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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.