= Complex geometry
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<Complex geometry> studies spaces locally described by <complex coordinates> with <holomorphic> transition maps. A complex $n$-dimensional <complex manifold> has real <dimension> $2n$, but its structure is more restrictive than a <smooth atlas>: the transition maps satisfy the <Cauchy-Riemann equations>. The <complex projective line> is a compact example, described by two complex charts related by $w=1/z$. <Hermitian metrics> and <Kähler manifolds> connect this analytic structure to <Riemannian geometry>, while <complex algebraic varieties> connect it to <algebraic geometry>.
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