Complex analytic varieties are mathematical objects that arise in the study of complex numbers and functions. They are defined as the zero sets of holomorphic functions, which are functions that are complex differentiable in a neighborhood of every point in their domain. These varieties can be thought of as higher-dimensional generalizations of algebraic curves and surfaces. These varieties are studied in the field of algebraic geometry and have applications in various areas, including string theory and complex geometry. They provide a framework for understanding the geometric properties of solutions to complex equations, allowing mathematicians to explore their structure and relationships.