Several complex variables

Several complex variables studies holomorphic functions on domains in ℂⁿ and the geometric spaces defined by them. A function f(z₁,…,zₙ) is holomorphic when it is complex-differentiable in the relevant multidimensional sense; equivalently, its restrictions in each coordinate direction are holomorphic under suitable regularity conditions. Power series, analytic continuation, zeros, and singularities remain central, but the geometry of the domain becomes equally important. A domain may be convex, pseudoconvex, Stein, or bounded by a complicated real hypersurface. The subject therefore connects classical analysis with differential geometry, algebraic geometry, topology, and partial differential equations. Its standard setting is ℂⁿ, although many results extend to complex manifolds, where local complex coordinates replace a single global coordinate system.

Unlike one-variable theory, there is no direct analogue of a simple ordering of singularities or of arbitrary isolated poles. The extra dimensions allow analytic sets to have positive dimension, creating geometric objects rather than merely discrete exceptional points.

The defining phenomenon of the field is that multidimensional holomorphicity imposes unexpectedly strong extension and rigidity properties. Hartogs' extension theorem states, in a standard form, that a holomorphic function on a domain with a compact hole can often extend across the hole when the ambient complex dimension is at least two.² Thus isolated singularities behave very differently in ℂⁿ from those in ℂ. The Cauchy integral formula survives through iterated integrals on product regions, while power-series expansions are organized by multi-indices rather than a single exponent.

Modern proofs frequently use the ∂̄-equation, whose solvability controls whether a differential form is locally or globally the derivative of a holomorphic object. Estimates for this equation, especially those associated with pseudoconvex domains, provide methods for approximation, extension, and the construction of holomorphic functions.³ Analytic continuation is consequently governed by both function theory and domain geometry.

Complex analytic geometry treats the common zero sets of holomorphic functions as geometric spaces called analytic sets. Their local structure can include smooth manifolds, singular points, branches, and components of different dimensions. A complex manifold is locally modeled on open subsets of ℂⁿ, with holomorphic transition maps; compact complex manifolds include important examples such as complex tori and projective varieties.¹

Stein manifolds form a particularly useful class: they support many global holomorphic functions and resemble affine spaces from the viewpoint of complex analysis. The Oka principle reveals a striking relationship between topology and holomorphic geometry: for broad classes of problems, a continuous or topological solution can be deformed into a holomorphic one, subject to the relevant hypotheses. This connection helps explain why several complex variables is closely linked to algebraic geometry, sheaf theory, and homotopy theory.

Several complex variables contains edge cases that reshape familiar one-variable intuition. Meromorphic functions on compact complex manifolds need not provide enough coordinates to describe the manifold, and some compact complex manifolds are not algebraic varieties. Boundary regularity is also subtle: a domain may admit plentiful holomorphic functions while solutions to the ∂̄-equation lose smoothness at a nonsmooth or weakly pseudoconvex boundary.

The field has practical reach beyond pure function theory. Several-complex-variable methods appear in complex dynamical systems, CR geometry, integral geometry, inverse problems, and quantum field theory. CR manifolds, which model boundaries of complex domains, carry partial complex structures and have their own extension and embedding questions. Another distinctive feature is the Hartogs phenomenon for holomorphic functions, which can fail for other classes of functions or on spaces with different geometric structure; the theorem is therefore both an extension result and a diagnostic of complex dimension.

Holomorphic function

A complex-valued function that is complex-differentiable in the multidimensional sense on an open domain.

Analytic set

A subset locally described as the common zero set of finitely many holomorphic functions.

Pseudoconvex domain

A domain satisfying a complex-geometric convexity condition closely associated with solvability of the ∂̄-equation.

Stein manifold

A complex manifold with sufficiently many global holomorphic functions and strong analytic separation properties.

CR manifold

A real manifold equipped with a partial complex structure, often occurring as the boundary of a complex domain.