A complex algebraic variety is an algebraic variety over the complex numbers. In the scheme convention used here, it is a reduced separated scheme of finite type over ; irreducibility is an additional condition when needed. Affine examples are zero sets of complex polynomials equipped with their regular functions.
For example, is the union of two complex coordinate lines, while is an irreducible singular curve. The smooth locus of a complex variety has the structure of a complex manifold, but a singular point need not be smooth as a complex analytic space.
One must also distinguish topologies. The complex affine line is as a point set; its proper Zariski-closed subsets are finite, whereas its usual analytic topology has many more closed and open subsets. Passing from an algebraic variety to its analytic space retains geometric information while changing which local functions and open sets are allowed.
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A **complex algebraic variety** is a fundamental concept in algebraic geometry, which is the study of geometric objects defined by polynomial equations. Specifically, a complex algebraic variety is defined over the field of complex numbers \(\mathbb{C}\). ### Definitions: 1. **Algebraic Variety**: An algebraic variety is a set of solutions to one or more polynomial equations. The most common setting is within affine or projective space.