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## Project Home PagePhoto by http://mustseeplaces.eu News:Project Summary:
This project's aim is two fold. 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 extension is not possible. In recent years, many advances in the study of the geometry of complex varieties, and many results of classical flavor, 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, and Orlov (for what it 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 Bridgland and Maciocia. All these results could be easily be extended to the case of an algebraically closed field of characteristic 0, thanks to the Lefshetz principle and the nice behavior of the derived category with respect of base change. However the landscape of varieties defined over positive characteristic fields is still widely unexplored. Our results, together with a recent preprint of Lieblich and Olsson, and the work of Ward, lead to think that in many cases it should be possible to extend the known results in a more algebraic setting. Thus we propose to investigate when such an extension is feasible. On the other side, the geometry in positive characteristic is richer, therefore we also propose to provide examples when the statement in characteristic zero do not hold in an algebraic setting. |