Complex projective or compact kӓhler manifolds with trivial canonical class are up to a finite unramified cover, a product of hyperkӓhler manifolds, Calabi-Yau manifolds and compact complex tori, the former two types being simply connected. Not much is known concerning the lack of hyperbolicity of such manifolds that are simply connected besides the case of K3-surfaces, even the case of Calabi-Yau threefolds remains deeply mysterious. One strong expectation is that entire holomophic curves can be Zariski dense in any such variety, i.e. fails to be algebraically degenerate. The only known method for showing this failure is to construct meromorphic or holomorphic dominating maps from and so far this has only been done for certainly classes of K3 surfaces among such varieties that are simply connected.

In this talk, I will show in analogy with my related previous works on K3 surfaces how to construct such dominating maps for certain well-known classes of hyperkahler manifolds. Each of these classes being dense among hyperkahler manifolds that satisfy the SYZ conjecture, which has been verified for all known hyperkӓhler manifolds. I will mainly focus on some key complex algebraic geometric ingredients in the proof. This will entail a basic introduction to hyperkӓhler manifolds, their structure and moduli, which I will do starting with some basic examples.

This is joint work with Ljudmila Kamenova.

 

Video (Click here)