  This paper is devoted to the complete algebraic and geometric classification
of complex $4$-dimensional nilpotent left-symmetric algebras. The corresponding
geometric variety has dimension $15$ and decomposes into $3$ irreducible
components determined by the Zariski closures of two one-parameter families of
algebras and a two-parameter family of algebras (see Theorem B). In particular,
there are no rigid $4$-dimensional complex nilpotent left symmetric algebras.
