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

Descrizione del progetto

Forme differenziali su spazi analitici non archimedei

Nei primi anni 2000, i matematici Kontsevich e Soibelman hanno presentato due varianti della congettura SYZ scaturite dalla teoria delle stringhe, di cui una era non archimedea e l’altra era di carattere differenziale-geometrico. Le due congetture ipotizzano l’esistenza di una varietà affine singolare. Entrambi gli approcci dovrebbero produrre il medesimo risultato, dove le varietà affini singolari corrispondenti sono naturalmente isomorfe; purtroppo, la presenza di tale isomorfismo è ora messo in discussione. Il progetto nalimdif, finanziato dal programma di azioni Marie Skłodowska-Curie, sta lavorando a nuovi strumenti che dovrebbero permettere l’analisi dell’ultima congettura e approfondire il crollo dei limiti Gromov-Hausdorff. Lo studio proposto si fonda sulla teoria delle forme differenziali su spazi analitici non archimedei sviluppati da Chambert-Loir e Ducros.

Obiettivo

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.

Meccanismo di finanziamento

MSCA-IF-EF-ST - Standard EF

Coordinatore

KATHOLIEKE UNIVERSITEIT LEUVEN
Contribution nette de l'UE
€ 166 320,00
Indirizzo
OUDE MARKT 13
3000 Leuven
Belgio

Mostra sulla mappa

Regione
Vlaams Gewest Prov. Vlaams-Brabant Arr. Leuven
Tipo di attività
Higher or Secondary Education Establishments
Collegamenti
Costo totale
€ 166 320,00