CORDIS - Forschungsergebnisse der EU
CORDIS

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

Projektbeschreibung

Differentialformen nicht-archimedischer analytischer Räume

In den frühen 2000er Jahren stellten die Mathematiker Konzewitsch und Soibelman zwei Varianten der SYZ-Vermutung mit Ursprung in der Stringtheorie vor: eine nicht-archimedische und eine differentialgeometrische Variante. Beide Vermutungen setzen die Existenz einer singulären affinen Mannigfaltigkeit voraus. Beide Ansätze sollten identische Ergebnisse liefern, wobei entsprechende singuläre affine Mannigfaltigkeiten natürlich isomorph sind. Leider ist die Existenz eines derartigen Isomorphismus nun in Frage gestellt. Das im Rahmen der Marie-Skłodowska-Curie-Maßnahmen finanzierte Projekt nalimdif arbeitet an neuen Werkzeugen, die es ermöglichen sollen, die letztgenannte Vermutung zu untersuchen und die kollabierenden Gromov-Hausdorff-Grenzwerte besser zu verstehen. Die vorgeschlagene Untersuchung basiert auf der von Chambert-Loir und Ducros entwickelten Theorie der Differentialformen auf nicht-archimedischen analytischen Räumen.

Ziel

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.

Koordinator

KATHOLIEKE UNIVERSITEIT LEUVEN
Netto-EU-Beitrag
€ 166 320,00
Adresse
OUDE MARKT 13
3000 Leuven
Belgien

Auf der Karte ansehen

Region
Vlaams Gewest Prov. Vlaams-Brabant Arr. Leuven
Aktivitätstyp
Higher or Secondary Education Establishments
Links
Gesamtkosten
€ 166 320,00