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Paris-Barcelona-Kyoto seminar on Arakelov geometry

Lecture Room 110, Department of Mathematics, Kyoto University

September 18-21, 2012

Titles and abstracts of the talks

José Ignacio Burgos Gil (CSIC & ICMAT). "The arithmetic Riemann-Roch theorem for projective morphism"
Abstract:    In this talk I will explain how to generalize the Arithmetic Riemann Roch theorem to projective morphisms that are not necessarily generically smooth by using the generalized analytic torsion forms.

Huayi Chen (Inst. Math. Jussieu). "Degree function on vector subspaces and tensorial semistability"
Abstract:    In this talk, I will discuss natural conditions on a function defined on the set of vector subspaces of a given vector space under which the function behaves like the Arakelov degree function. This vision will help us to better understand the problem of tensorial semistability and leads to non-trivial applications. This is a joint work with Jean-BenoĆ®t Bost.

Hideaki Ikoma (Kyoto Univ.). "On the existence of strictly effective basis on an arithmetic variety"
Abstract:    Let $X$ be a projective arithmetic variety, $\overline{L}$ a hermitian line bundle on $X$, and $h_{\overline{L}}:X(\overline{\mathbb{Q}})\to \mathbb{R}$ the height function associated to $\overline{L}$. It has been known that the height function closely reflects the positivity properties of $\overline{L}$. For example, ShouWu Zhang has proved that if $L$ is relatively ample and the metric is semipositive, then $\overline{L}$ is ample if and only if $h_{\overline{L}}$ takes positive values on $X(\overline{\mathbb{Q}})$. Later, Moriwaki has partially generalized Zhang's result to show that, if the strictly effective sections of a certain power of $\overline{L}$ has no base-point, $H^0(X,mL)$ has a free basis consisting of strictly effective sections for $m\gg 1$. In this talk, I would like to talk about a further generalization of these results and some applications of it.

Carlo Gasbarri (Univ. Strasbourg). "On the Vojta conjecture over function fields"
Abstract:    Let $K$ be a number field or a function field. If $X$ is a smooth projective variety defined over $K$, the Vojta conjecture preview that if $\epsilon>0$ and $d>0$, there exists a proper closed subset $Z\subset X$ such that, for every algebraic point $p\in X\setminus Z$ of degree less then $d$ the following inequality holds: $$h_K(p)\leq (1+\epsilon)(2g(p)-2)+O(1)$$ where $h_K(p)$ is an height associated to the canonical line bundle on $X$ and $g(p)$ is the genus of the field where $p$ is defined. In this talk I will make some comments on the freedom of the choice of $\epsilon$ and on the possible proof of it for subvarieties of abelian varieties over function fields.

Henri Gillet (Univ. Illinois at Chicago). "Some Remarks on Arithmetic Singular Riemann Roch"
Abstract:    Baum, Fulton, and MacPherson extended the Grothendieck Riemann Roch theorem to projective morphisms between singular varieties. I will discuss what might be done in the arithmetic case.

Kai Köhler (Univ. Düsseldorf). "Laplacians and twistor fibrations"

Damian Rössler (Univ. Toulouse III). "A direct proof of the equivariant Gauss-Bonnet formula on abelian schemes"
Abstract:    We shall present a proof of the relative equivariant Gauss-Bonnet formula for abelian schemes, which does not rely on the Grothendieck-Riemann-Roch theorem. This proof can be carried through in the arithmetic setting (ie in Arakelov theory) and leads to interesting analytic questions.

Martin Sombra (ICREA and Universitat de Barcelona). "Arithmetic positivity on toric varieties", Part I and Part II
Abstract:    There is a dictionary between the algebro-geometric properties of toric varieties and the combinatorial properties of lattice polytopes and fans. This dictionary can be extended to relate the Arakelov theory of toric varieties with convex analysis.
In this talk, I will explain some recent additions to this dictionary concerning arithmetic positivity. In particular, I will give criteria for positivity of metrized line bundles on toric varieties, formulae for the arithmetic volume of a toric variety, a generalization of Dirichlet's units theorem to toric varieties, and a criterion for the existence of an equivariant Zariski decomposition.
This is joint work with J. I. Burgos, A. Moriwaki and P. Philippon.

Kazuhiko Yamaki (Kyoto Univ.). "Strict supports of canonical measures and applications to the geometric Bogomolov conjecture"
Abstract:    We investigate the canonical measure on a subvariety of an abelian variety, applying it to the geometric Bogomolov conjecture. In fact, we show that the geometric Bogomolov conjecture holds for a large class of abelian varieties. Further, we show that the conjecture in full generality holds true if the conjecture holds true for abelian varieties which is nowhere degenerate.

Last modified: September 10, 2012