Abstracts for Seminar Algebraic Geometry (SAG)

Determinantal Barlow surfaces and phantom categories I will report on the construction of an exceptional collection of maximal length 11 on the Barlow surface S. This can be used to show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is joint work with Christian B?hning, Hans-Christian Graf von Bothmer and Ludmil Katzarkov.

Using a codimension-1 algebraic cycle obtained from the Poincare line bundle, Beauville defined the Fourier transform on the Chow ring of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring CH_*(A) which is compatible with its ring structure. We prove that a similar decomposition exists for certain hyperkaehler fourfolds by using a codimension-2 algebraic cycle representing the Beauville--Bogomolov bilinear form. This is joint work with Mingmin Shen.