Example
- \(\mathbb C^n\) with the usual structure is of course a Kahler manifold.
- \(\mathbb {CP}^n\) with \(g_{FS}\) induced by \(S^{2n + 1}\to \mathbb {CP}^n\) is a Kahler manifold.
- In \(n = 2\), if \((MJ, g)\) is a Kahler surface, you can blow it up by replacing a point with a holomorphic \(S^2\). This operation can be repeated indefinitely to obtain distinct non-isomorphic Kahler surfaces, hence Kahler surfaces are not classified (but minimal models are).