= Kähler geometry
{c}
{title2=$d\omega=0,\quad\nabla J=0$}
<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>.<Lectures on Kähler Geometry by Andrei Moroianu>[sections on Hermitian and Kähler structures]{ref}
On $\mathbb C^n$, the Euclidean <metric> gives $\omega=\sum_jdx_j\wedge dy_j$, which is closed and nondegenerate. Locally one can express a <Kähler form> as $\omega=i\partial\bar\partial\varphi$ for a real <Kähler potential> $\varphi$, using a stated normalization. On <Complex projective space>, $\varphi=\log(1+\sum_j|z_j|^2)$ 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.
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