Project description DEENESFRITPL 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. Show the project objective Hide the project objective 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. Fields of science natural sciencesphysical sciencestheoretical physicsstring theory Programme(s) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Topic(s) MSCA-IF-2018 - Individual Fellowships Call for proposal H2020-MSCA-IF-2018 See other projects for this call Funding Scheme MSCA-IF-EF-ST - Standard EF 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 Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 166 320,00