Degenerations of Calabi-Yau Manifolds and Related Geometries

• Applicant: Professor Dr. Helge Ruddat
• Subject Area: Mathematics
• Term: from 2015 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 272561367

Degenerations of Calabi-Yau Manifolds and Related Geometries.Calabi-Yau manifolds form a central geometric class with a plethora of connections and applications to other mathematical areas and mathematical physics. Various structural questions about the particularly interesting three-dimenensional such geometries could not be answered to date, e.g. whether the number of deformation types is finite, whether all deformation types are connected by extremal transitions or to which extent mirror symmetry applies. The goal of this proposal is the development of methodology to study these questions.The basic approach of the project is the maximal degeneration of the Calabi-Yau geometry. In the past decade, methods controlling such degenerations were developed in particular by Gross and Siebert. These employ logarithmic and tropical geometry. The motivation for degenerating stems from mathematical physics. Around 1990, string theorists discovered mirror symmetry, a deep relationship between different Calabi-Yau geometries featuring degenerations. Mirror symmetry relates the complex geometry of one Calabi-Yau manifold to the symplectic geometry of another Calabi-Yau manifold. The structural data on the symplectic side are governed by holomorphic curves, that on the complex side by variations of the complex structure. The proposed project aims at extending this relationship to curves of higher genus and the corresponding complex data. For this purpose, existing methods by Costello-Li and Barannikov-Kontsevich shall be translated into logarithmic geometry and then be enhanced. Also tropical methods shall be extended by studying tropical deformations. These directly relate to Morrison's conjecture stating that mirror symmetry is compatible with extremal transitions. An extremal transitions connects two different Calabi-Yau manifolds and Reid conjectured that all three-dimensional Calabi-Yau manifolds are connected by such transitions. We seek to make progress towards these conjectures. Finally, related structures like homological mirror symmetry relative to a divisor and non-compact Calabi-Yau manifolds based on a spectral curve shall be analysed as this extends the scope of the applications for the developed techniques.

• DFG Programme: Independent Junior Research Groups
• International Connection: Italy, USA
• Cooperation Partners: Privatdozent Dr. Diego Matessi; Professor Dr. Arthur Ogus; Professor Dr. Bernd Siebert; Professor Dr. Eric Zaslow