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Authors
Eckart Viehweg
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We discuss several numerical conditions for families of projective varieties or variations of Hodge structures.
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In this article, we construct a $theta$-density for the global sections of ample Hermitian line bundles on a projective arithmetic variety. We show that this density has similar behaviour to the usual density in the Arakelov geometric setting, where only global sections of norm smaller than $1$ are considered. In particular, we prove the analogue by $theta$-density of two Bertini kind theorems, on irreducibility and regularity respectively.
This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesnt require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with exotic objects such as F_1 (field with one element), Z_infty (real integers), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models.
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
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. A main result of this paper is a bound on this invariant which generalizes both the Milnor-Wood inequality of the Hermitian case and 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.
Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank >1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds. The revised version corrects several mistakes and ambiguities, pointed out by the referee. Following suggestions of the referee the presentation of the results was improved.
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