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For a split reductive group defined over a number field, we first introduce the notations of arithmetic torsors and arithmetic Higgs torsors. Then we construct arithmetic characteristic curves associated to arithmetic Higgs torsors, based on the Chevalley characteristic morphism and the existence of Chevalley basis for the associated Lie algebra. As to be expected, this work is motivated by the works of Beauville-Narasimhan on spectral curves and Donagi-Gaistgory on cameral curves in algebraic geometry. In the forthcoming papers, we will use arithmetic characteristic curves to construct arithmetic Hitchin fibrations and study the intersection homologies and perverse sheaves for the associated structures, following Ngos approach to the fundamental lemma.
We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can be writte
In this paper we study local-global principles for tori over semi-global fields, which are one variable function fields over complete discretely valued fields. In particular, we show that for principal homogeneous spaces for tori over the underlying
Let $G$ be a reductive group, and let $X$ be an algebraic curve over an algebraically closed field $k$ with positive characteristic. We prove a version of nonabelian Hodge correspondence for $G$-local systems over $X$ and $G$-Higgs bundles over the F
We study the normal map for plane projective curves, i.e., the map associating to every regular point of the curve the normal line at the point in the dual space. We first observe that the normal map is always birational and then we use this fact to
This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P^2 or P^1 x P^1 with a cusp or a prescribed t