Arakelov Geometry over Adelic Curves

@inproceedings{Chen2020ArakelovGO,
  title={Arakelov Geometry over Adelic Curves},
  author={Huayi Chen and Atsushi Moriwaki},
  year={2020}
}
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. 
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15 Citations
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Background Citations
4
Methods Citations
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15 Citations

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Arithmetic over trivially valued field and its applications
By some result on the study of arithemtic over trivially valued field, we find its applications to Arakelov geometry over adelic curves. We prove a partial result of the continuity of arithmetic…
Volume function over a trivially valued field
We introduce an adelic Cartier divisor over a trivially valued field and discuss the bigness of it. For bigness, we give the integral representation of the arithmetic volume and prove the existence…
The theta invariants and the volume function on arithmetic varieties
We introduce a new arithmetic invariant for hermitian line bundles on an arithmetic variety. We use this invariant to measure the variation of the volume function with respect to the metric. The main…
A relative bigness inequality and equidistribution theorem over function fields
. — For any line bundle written as a subtraction of two ample line bundles, Siu’s inequality gives a criterion on its bigness. We generalize this inequality to a relative case. The arithmetic meaning…
The continuity of $\chi$-volume functions over adelic curves
In the setting of Arakelov geometry over adelic curves, we introduce the $\chi$-volume function and show some general properties. This article is dedicated to talk about the continuity of…
Arithmetic Okounkov bodies and positivity of adelic Cartier divisors
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov…
Toward Fermat's conjecture over arithmetic function fields.
Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show…
Infinite rank Hermitian Lattices and Loop Groups
— In this paper, we associate a family of infinite-rank pro-Hermitian Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac-Moody…
Successive minima and asymptotic slopes in Arakelov Geometry
Let X be a normal and geometrically integral projective variety over a global field K and let D be an adelic Cartier divisor on X. We prove a conjecture of Chen, showing that the essential minimum…
Categorification of Harder-Narasimhan Theory via slope functions
where the subquotients Ei/Ei−1 are semi-stable vector bundles with strictly decreasing slopes. Later Harder-Narasimhan filtration has been generalized to the setting of pure sheaves over higher…
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References

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