Hironaka's example

In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Hironaka (1960, 1962). Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at most 2 fail for smooth varieties of dimension at least 3.

Take two smooth curves C and D in a smooth projective 3-fold P, intersecting in two points c and d that are nodes for the reducible curve C∪D. For some applications these should be chosen so that there is a fixed-point-free automorphism exchanging the curves C and D and also exchanging the points c and d. Hironaka's example V is obtained by blowing up the curves C and D, with C blown up first at the point c and D blown up first at the point d. Then V has two smooth rational curves L and M lying over c and d such that L+M is algebraically equivalent to 0, so V cannot be projective.

For an explicit example of this configuration, take t to be a point of order 2 in an elliptic curve E, take P to be E×E/(t)×E/(t), take C and D to be the sets of points of the form (x,x,0) and (x,0,x), so that c and d are the points (0,0,0) and (t,0,0), and take the involution σ to be the one taking (x,y,z) to (x + t, z,y).

Hironaka's variety is a smooth 3-dimensional complete variety but is not projective as it has a non-trivial curve algebraically equivalent to 0. Any 2-dimensional smooth complete variety is projective, so 3 is the smallest possible dimension for such an example. There are plenty of 2-dimensional complex manifolds that are not algebraic, such as Hopf surfaces (non Kähler) and non-algebraic tori (Kähler).

In a projective variety, a nonzero effective cycle has non-zero degree so cannot be algebraically equivalent to 0. In Hironaka's example the effective cycle consisting of the two exceptional curves is algebraically equivalent to 0.

If one of the curves D in Hironaka's construction is allowed to vary in a family such that most curves of the family do not intersect D, then one obtains a family of manifolds such that most are projective but one is not. Over the complex numbers this gives a deformation of smooth Kähler (in fact projective) varieties that is not Kähler. This family is trivial in the smooth category, so in particular there are Kähler and non-Kähler smooth compact 3-dimensional complex manifolds that are diffeomorphic.

...
Wikipedia