Toric degenerations and Calabi-Yau orbifolds
Final Report Abstract
In short, before coming to Berkeley, my original plan had been to study (1) Transitions between orbifolds and non-orbifolds on Calabi-Yau degenerations, (2) Conifold transitions, (3) A cohomology theory for stringy Hodge numbers which all strongly relate to mirror symmetry. When the opportunity to contribute to a project with Ludmil Katzarkov and Mark Gross opened up to me shortly after my arrival in Berkeley, I postponed my original plans in order to join this project. Proposing a mirror duality for singularities, this would also provide useful input for (3). The project was still at an early stage and had been lying around unfinished for 6 years already. I became the leading author and implemented the Hodge theoretic technology in order to prove the main conjecture aboul a mirror duality of Hodge numbers for mutually dual Landau-Ginzburg models. After one year of intense work, I could finally prove this. In our paper Towards Mirror symmetry for varieties of general type, we give a duality construction for Landau-Ginzburg models. This enables us to produce a mirror of a variety of any Kodaira dimension. Previously, one had only seen mirror partners for Calabi-Yau or Fano varieties. The crucial observation was that the sheaf of vanishing cycles of the potential of the Landau-Ginzburg model contains the essential geometric information. On the basis of the results, there will be a series of further investigations in additional projects. In particular, our construction generalizes the Berglund-Hübsch-Krawitz mirror construction. There has been some progress towards my original plan also. Concerning (2), I discussed with Diego Matessi who collaborates with Ricardo Castano-Bernard on a tropical criterion for conifold transitions. We agreed on a collaboration in a sequel of their work. My input will be useful to them as it comes to tropical cohomological questions. Concerning (1), I have on-going discussions with Ludmil Katzarkov and David Favero. I believe that the transitions that I observed are instances of several Landau-Ginzburg models localized at the singularities of the log structure of the Calabi-Yau degeneration. In the recent paper by Ballard-Favero-Katzarkov, the authors analyze the change of grading for derived categories. This is strongly related to my observation and will be subject to further investigation.
