The Arithmetic of Derived Categories

Dette er et prosjekt i algebraisk geometri, grenen i matematikk som studerer geometriske objekter som kan beskrives ved hjelp av polynomielle ligninger. Disse kalles (algebraiske) varieteter. Hovedmålet er å utvikle nye teknikker for å studere geometrien til varieteter definert over kropper av positiv karakteristikk. Vi ønsker å gjøre dette ved å studere oppførselen av den deriverte kategorien til en varietet, et redskap introdusert av Verdier på 60-tallet hvis anvendelser har brakt mange fremskritt i forskningen på varieteter over de komplekse tall. Forskningen som blev produserad i dette studiet har bidratt betydelig til å løse både nye og gamle problemer som angår geometrien og aritmetikken til varieteter i positiv karakteristikk. I tilleg har dette projektet stort forbedret synligheten til norsk forsker og institusjon i det bredere internasjonale rammeverket.

During the project period ,43 publications in peer reviewed journals, both in Nivå 1 and 2 , have been produced by the team members. More than 65 seminars have been given. Some articles on the research produced with the project funding are currently under review and will hopefully soon be published in peer reviewed journals.  In addition, there are still 5 ongoing projects that could not be completed under the grant period which will result in at least as many publications. Thus the bibliometric impact of this grant is at the moment widely underestimated. We organized 3 conferences and  a workshops, as well invited scholars to give seminars in Bergen. This ncreased the visibility of the mathematics department at UiB in the wider international setting, making a more attractive place for students and researchers. In general, the visibility of Norwegian research and of Norwegian research institutions was greatly improved by the grant. 

This project's aim is twofold. On one side we propose to investigate to which extent it is possible to extend known theorems about derived categories of smooth complex varieties to varieties defined over (possibly finite) fields of positive characteristic. On the other hand, we want to provide examples in which such extensions are not possible. In recent years, many advances in the study of the geometry of complex varieties, and many results of classical flavour, have been brought forth with non-classical tools, such as the study of the derived categories of coherent sheaves and the Fourier--Mukai transform. One of the central problems of the theory is to see to which point the bounded derived category of coherent sheaves on a variety carries information on the geometry. For complex projective varieties this was studied by Bondal-Orlov (for what concerns varieties with ample, or anti-ample, canonical bundle or abelian varieties), Kawamata (who studied complex varieties of general type), and Mukai (who investigated the derived categories of K3 surfaces). A complete answer for the case of complex surfaces was given in 2000 by Bridgeland and Maciocia. All these results can easily be extended to the case of an algebraically closed field of characteristic 0. However, the landscape of varieties defined over positive characteristic fields is still widely unexplored. Some recent joint results by a part of the team members, together with the work of Lieblich-Olsson, suggest that in many cases it is possible to extend the known statements in a more algebraic setting. However, the geometry in positive characteristic is richer, thus we also expect to find many discrepancies. A thorough analysis of the similarities and differences between the two settings will yield new insights on the study of positive characteristic algebraic geometry. Possible applications will range from the study of the variational crystalline Hodge conjecture to the proof of Torelli-like theorems.