Several complex variables

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.