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

Periodic Reporting for period 1 - nalimdif (Non-Archimedean limits of differential forms, Gromov-Hausdorff limits and essential skeleta)

Reporting period: 2019-10-01 to 2021-09-30

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.
During the course of the project, a novel way to describe the singular affine strucutures on the base of the non-archimedean SYZ fibration was developed. With this description at hand one can on the one hand compute certain properties of the families of Calabi-Yau manifolds (a subspace of nearby fibre cohomology) pertinent in the non-archimedean picture, and on the other hand introduce an object called differential superforms that can used to study metric aspects of the SYZ conjecture. Some foundational theorems about this new way of looking at singular affine structures have been proved.

These theorems were applied to a particular class of families of Calabi-Yau manifolds called Kulikov degenerations, some new results about their properties were obtained. These results were reported to experts at scientific seminars, and published on an open publication pre-print server.

The technical methods underlying these results are closely related to recently developed theory of tropical homology. As a part of the project an international summer school for Masters and early PhD students on the subject of tropical homology was held (online due to COVID pandemic).
The main contribution of the project to the state of the art is the foundational results about differential superforms, their relationship to the singular affine structure on the base of the SYZ fibration and to the invariants of the cohomology of the nearby fibre of a degeneration of Calabi-Yau manifolds.