 Kähler geometry

Kähler geometry studies complex manifolds with a Hermitian metric $g$ whose associated real two-form $\omega (X,Y)=g(JX,Y)$ is closed. Here $J$ is the complex structure. Equivalently, the Levi-Civita connection preserves $J$. Thus the same metric simultaneously supports complex geometry, Riemannian geometry and symplectic geometry.^[ref]

On ${\mathbb{C}}^n$, the Euclidean metric gives $\omega ={\sum }_{j}d{x}_j\wedge d{y}_j$, which is closed and nondegenerate. Locally one can express a Kähler form as $\omega =i\partial \bar{\partial }\phi$ for a real Kähler potential $\phi$, using a stated normalization. On Complex projective space, $\phi =\log (1+{\sum }_j|z_j{|}^2)$ gives the Fubini-Study metric in an affine chart. The potentials need not glue as one global function; their forms glue. This is how locally elementary differentiation can define nontrivial global geometry.