analytic geometry (complex, rigid, global)
geometry+analysis/analytic number theory
analytic space, analytic variety, Berkovich space
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Analytic varieties form an analogue of algebraic varieties in analytic context; they are more general than analytic manifolds in allowing singularities.
While an algebraic variety is the loci of zeros of some set of polynomials, an analytic varieties is the loci of zeros of some set of analytic functions. By Chow's theorem every complex projective analytic variety is algebraic; this is based on the machinery of Weierstrass (the Weierstrass preparation theorem etc.).