On the topology of compact hyperkahler manifolds

In this talk we will describe some results about Betti, Hodge, and characteristic numbers of compact hyperkahler manifolds. In (complex) dimension four one can find universal bounds for all of these invariants (Beauville, Guan); in higher dimensions it is still possible to find some bounds. We also describe how these bounds are related to the question: are there finitely many hyperkahler manifolds in each dimension, up to deformation?