Blanc SIMI 1 - Sciences de l'information, de la matière et de l'ingénierie : Mathématiques et interactions

Positivity in Arithmetic, Algebraic and Analytic Geometry – POSITIVE

Submission summary

The members of this project are mainly united by questions in
arithmetic geometry whose solution resides in positivity properties
of certain geometric structures. This positivity is most often
expressed in analytic conditions (existence of particular metrics
on bundles, inequalities of Schwarz lemma type), in complex geometry
and in non-archimedean geometry.

Four directions can be seen to structure our works:
1) Hyperbolicity and rational points
2) Transcendence and algebraic geometry
3) Pluriharmonic analysis, in the complex and non-archimedean settings
4) Algebraic and arithmetic volumes

1) We will study how the positivity of the canonical bundle impacts
the existence of rational or entire curves in a projective algebraic
variety. We of course have in view the (inaccessible?) conjectures
of Bombieri-Lang, Vojta, Green-Griffiths (when the canonical bundle
is positive) who predict the scarcity of rational points or the
degeneracy of entire curves, but also conjectures who predict their
existence when the canonical bundle is trivial. We are also interested
in variants concerning integral points, and underlying questions
of heights.

2) Since twenty years and the birth of Arakelov geometry, transcendence
methods have become more geometric and we now can try to study more
involved geometric situations. It appeared efficient to formulate
problems in terms of algebraization of geometric objets from formal
geometry and to prove theorems of Lefschetz or GAGA type. The
transcendence method requires precise informations studied in themes
4) — existence of small auxiliary functions — and 3) — estimation
of their derivatives. In almost all interesting cases, all places,
archimedean and non-archimedean, require a subtle analysis.

3) Pluriharmonic analysis is essential for the future developments
of Arakelov geometry where privileged metrics on bundles, and
currents, are one of the basic tools. Bringing in singular metrics
is often necessary; except in dimension 1, where fundamental progress
has been done since 2005, its development in the non-archimedean
setting is at its very beginning.

4) Spaces of sections of powers of metrized line bundles are a basic
tool in Diophantine Geometry; the existence of small auxiliary
functions is conditioned by the thorough understanding of the
algebraic volume (growth of the dimension of these spaces) and of
the arithmetic volume (growth of the covolume of a canonical lattice).
The use of more subtle invariants (like the Harder-Narasimhan
polygons) is promising.

Project coordination


The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.



Help of the ANR 120,000 euros
Beginning and duration of the scientific project: - 36 Months

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