Scientific Scope
In recent years, significant advances have been made in complex algebraic geometry. This research group aims to adjust its themes to accompany the most spectacular recent developments in the field.
Here are some areas currently in development:
The Mori program, its analogues in Kähler geometry, in positive characteristic, and in the theory of holomorphic foliations are particularly active subjects. The group has contributed to advancing birational geometry in France, and we wish to continue promoting its dissemination.
The study of special varieties is thriving, notably the geometry of rationally connected varieties, particularly the problem of their rationality. The study of holomorphic symplectic varieties (algebraic cycles, moduli spaces, Lagrangian fibrations) is a traditionally represented theme within the group. The geometry of mildly singular spaces with Kodaira dimension zero, their foliated analogues, and the hyperbolicity of varieties (especially the Green-Griffiths and Lang conjectures) are also research areas with significant progress.
Among particularly active topics, we can also mention questions surrounding moduli spaces of higher-dimensional varieties, especially their compactifications (including positive characteristic analogues), their birational geometry, and more.
Additionally, a range of problems where analytical methods have proven very effective is gaining attention. These include birational geometry (the Mori program via Ricci flow), questions related to Kähler-Einstein metrics on singular varieties, particularly their behavior near singularities, and issues of hyperbolicity. The group’s expertise in these areas is strong and should be further developed.
Simultaneously, the group seeks to support emerging interactions with closely related disciplines, such as non-Archimedean analytic geometry, algebraic topology for the use of homological methods in algebraic geometry (derived categories of coherent sheaves on algebraic varieties, derived algebraic geometry), and areas that have only grown stronger, like holomorphic dynamics for studying birational transformations or endomorphisms of certain varieties, or arithmetic, concerning both hyperbolicity and Arakelov geometry or foliation theory. We also wish to strengthen interactions with specialists in representation theory to study algebraic transformation groups and, more broadly, all questions at the interface between representation theory and geometry.
The group aims to enable its members to benefit from this scientific dynamism while continuing to support traditionally well-represented themes within it.
Genesis of the GAGC Axis
The GAGC group (Géométrie Algébrique et Géométrie Complexe) follows the former GDR (Groupement de Recherche) on Algebraic Geometry and Complex Geometry, itself stemming from the GDR on Complex Algebraic Geometry (GDR GAC), led by Arnaud Beauville, which ended on December 31, 2005. It was the French hub for the European EAGER network in algebraic geometry. The GDR GAGC was created on January 1, 2007, under the direction of Olivier Debarre, then renewed on January 1, 2011, for a four-year term with Laurent Manivel as director. It was subsequently led by Christophe Mourougane until December 2019, and then by Stéphane Druel until December 2023.