Analytic varieties are mathematical objects studied in the field of algebraic geometry. They are defined as the zero sets of collections of analytic functions, which are functions that are locally represented by convergent power series. These varieties can be thought of as the geometric manifestations of solutions to polynomial equations in complex spaces. In contrast to algebraic varieties, which are defined over algebraically closed fields, analytic varieties are studied in the context of complex numbers. This allows for a rich interplay between geometry and analysis, particularly in understanding the properties of complex manifolds and their singularities, often involving concepts from complex analysis and differential geometry.