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H2020Individual fellowship2019–2021

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

Horizon 2020 — Marie Skłodowska-Curie Actions

Duration
2019-10-01 → 2021-09-30
EU contribution
€166,320
Participants
1
Scheme
MSCA-IF-EF-ST

Lines connect the coordinator with its partners.

Results in brief

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

This project is concerned with the mathematical underpinnings of the Strominger-Yau-Zaslow conjecture originating in the physical field of mirror symmetry. Mirror symmetric is a phenomenon in high energy physics when two theories which explain interaction of elementary particles of high energies complement each other in a certain way. Each such physical theory is based on a mathematical object, a family of complex Calabi-Yau manifolds, an when two such families correspond to the complementary theories, they are called mirror partners. The Strominger-Yau-Zaslow (SYZ) conjecture seeks to give a mathematical explanation to the phenomen of mirror symmetry, but mathematical questions that arise in it are interesting in their own right, from a purel mathematics point of view, and they make sense even for families of Calabi-Yau manifolds that do not have physical significance. In the mid-2000s two approaches to the SYZ conjecture --- metric and non-archimedean --- were put forward by Kontsevich and Soibelman. Each approach proposed certain mathematical construction that would take a family of Calabi-Yau manifolds of dimension 2n and produce an n-dimensional sphere ("base of the SYZ fibration") endowed with additional mathematical structures. The structures produced by the two approaches are different, but are both underlied by a common geometric strucuters called singular integral affine structure. Both approaches have been extensively studied each on its own, but the relationship between the two remains mysterious. Konstevich and Soibelman conjectured that the singular affine structures arising from both approaches should be related in a certain way. The aim of this project is to develop mathematical tools that would allow to describe this relationship and study the interaction of mathematical structures in both approaches via the singular integral affine structure that underlies them. Development of these tools will have impact on several fields of pure mathematics that intersect in the mathematical treatment of mirror symmetry.

Data: CORDIS, © European Union

Project 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.

Original text from CORDIS.

Participants

  • KATHOLIEKE UNIVERSITEIT LEUVEN · LeuvenCoordinatorBelgium

Links

Data: CORDIS, © European Union