What are some examples of compact Kaehler manifolds (in particular,or smooth complex projective varieties) that are not isomorphic as complex manifolds (or as varieties), but are isomorphic as symplectic manifolds (with the symplectic structure induced from the Kaehler structure)? Elliptic curves should be an example, but I can't think of any others. I'm sure there should be lots...
In the other direction, if I have two compact Kaehler manifolds (or, again, smooth complex projective varieties) that are isomorphic as complex manifolds (or as varieties), then are they necessarily isomorphic as symplectic manifolds?
And one last question that just came to mind while typing the previous one: ifIf two smooth complex (projective, if need be) varieties are isomorphic as complex manifolds, then they are isomorphic as varieties?
