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Creators/Authors contains: "Collier, Brian"

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  1. ; ; ; ; (, Annals of Mathematics)
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  2. ; (, Transactions of the American Mathematical Society)
    The space H 4 , 2 \mathbf {H}^{4,2} of vectors of norm −<#comment/> 1 -1 in R 4 , 3 \mathbb {R}^{4,3} has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form G 2 ′ \mathsf {G}_2’ . In this paper we consider a class of holomorphic curves in H 4 , 2 \mathbf {H}^{4,2} which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group is naturally parameterized by certain G 2 ′ \mathsf {G}_2’ -Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichmüller space with a holomorphic action of the mapping class group. Using a generalization of Labourie’s cyclic surfaces, we then show that equivariant alternating holomorphic curves are infinitesimally rigid. 
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  3. ; ; ; (, Compositio Mathematica)
    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. 
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  4. ; (, Advances in Mathematics)
    null (Ed.)
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