Skip to main content
European Commission logo print header

Non-Archimedean limits of differential forms, Gromov-Hausdorff limits and essential skeleta

Project description

Differential forms on non-Archimedean analytic spaces

In the early 2000s, mathematicians Kontsevich and Soibelman introduced two variants of the SYZ conjecture originating from string theory: a non-Archimedean one and a differential–geometric one. Both conjectures posit the existence of a singular affine manifold. Both approaches should give the same result, where corresponding singular affine manifolds are naturally isomorphic; unfortunately, the existence of such an isomorphism is now open to question. Funded by the Marie Skłodowska-Curie Actions programme, the nalimdif project is working on new tools that should allow us to investigate the latter conjecture and further understand the collapsing Gromov-Hausdorff limits. The proposed study is based on the theory of differential forms on non-Archimedean analytic spaces developed by Chambert-Loir and Ducros.

Objective

In the beginning of 2000s Kontsevich and Soibelman have introduced two variants of the SYZ conjecture originating from string theory: a non-Archimeadean one and a differential-geometric one. Both of these conjectures posit existence of a singular affine manifold (the base of the SYZ fibration) that can be obtained either as a subset of the non-Archimedean analytic space associated to a family of complex projective Calabi-Yau varieties with maximally unipotent monodromy, or as a Gromov-Hausdorff limit of fibres of the family with Ricci-flat metric in the polarization class and normalized diameter (the latter was also independently conjectured by Gross, Wilson, and Todorov). Recent years have seen active developments in both of these conjectures through work of de Fernex, Kollár, Mustaţa, Nicaise, Xu, Gross, Tosatti, Zhang and others. Kontsevich and Soibelman have also conjectured that both approaches give the same result, with corresponding singular affine manifolds naturally isomorphic; unfortunately, the existence of such an isomorphism is open as of now.

The aim of this project is to build tools that will allow both to attack the comparison conjecture and to make progress in the understanding of the collapsing Gromov-Hausdorff limits in the odd-dimensional case (hypekähler case having been extensively studied). The proposed approach is based on the theory of differential forms on non-Archimedean analytic spaces due to Chambert-Loir and Ducros. Firstly, a notion of a non-Archimedean limit of a degenerating family of real forms with values in Chambert-Loir-Ducros forms will be defined. Secondly, the metric structure of the collapsing limit will be described in terms of such non-Archimedean limits of Kähler forms. Thirdly, the canonical affine structure on the limit space conjectured to exist in the metric picture will be studied using non-Archimedean methods, assuming a natural statement about the limits of the solutions of Monge-Ampere equations.

Coordinator

KATHOLIEKE UNIVERSITEIT LEUVEN
Net EU contribution
€ 166 320,00
Address
Oude markt 13
3000 Leuven
Belgium

See on map

Region
Vlaams Gewest Prov. Vlaams-Brabant Arr. Leuven
Activity type
Higher or Secondary Education Establishments
Links
Other funding
€ 0,00