Examples of non-algebraic compact Kahler surface?

Question

We call a Kahler manifold of dimension 2 a Kahler surface.
Kodaira has proved a famous theorem:every compact Kahler surface is a deformation of an algebraic surface. We know every algebraic surface is compact Kahler surface, but inversely, not every compact Kahler surface is an algebraic surface. I don't know any example of such surface, so can anybody provide me an example of non-algebraic compact Kahler surface？The simpler the better, thanks!

Answer 1

A lattice $\Lambda \subset \mathbb C^n$ is a discrete abelian subgroup of rank $2n$ generated by a basis $\lambda_1,...,\lambda _{2n}$ of $\mathbb C^n$ seen as a real vector space.
The quotient $T_\Lambda=\mathbb C^n/\Lambda$ is a complex compact manifold, called a complex torus and that torus is always Kähler because the standard Kähler structure on $\mathbb C^n$ descends to $T_\Lambda$.
For $n=1$ that torus is a projective variety that can always be imbedded in $\mathbb P^2$ (and is then called an elliptic curve).
For $n\geq 2$ however the torus $T_\Lambda$ is not projective (nor even algebraic) for a general choice of the lattice $\Lambda$.
The necessary and sufficient condition on $\Lambda$ for $T_\Lambda$ to be projective is due to Riemann.
In modern language that condition is that there exist a hermitian structure $H=G+iA:\mathbb C^n\times \mathbb C^n\to \mathbb C$ such that $A(\Lambda \times \Lambda)\subset \mathbb Z$ .