Hodge Theory, Motivic Homotopy Theory and Derived Categories

Hodge theory is an extremely useful tool for the study of abelian varieties and K3 surfaces. It also has important applications to Calabi-Yau threefolds, but unfortunately it is very hard to characterise the Hodge structures on the middle cohomology groups of these threefolds.

Several fellows in our group do actively research on these threefolds and their Hodge structures. The rich harvest of examples collected by theoretical physicists has already led to some interesting results. Moreover, we expect that classical algebraic geometry (also computer assisted) will be very helpfull to uncover new examples and phenomena. They intend to use point counting over finite fields and thus L series of Calabi-Yau varieties to determine the motives of some key examples.

Our group will also continue its research on moduli spaces of curves, K3 surfaces and abelian varieties as well as automorphisms of K3 surfaces.

Derived categories of coherent sheaves have been at the forefront of research in algebraic geometry in recent years. They are also of great importance in string (and brane) theory in theoretical physics. Our group is active on this field with particular attention to stability conditions, deformations of objects in the derived categories and representability of functors. Recently we started studying the relation between derived categories of cubic hypersurfaces and classical geometric problems.

Finally, another active reserac area in our group is motivic homotopy theory.

Projective Algebraic Geometry

A wide variety of topics studied by some fellows of our group is related to classical aspects of projective algebraic geometry.

Among them, we can certainly mention extension properties of rationally connected fibrations from a submanifold Y to its ambient variety. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.

Other topics are the classification of ample vector bundles with implications to various problems in the classification of projective varieties and the relation between projective bundle in classical and in the adjunction theoretic sense. Moreover, the discriminant locus of a linear system has been investigated leading to generalisations of classical results on dual varieties of Ein and Zak. Hyperplane sections were studied (following ideas of Chandler, Howard and Sommese) as long as subvarieties of the Grassmannian of lines.

Some other topics are the study of rational maps in a wide sense, the study of tangent and normal bundles related to rational curves in projective spaces and the study of the zero-loci of general sections of special vector bundles.

Some fellows of our group study the relations between problems in computer vision and projective algebraic geometry.