Complex analytic space

A complex analytic space is a ringed space (X, O_X) locally isomorphic to the zero set of finitely many holomorphic functions in a domain of C^n, with the sheaf of holomorphic functions on that zero set. More precisely, each point has an open neighborhood U such that (U, O_X|U) is isomorphic to (V(f_1,...,f_k), O_{C^n}/(f_1,...,f_k)) for holomorphic functions f_i on an open set in C^n. This local model allows the space to have singularities, points where the local ring is not a regular local ring. The structure sheaf O_X is a sheaf of C-algebras, and the space is called reduced if the local rings have no nilpotent elements; otherwise it is non-reduced, carrying infinitesimal information. This definition, due to Jean-Pierre Serre and Henri Cartan, unifies complex manifolds (where the local model is just an open set in C^n) and complex algebraic varieties with singularities.

Morphisms between complex analytic spaces are morphisms of ringed spaces that are local homomorphisms of C-algebras. This category is closed under fiber products, allowing the construction of families and moduli spaces. The theory of coherent sheaves, developed by Cartan and Serre, applies to complex analytic spaces, yielding cohomology groups H^i(X, F) that are finite-dimensional for compact spaces. The deep analogy with algebraic geometry is formalized by the GAGA principle (Serre, 1956), which states that for projective algebraic varieties, analytic and algebraic objects coincide. Analytic spaces also support the theory of currents and L^2 cohomology, essential for Hodge theory and the study of singularities. The local structure of singularities is classified by invariants such as the Zariski tangent space, the multiplicity, and the analytic spectrum of the local ring.

A central theme is the resolution of singularities: given a complex analytic space, one seeks a proper surjective morphism from a smooth complex manifold that is an isomorphism over the regular locus. Hironaka's theorem (1964) establishes resolution for algebraic varieties over C, and subsequent work extended it to analytic spaces in many cases, though the general analytic case remains subtle. Singularities are classified by their local rings; for example, normal surface singularities have a finite fundamental group of the link, and rational singularities are characterized by the vanishing of higher direct images of the structure sheaf. The study of deformations of singularities, via the cotangent complex and versal deformations, is a rich area. The Milnor number and the monodromy of the Milnor fibration provide topological invariants that distinguish singularity types.

Beyond the standard theory, complex analytic spaces appear in unexpected contexts. The notion of a Stein space generalizes Stein manifolds to the singular setting, and the Cartan theorems B and A hold for coherent sheaves on Stein spaces, enabling global sections to generate stalks. The analytic space of a Riemann surface with nodes is a key example in Teichmüller theory and the compactification of moduli spaces. In mathematical physics, complex analytic spaces arise as the target spaces of nonlinear sigma models and in the study of D-branes in string theory, where singularities correspond to enhanced gauge symmetries. The Hilbert scheme of points on a complex surface is a complex analytic space that is smooth for the punctual Hilbert scheme, but singular in general. The analytic space of a complex torus is a compact complex manifold, but its moduli space is a complex analytic space with orbifold singularities. The theory of analytic stacks extends complex analytic spaces to include quotient singularities, essential for moduli problems with automorphisms.

Ringed space

A topological space equipped with a sheaf of rings, here the sheaf of holomorphic functions.

Singularity

A point of a space where the local ring is not regular, meaning the space is not locally a manifold.

Coherent sheaf

A sheaf of modules that is locally finitely presented and satisfies a closure condition, enabling cohomology theory.

GAGA principle

The equivalence between analytic and algebraic objects on projective varieties, established by Serre.

Resolution of singularities

A proper birational morphism from a smooth space that is an isomorphism over the regular locus.

Stein space

A complex analytic space that is holomorphically convex and separable, generalizing Stein manifolds.