Many examples of Calabi-Yau manifolds are constructed as algebraic varieties in weighted projective space, or more generally as complete intersection Calabi-Yau (CICY) manifolds. Are there such realizations of compact hyperkahler manifolds besides K3? If no, is there a fundamental obstruction?



Compact hyperkahler manifold as algebraic variety in weighted projective space?
If $X$ is a hypersurface and $\dim(X) > 2$ then by Lefschetz hyperplane theorem $H^{2,0}(X) = 0$, hence $X$ can't by hyperkahler.


