure. They are part of the global nilpotent cone, and coincide with critical points of a natural energy function on the moduli space. Another importance of the C*-fixed points stems from the fact that, roughly speaking, the subvariety of Hodge bundles determines the topology of the moduli space of Higgs bundles via different localization methods.

In this paper, the authors establish some basic properties of Hodge bundles and their moduli spaces. They introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. A main 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, the authors establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Toledo Higgs bundles and their associated maximal representations in the Hermitian case. The theory developed in this paper opens the door to a systematic study of the topology of moduli spaces of Hodge bundles, and hence the topology of the moduli spaces of Higgs bundles for arbitrary reductive groups.