Speaker: Vladimir Guletskii (Liverpool)

Title: Transcendence degree of algebraic cycles and non-trivial elements in Abel-Jacobi kernels

Abstract: In the talk I will give an overview of my recent joint work with Sergey Gorchinskiy on constructing of non-trivial elements in the Abel-Jacobi kernels for surfaces, threefolds and higher-dimensional varieties. Our method relies upon an interplay between algebraic cycles of non-zero transcendence degree (over the primary subfield), and correspondences over complex numbers. In brief, any algebraic cycle can be spread out over a base (defined over a smaller field) giving a correspondence over complex numbers which is a geometrical gadget with much more powerful Hodge characteristics. This approach provides a relation between algebraic cycles of non-zero transcendence degree in Abel-Jacobi kernels and the transcendental part in cohomology. If the transcendental cohomology vanishes, say on a surface, then a single algebraic cycle with big enough transcendence degree controls the behaviour of the corresponding kernel at large. These things fit well into the conjectural picture due to Green and Griffiths on their construction of the Bloch-Beilinson filtration on Chow groups.