What is the simplest example of a projective variety for which it is not known if Grothendieck's Standard Conjectures hold?
What is the simplest example of a flag manifold for which it is not known if Grothendieck's Standard Conjectures hold?
What is the simplest example of a projective variety for which it is not known if Grothendieck's Standard Conjectures hold? What is the simplest example of a flag manifold for which it is not known if Grothendieck's Standard Conjectures hold? 


It seems that for flag manifolds everything is very easy.:) For other varieties much less is known. For example, recently there was an attempt to construct a counterexample to the Hodge conjecture in the Cartesian square of a K3 surface: http://adsabs.harvard.edu/abs/2006math......8265K It seems that a mistake was found later (I'm not quite sure); yet probably nobody at the moment knows how to prove the conjecture in this case. 

