  We study characteristic classes on hyperk\"ahler manifolds with a view
towards the Verbitsky component. The case of the second Chern class leads to a
conditional upper bound on the second Betti number in terms of the
Riemann--Roch polynomial, which is also valid for singular examples. We discuss
the general structure of characteristic classes and the Riemann--Roch
polynomial on hyperk\"ahler manifolds using among other things Rozansky--Witten
theory.
