Several complex variables is not one-variable complex analysis repeated coordinate by coordinate. New rigidity appears, singularities behave differently, zero sets gain geometry, and the shape of the domain becomes part of the analytic problem.
This guide introduces holomorphic functions on Cⁿ while keeping the contrasts with one complex variable explicit. It develops complex derivatives, polydiscs, iterated Cauchy formulas, Hartogs phenomena, analytic sets, domains of holomorphy, plurisubharmonicity and the ∂̄ viewpoint. The goal is a working map, not a complete graduate course.
Prerequisites: one-variable holomorphic functions, Cauchy integral formula, power series, multivariable calculus and basic topology.
Reading route: Cⁿ geometry → complex derivatives → Wirtinger operators → separate versus joint holomorphy → power series → polydiscs and Cauchy formula → identity and zero sets → Hartogs extension → failure of isolated singularity intuition → domains of holomorphy → pseudoconvexity → plurisubharmonic functions → Levi form → ∂̄ equations → analytic varieties → inverse and implicit theorems → practice and solutions.
1. Cⁿ is a real 2n-dimensional space with extra complex structure
A point in Cⁿ is z=(z₁,…,z_n), with z_j=x_j+iy_j.
As a real vector space, Cⁿ≅R^{2n}. But multiplication by i acts simultaneously on each complex coordinate, and holomorphicity requires compatibility with this complex-linear structure.
That extra structure is why complex differentiability remains much more rigid than ordinary differentiability in R^{2n}.
2. Open sets and domains are still the stage
A domain Ω⊂Cⁿ means a connected open set.
Common examples include balls, polydiscs, product domains, tubes, Reinhardt domains and complements of analytic sets.
Unlike one variable, not every simply connected-looking domain behaves like a ball under biholomorphic maps. Domain geometry becomes substantially richer.
3. Complex differentiability means a complex-linear first derivative
For f:Ω→C, real differentiability at a point gives a real-linear derivative Df:R^{2n}→R².
Holomorphicity requires this derivative to be compatible with multiplication by i. Equivalently, the antiholomorphic derivatives ∂f/∂z̄_j vanish under suitable differentiability hypotheses.
4. Wirtinger derivatives separate holomorphic and antiholomorphic directions
For each coordinate z_j=x_j+iy_j, define
∂/∂z_j = 1/2(∂/∂x_j − i∂/∂y_j),
∂/∂z̄_j = 1/2(∂/∂x_j + i∂/∂y_j).
A C¹ function is holomorphic exactly when ∂f/∂z̄_j=0 for every j.
5. Worked Wirtinger example
Let f(z₁,z₂)=z₁²+z₁z₂+e^{z₂}.
The formula depends only on z₁ and z₂, not on their conjugates. Thus
∂f/∂z̄₁=∂f/∂z̄₂=0.
Its complex partial derivatives are 2z₁+z₂ and z₁+e^{z₂}.
6. A conjugate term destroys holomorphicity
For g(z₁,z₂)=z₁z̄₂,
∂g/∂z̄₂=z₁.
Hence g is not holomorphic on any open set where z₁ varies nontrivially.
This is the several-variable version of the one-variable warning that conjugation is real-smooth but not holomorphic.
7. Separate holomorphy is unexpectedly strong
A function f(z₁,…,z_n) is separately holomorphic if, after fixing all variables except one, it is holomorphic in the remaining variable.
Hartogs’ theorem says that on an open domain, separate holomorphy already implies joint holomorphy.
This is a distinctly several-variable rigidity phenomenon. For arbitrary real differentiability notions, coordinatewise regularity is far weaker.
8. Holomorphic functions have convergent multivariable power series
Near any a∈Ω, a holomorphic function has an expansion
f(z)=Σ_{α∈N^n} c_α (z−a)^α,
where α=(α₁,…,α_n), |α|=α₁+⋯+α_n, and (z−a)^α=∏(z_j−a_j)^{α_j}.
Multi-index notation is not decorative; it keeps derivatives, Taylor coefficients and degree structure manageable.
9. Polydiscs are natural product neighbourhoods
A polydisc centred at a is
P(a;r₁,…,r_n)={z:|z_j−a_j|<r_j for every j}.
Unlike a Euclidean ball, a polydisc is a product of one-variable disks. This product structure makes iterated Cauchy integration immediate.
10. The Cauchy formula iterates over a distinguished boundary torus
If f is holomorphic near the closure of a polydisc, then at a point z inside,
f(z)=1/(2πi)^n ∫⋯∫ f(ζ)/[(ζ₁−z₁)…(ζ_n−z_n)] dζ₁…dζ_n,
where each ζ_j runs around the corresponding coordinate circle.
Repeated one-variable Cauchy formulas drive many basic results in Cⁿ.
11. Cauchy estimates extend to multi-indices
If |f|≤M on the distinguished boundary of a polydisc, then derivatives satisfy estimates of the form
|∂^α f(a)|≤α! M/(r₁^{α₁}⋯r_n^{α_n}).
As in one variable, boundary control yields derivative control and compact-family consequences.
12. The identity theorem changes form in several variables
In one variable, zeros of a nonzero holomorphic function are isolated. Therefore an accumulation point of zeros inside the domain forces the function to vanish identically.
In several variables, a nonzero holomorphic function can vanish on a whole hypersurface. For example f(z₁,z₂)=z₁ vanishes on {z₁=0}.
Thus “zeros have an accumulation point” is no longer enough. Agreement on a nonempty open set still forces global agreement on a connected domain.
13. Zero sets become complex analytic geometry
The zero set of one holomorphic function in Cⁿ is typically a complex hypersurface away from singular points.
Systems f₁=⋯=f_k=0 create analytic sets whose dimensions depend on rank and dependence among the equations.
This is where several complex variables meets algebraic and analytic geometry.
14. The complex Jacobian controls local invertibility
For F=(f₁,…,f_n):Cⁿ→Cⁿ, define the complex Jacobian matrix
J_C F=(∂f_i/∂z_j).
If det J_C F(a)≠0, the holomorphic inverse function theorem gives a local biholomorphic inverse near a.
This is the natural Cⁿ analogue of f'(a)≠0 in one variable.
15. Holomorphic implicit functions are similarly rigid
Suppose F(z,w)=0 with complex variables z∈C^m and w∈C^k. If the Jacobian with respect to w is invertible at a solution, then locally w is a holomorphic function of z.
This constructs complex submanifolds from regular analytic equations.
16. Hartogs extension destroys the isolated-pole intuition in dimension at least two
For n≥2, holomorphic functions often extend across compact holes that would support genuine singularities in one variable.
A standard Hartogs extension theorem says: if Ω⊂Cⁿ is a domain, K⊂Ω is compact, Ω\K is connected, and n≥2, then every holomorphic function on Ω\K extends holomorphically to Ω under the standard theorem hypotheses.
In particular, an isolated missing point in Cⁿ for n≥2 cannot behave like the pole of 1/z in one variable.
17. Why 1/z₁ is not a counterexample to Hartogs extension
The function 1/z₁ is holomorphic only where z₁≠0. Its singular set is the whole hypersurface {z₁=0}, not one compact isolated point.
Hartogs extension does not say every hypersurface singularity is removable. It says sufficiently thin compact holes cannot support arbitrary holomorphic singularities when n≥2.
18. Hartogs figures illustrate analytic filling
A Hartogs figure contains a shell-like region that surrounds a missing core in some coordinates while remaining connected through another coordinate direction.
Power-series arguments show that holomorphic data on the figure determines values in the missing interior.
The geometry makes the extension phenomenon visible: extra complex directions provide analytic access around what would be a barrier in one variable.
19. Not every domain is the natural domain of some holomorphic function
Because Hartogs extension can force functions beyond the visible boundary, some domains are too small to be genuine maximal domains of holomorphy.
A domain of holomorphy is, roughly, a domain that is the maximal natural holomorphic domain for at least one holomorphic function.
This concept has no equally dramatic role in basic one-variable theory, where every plane domain is a domain of holomorphy.
20. Holomorphic convexity detects analytic escape
Given a compact K⊂Ω, its holomorphic hull consists of points z∈Ω satisfying
|f(z)|≤sup_K|f| for every holomorphic f on Ω.
A domain is holomorphically convex if these hulls remain compact in Ω.
Holomorphic convexity is one way to formalise whether holomorphic functions can separate the domain from its boundary.
21. Plurisubharmonic functions test every complex line
A real-valued upper-semicontinuous function u on Ω is plurisubharmonic if its restriction to every complex line is subharmonic.
For C² functions, this can be tested by positivity of the complex Hessian, also called the Levi form.
Plurisubharmonicity is the correct multidimensional convexity-like notion for many complex-analytic problems.
22. The Levi form is a Hermitian second-derivative matrix
For a C² real-valued function ρ, form
L_ρ(v)=Σ_{j,k} (∂²ρ/∂z_j∂z̄_k) v_j overline{v_k}.
If this Hermitian form is nonnegative for all v, then ρ is plurisubharmonic.
Strict positivity corresponds to strict plurisubharmonicity.
23. |z|² is strictly plurisubharmonic
For ρ(z)=|z₁|²+⋯+|z_n|²,
∂²ρ/∂z_j∂z̄_k=δ_{jk}.
Hence L_ρ(v)=|v|²>0 for v≠0.
This simple computation is the model for strongly pseudoconvex geometry.
24. Pseudoconvexity is the analytic boundary condition behind domains of holomorphy
One formulation describes a domain by ρ<0 and examines the Levi form of ρ along complex tangent directions to the boundary.
Nonnegative Levi curvature corresponds to pseudoconvexity under suitable smoothness conditions.
The Levi problem, solved through twentieth-century work, states that pseudoconvex domains in Cⁿ are domains of holomorphy, with equivalent formulations requiring technical care.
25. The unit ball and polydisc are not biholomorphically equivalent for n>1
In one variable, the disk is the standard simply connected bounded domain. In several variables, the unit ball and unit polydisc have different biholomorphic geometry.
Their automorphism groups and boundary geometries differ; no biholomorphism exists between them when n>1.
Thus there is no direct higher-dimensional Riemann Mapping Theorem turning every nice simply connected domain into one universal ball.
26. Pluriharmonic means locally the real part of a holomorphic function
A real-valued function u is pluriharmonic if its restriction to every complex line is harmonic.
Locally, pluriharmonic functions are real parts of holomorphic functions under standard simply connected/local hypotheses.
Pluriharmonicity is stronger than ordinary harmonicity on R^{2n}.
27. Ordinary harmonic does not imply pluriharmonic
In C², real harmonicity means the real 4-dimensional Laplacian vanishes.
Pluriharmonicity imposes vanishing of the complex Hessian ∂∂̄u, a stronger directional condition.
This is another warning that one-variable harmonic intuition does not lift unchanged.
28. The ∂̄ operator measures failure of holomorphicity
For a smooth function f,
∂̄f=Σ_j (∂f/∂z̄_j)d z̄_j.
Thus f is holomorphic exactly when ∂̄f=0.
Solving ∂̄u=α is one of the central analytic problems of several complex variables. Existence and estimates connect directly to pseudoconvexity and complex geometry.
29. Differential forms split into (p,q)-types
Complex differential forms decompose according to numbers of dz_j and d z̄_k factors.
A form of type (p,q) has p holomorphic and q antiholomorphic differentials.
The exterior derivative splits as
d=∂+∂̄.
This Dolbeault decomposition is foundational for complex manifolds and complex geometry.
30. Weierstrass preparation gives polynomial-like local structure
Near a point where a holomorphic function has finite order in one chosen variable, the Weierstrass Preparation Theorem factors it into a nonvanishing holomorphic unit times a monic polynomial in that variable whose coefficients are holomorphic in the others.
This theorem turns local analytic zero sets into controlled polynomial-like geometry.
31. Several-variable singularities are often sets, not isolated points
Meromorphic functions in Cⁿ typically have poles along analytic hypersurfaces. Branching and analytic-set singularities also occur along higher-dimensional loci.
Therefore singularity analysis asks not only “what type of point is this?” but “what is the geometry and codimension of the singular set?”
32. Common theorem-boundary errors
- One-variable isolated-zero reasoning does not transfer directly: zero sets can be hypersurfaces.
- An accumulation point of zeros does not alone force a several-variable holomorphic function to vanish identically.
- Hartogs extension does not remove singular hypersurfaces such as {z₁=0}.
- Simple connectedness is not the whole story for biholomorphic classification in Cⁿ.
- The unit ball and polydisc are not interchangeable when n>1.
- Ordinary harmonicity is weaker than pluriharmonicity.
- Pseudoconvexity and domains of holomorphy require precise definitions; they are not merely visual convexity.
33. A dependable Cⁿ workflow
First decide whether the problem is local or global. Locally, use Wirtinger derivatives, power series, Jacobians and Weierstrass structure.
For extension questions, inspect the dimension and codimension of the missing set before importing one-variable singularity language.
For domain questions, test holomorphic convexity, pseudoconvexity or plurisubharmonic exhaustion rather than relying only on topology.
For geometric PDE questions, express failure of holomorphicity through ∂̄ and identify which estimates or domain hypotheses make the equation solvable.
34. Independent practice: twenty questions
- Write a point of C² in real coordinates.
- Define ∂/∂z_j and ∂/∂z̄_j.
- State the C¹ Wirtinger criterion for holomorphicity.
- Is f(z₁,z₂)=z₁²+e^{z₂} holomorphic?
- Why is z₁z̄₂ not holomorphic?
- What does separate holomorphy mean?
- State Hartogs’ separate-holomorphy theorem in words.
- Write the general multi-index power-series form.
- Define a polydisc.
- What geometric object is the distinguished boundary of a bidisc?
- Why does the one-variable accumulation-of-zeros criterion fail in C²?
- What does det J_CF(a)≠0 imply?
- State the key Hartogs extension phenomenon for n≥2.
- Why is 1/z₁ not a counterexample to isolated-hole extension?
- What is a domain of holomorphy?
- Define plurisubharmonicity by restriction to complex lines.
- What is the Levi form of |z|²?
- Why are the unit ball and polydisc not universal equivalents in n>1?
- What does ∂̄f=0 mean?
- What is the decomposition d=∂+∂̄ used for?
35. Worked solutions and checks
1. (z₁,z₂)=(x₁+iy₁,x₂+iy₂), corresponding to (x₁,y₁,x₂,y₂)∈R⁴.
2. ∂/∂z_j=1/2(∂/∂x_j−i∂/∂y_j), ∂/∂z̄_j=1/2(∂/∂x_j+i∂/∂y_j).
3. A C¹ function is holomorphic iff all ∂f/∂z̄_j vanish.
4. Yes, it is entire on C².
5. ∂/∂z̄₂ gives z₁, which does not vanish identically.
6. Fix all but one coordinate; the function is holomorphic in each remaining coordinate separately.
7. On an open domain in Cⁿ, separate holomorphy implies joint holomorphy.
8. Σ_{α∈N^n}c_α(z−a)^α.
9. The product of coordinate disks |z_j−a_j|<r_j.
10. A product of two circles, topologically a 2-torus.
11. A nonzero function such as z₁ vanishes on the whole hypersurface {z₁=0}, which has accumulation everywhere.
12. A local biholomorphic inverse exists.
13. Compact holes with connected complement inside a domain often cannot support holomorphic singularities; holomorphic functions extend across them.
14. Its singularity is not isolated; it lies on the hypersurface z₁=0.
15. Roughly, a domain that is the maximal holomorphic domain of some holomorphic function.
16. Its restriction to every complex line is subharmonic.
17. The identity Hermitian form, so L(v)=|v|².
18. Several-variable biholomorphic geometry has invariants beyond simple connectedness; ball and polydisc have different boundary and automorphism structures.
19. All antiholomorphic first derivatives vanish, so f is holomorphic under the smoothness hypothesis.
20. It separates exterior differentiation into holomorphic and antiholomorphic type components and underlies Dolbeault theory.
36. Where this guide hands off
R26.08 returns the theory to use and verification: transform methods, zero counting, boundary models, numerical contour checks and application handoffs into BTT’s existing signal-processing, control, quantum and applied-mathematics owners.
Sources and further study
For deeper study, standard references include Lars Hörmander’s complex-analysis texts and Steven Krantz’s introductions to several complex variables. BTT’s Schemes and Local Algebra and Manifolds and Geometric Topology provide neighbouring algebraic and geometric routes without replacing the analytic theory here.