Birational Geometry of Fano 4-folds

This proposal aims to make inroads into the subject of `complex algebraic 4-folds'. The historyleading up to this point is instructive. Curves have been studied for hundreds of years, withthe modern treatment evolving in the 19th century. The classification of surfaces was achievedby the Italian masters at the beginning of the 20th century, and these have been the subject ofcontinued intensive study until the present day. The next dimension up, 3-folds, was immediately considered, and in the 1930s another Italian master, Gino Fano, presented an analysis of the firstclass of 3-folds, nowadays known as `Fano 3-folds'. Nevertheless, serious conceptual problemshounded the subject of 3-folds for 70 years, until developments in the 1980s opened the floodgates and they entered the mainstream. Early work of Iskovskikh in the 1970s (reworking and correcting the work of Fano) and then of Mori and Mukai in the 1980s (incorporating the new orthodoxy, nowadays known as `Mori theory') provided important impetus. The influence of physics was also particularly important, perhaps especially in relation to the class of so-called Calabi--Yau 3-folds.Our project is to work with the next dimension up again, the 4-folds. Once more, we propose to investigate the natural initial case, the Fano 4-folds. There are three key influences that maketoday an especially exciting time to start such a project. First, the recent work in the foundationsof 4-folds---the existence of 4-fold flips---echoes Mori's central result of the 1980s on 3-fold flips.Second, Calabi--Yau 3-folds are intimately connected with Fano 4-folds, giving a direct connectionto the ambitions of contemporary physicists. And finally, we propose to employ recent work of Reid, one of the founders of the modern approach to 3-folds in the 1980s, that hints how we may apply the battery of 3-fold methods to the study of 4-folds. It is a very exciting time to be exploring this new field.