Good structures in higher dimensional birational geometry

• Applicant: Professor Dr. Vladimir Lazic
• Subject Area: Mathematics
• Term: from 2013 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 239673722

The goal of this proposal is to make ground-breaking progress in geometry of higher dimensional varieties. Good minimal models: The aim of the Minimal Model Program is to classify higher dimensional algebraic varieties, generalising the classification of curves and surfaces. The purpose of the classification is to give a rough understanding of the structure of projective manifolds, and the programme was completely resolved only in dimension 3 in the 1980s. Recently there has been spectacular progress in the field, partially due to me and my coauthors. However, the programme remains far from being complete, and the main open problems left are Abundance conjecture and existence of good models. The goal of this project is to prove these two conjectures by applying recent techniques pioneered by me and my coauthors, and by establishing a new extension result for pluricanonical forms. Calabi-Yau manifolds: Calabi-Yau manifolds represent one of the most important classes of manifolds, and they are notoriously difficult to study. Foundational work on the structure of the K¨ ahler cone of a Calabi-Yau and the existence of rational curves was done in the 1990s. An important and extremely hard Cone conjecture of Morrison and Kawamata, motivated by mirror symmetry, predicts that the nef and movable cones of a Calabi-Yau are, vaguely speaking, close to being rational polyhedral. The conjecture implies existence of rational curves on Calabi-Yau threefolds with Picard number 2. I plan to prove this conjecture for Calabi-Yau threefolds of small Picard rank, which would be the biggest breakthrough to date. The approach uses recent techniques introduced by me and my coauthors, and reduction to positive characteristic.

• DFG Programme: Independent Junior Research Groups
• International Connection: France, United Kingdom
• Participating Persons: Dr. Paolo Cascini; Dr. Simone Diverio; Dr. Anne-Sophie Kaloghiros; Professor Dr. Thomas Peternell