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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 $\mathbb{C}$; irreducibility is an additional condition when needed. Affine examples are zero sets of complex polynomials equipped with their regular functions.

For example, $xy=0$ is the union of two complex coordinate lines, while $y^2=x^3$ 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 $\mathbb{C}$ 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.

Let $f:X\to S$ be a proper map of complex analytic spaces defined locally by holomorphic functions, with $S$ reduced, and let $\mathcal{F}$ be a coherent analytic sheaf flat over $S$. Ifis locally constant, then the higher direct image sheaf $R^q{f}_{\ast }\mathcal{F}$ is locally free and the canonical comparisonis an isomorphism. Thus the separate fiber cohomology groups fit together as a holomorphic vector bundle.^[ref]

Properness supplies finite-dimensional cohomology and coherent direct images; flatness controls the variation of the coefficients. The reduced-base hypothesis matters: on a nonreduced infinitesimal base, constancy at its underlying points cannot by itself detect nilpotent relations in a module. It must not be substituted for the full base change criterion there.

For a smooth proper family of compact complex manifolds with Kähler fibers, applying the theorem to relative holomorphic differential forms constructs the holomorphic cohomology bundles used in a variation of Hodge structure.

Several complex variables studies holomorphic functions on open subsets of ${\mathbb{C}}^n$ and their geometry, especially when $n\ge 2$. A scalar-valued function is holomorphic when it is complex differentiable in the multivariable sense; locally it has a convergent power series in $z_1,\dots ,z_n$. For a continuously differentiable function, the equations $\partial f/\partial {\bar{z}}_j=0$ for every $j$ give the corresponding Cauchy-Riemann equations.

For example, $f(z_1,z_2)=z_1{z}_2$ is holomorphic, while $|z_1{|}^2$ is not holomorphic on any open set where $z_1$ varies. A polydisc admits an iterated Cauchy integral formula, reconstructing interior values by integrating over one boundary circle for each coordinate.

The higher-dimensional theory is not just the one-variable theory repeated independently. Hartogs's extension theorem shows that isolated holes in complex dimension at least two can be removable for holomorphic functions under conditions that have no one-dimensional analogue: $1/z$ obstructs such extension in one variable. Complex analytic spaces and complex analytic hypersurfaces provide the natural geometric objects.