Local results of several complex variables

1. de Rham currents
2. Subharmonic and Plurisubharmonic functions
3. Resolution of \( \bar\partial \), Dolbeault-Grothendieck lemma
4. Extension theorems, Domain of holomorphy
- 4.1. Generalities
- 4.2. Case \( \Omega\subset \mathbb{C}^n \)
  - 4.2.1. Different notion of pseudoconvexity
  - 4.2.2. Richberg approximation theorem

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1 de Rham currents

Let \( M \) be a differential \( m \)-dimensional manifold and \( \mathcal{E}^p(M) \) be the vector space of smooth \( p \)-forms on \( M \) and \( \mathcal{D}^p(M) \) be the space of those with compact support. Then \( \mathcal{E}^p(M), \mathcal{D}^p(M) \) is a topological vector space with the pseudonorms \( p_{K,\Omega}^s(\omega) = \max_{K, |\alpha|\leq s}|D^\alpha u_I| \) where \( K\Subset \Omega \) an coordinated open set. The space of de Rham current with dimension \( p \) / degree \( m - p \) is defined as the dual space of \( \mathcal{D}^p(M) \), denoted by \( \mathcal{D}'^{m-p}(M) \) or \( D'_{p}(M) \)

We are still in \( \mathbb{R} \), but the definition expands to the complex case, denoted by \( \mathcal{D}'^{m-p, m-q}(M) = \mathcal{D}'_{p,q}(M) \) where \( m \) is the complex dimension of \( M \).
The degree is defined such that the current \( T_\omega: \eta \mapsto \int_M \omega\wedge\eta \) is of the same degree as \( \omega \). The dimension is defined so that the current \( T_{[Z]}: \eta \mapsto \int_Z \eta \) is of the same dimension as \( Z \).

One has the following operation on \( \mathcal{D}'^{m-p}(M) \):

Derivative: \( \langle dT, \omega \rangle = (-1)^{\deg{T}} \langle T, d\omega \rangle \)
Wedge product with a form: \( \langle T\wedge \eta,\omega \rangle = \langle T, \eta \wedge \omega \rangle \)
Pushforward: If \( F: X \longrightarrow Y \) proper on \( \supp T \) then \( \langle F_*T, \omega \rangle = \langle T, F^*\omega \rangle = \langle T, \chi F_* \omega \rangle \) where \( \chi \in C^{\infty}(M) \) identically 1 on \( \supp T \). The proper condition is such that the pullback of \( \omega \) is compactly support in \( \supp T \)
Pullback: Let \( F: X \longrightarrow Y \) submersion then the pushforward of a form on \( X \) is well-defined by Fubini. One has \( \langle F^* T, \omega \rangle = \langle T, F_* \omega \rangle \)

The sign of derivative is chosen so that \( dT_\omega = T_{d\omega} \).
Pushforward keeps the dimension, as the arguments are of the same degree.
Pullback keeps the codimension, meaning the degree (think \( F^* T_{[Z]} = T_{[F^{-1}(Z)]} \)).
Locally a current is of form \( T = \sum u_I dx^I \) where \( u_I \) are distribution. Note: Here distribution are indentified as a current of maximal degree and not zero degree as they naturally are. To be exact, the notation of \( u_I \) is contravariant and its action is \( \varphi dx^1\wedge\dots\wedge dx^N \mapsto \langle u_I, \varphi \rangle dx^1\wedge\dots\wedge dx^n/\text{vol} \) where \( \text{vol} \) is a canonical volume form.

The last two remarks explain the sign in the following proposition.

Let \( F: M_1 \longrightarrow M_2 \), submersion if needed, then

\( \supp F_* T \subset F(\supp T)\)
\(d(F_* T) = F_* dT\) (pushforward of a form is still that form)
\(F_*(T\wedge F^* g) = (F_*T)\wedge g\)

and

\(F^*(dT) = (-1)^{m_1-m_2} d(F^* T) \)
\(F^*(T\wedge g) = (-1)^{m_1-m_2 -\deg g}(F^* T)\wedge F^* g\)

2 Subharmonic and Plurisubharmonic functions

Some properties of holomorphic functions that remain in several variables.

Cauchy formula
Analyticity: series development. Therefore its zeroes never form an open set (except for constant)
Maximum modulus
Cauchy inequality and Montel's theorem

2.1 Subharmonic functions

We are now in the context of \( \mathbb{R}^n \).

Let \( \Omega \Subset \mathbb{R}^n \) be a smoothly bounded domain, then there exists uniquely a function \( G_\Omega :\bar\Omega\times\bar\Omega \longrightarrow [-\infty, 0] \), called the Green kernel of \( \Omega \), with the following properties:

Regular: \( G_\Omega \) is \( C^\infty \) on \( \bar\Omega\times\bar\Omega\setminus \Delta_\Omega \) where \( \Delta_\Omega \) denotes the diagonal,
Symetric: \( G_\Omega(x,y) = G_\Omega(y,x) \),
Negative: \( G_\Omega(x,y) <0 \) on \( \Omega\times\Omega \) and \( G_\Omega(x,y) = 0 \) on \( \partial\Omega\times \Omega \),
\( \Delta_x G_\Omega(x,y) = \delta_y \) on \( \Omega \) for every \( y\in \Omega \).

One can take \( G_{r}=N(x-y) - N(\frac{|y|}{r}(x-\frac{r^2}{|y|^2}y))\) where \( N \) is the Newton kernel (or Newtonian potential, the gravitational potential). Explicitly, one has

\begin{align} G_r(x,y) &= \frac{1}{4\pi}\log \frac{|x-y|^2}{r^2 -2 \langle x,y \rangle +\frac{1}{r^2}|x|^2|y|^2} &\text{ if } n=2\\ G_r(x,y) &=\frac{-1}{(m-2)\vol(S^{m-1})}(|x-y|^{2-m} - (r^2 - 2 \langle x,y \rangle + \frac{1}{r^2}|x|^2|y|^2)^{1-m/2}) &\text{ if } n\geq 3 \end{align}

For \( u\in C^2(\bar \Omega, \mathbb{R}) \) one has \[ u(x) = \int_{\Omega}G_\Omega(x,y) \Delta u(y) d\lambda(y) + \int_{\partial\Omega} u(y) \frac{\partial G_\Omega}{\partial \nu_y} d\sigma(y) \] In particular, for \( \Omega = B(0,r) \), one has \[ P_r(x,y) := \frac{\partial G}{\partial \nu_y} =\frac{1}{\vol(S^{m-1})r}\frac{r^2 - |x|^2}{|x-y|^m} \] called the Poisson kernel.

Use the Green-Riesz formula: \( \int_\Omega u\Delta v - v\Delta u = \int_{\partial \Omega}u \frac{\partial v}{\partial \nu} - v \frac{\partial u}{\partial \nu} \).

Let \( \Omega\subset \mathbb{R}^n \) be an open subset and \( u: \Omega \longrightarrow [-\infty,\infty) \) a upper semi-continuous function: \[ \limsup_{x \to x_0}u(x) \leq u(x_0) \] One notes by \( \mu_S(u,a,r) \) and \( \mu_B(u,a,r) \) the average of \( u \) in the sphere and the disk centered in \( a \) of radius \( r \). Then the following properties are equivalent and a function is called subharmonic if they are verified.

1): \( u(x) \leq P_{a,r}[u](x) \quad \forall a,r, x\in B(a,r)\subset \Omega\),
2): \( u(a) \leq \mu_S(u,a,r)\quad \forall B(a,r)\subset\Omega \),
2'): \( u(a) \leq \mu_S(u,a,r)\quad \text{ for } B(a,r_n)\subset\Omega, r_n\to 0 \),
3): \( u(a) \leq \mu_B(u,a,r)\quad \forall B(a,r)\subset \Omega \),
3'): \( u(a) \leq \mu_B(u,a,r)\quad \text{ for } B(a,r_n)\subset \Omega, r_n\to 0 \),
4): If \( u\in C^2 \), then \( \Delta u \geq 0 \).

The convex cone of subharmonic functions on a domain \( \Omega \) is denoted by \( Sh(\Omega) \).

It is obvious that \( (1) \to (2) \to (3) \to (3')\to (2') \). To prove \( (2')\to (1) \) one needs the following 2 facts:

Let \( u \) be a u.s.c. function on a compact metric space \( X \), then there exists a sequence \( u_n \) continuous function on \( X \) that decreases to \( u \) pointwise.

Proof. Let \( \tilde u_k(x) = \max \{u(x), -k \} \) to exclude the \( -\infty \) points. Then \( v_k(x) = \sup_{y\in X} \left(u(y) - kd(x,y)\right) \) works.

\( (2') \) implies strict maximum principle (see prop:subhar).

Proof. By restriction to smaller neighborhood, one can suppose that \( u \) attains global maximum at \( x_0 \) in \( \Omega \). Then \( W=\{x\in \Omega:\ u(x) < u(x_0)\} \) is an open set, and has a point \( y \) in its boundary if \( W \) nonempty. Then \( (2') \) is not satisfied at \( y \) since the measure of open arc is nonzero.

Note that if \( u \) is continue than \( (2') \to (1) \): Let \( h = P_{a,r}[u] \) harmonic then \( u-h \) satisfies \( (2') \), therefore the maximum principle, hence \( u-h\leq \restr{(u-h)}{S(a,r)} = 0 \).

If \( u \) is u.s.c, take a sequence \( v_k \) continuous that decreases to \( u \) and let \( h_k = P_{a,r}[v_k]\) then \( h_k\geq v_k\geq u \) and \( h_k\to P_{a,r}[u] \) by monotone convergence.

Let \( u\in Sh(\Omega) \) then

(Strict) maximum principle.: \( u \) cannot attain local maximum unless it is constant in the corresponding connected component,
Locally integrable.: \( u \) is \( L^1_{loc}\) on each connected component where \( u\not\equiv -\infty \),
Pointwise decreasing limit: The pointwise limit \( u \) of a decreasing sequence \( u_k \) of subharmonic functions is also subharmonic.
Regularisation.: \( \mu_S(u,a,\varepsilon),\mu_B(u,a,\varepsilon),\rho_\varepsilon \ast u \) increase in \( \varepsilon \). Moreover, \( \rho_\varepsilon * u \in Sh(\Omega) \) and decreases to \( u \) pointwise as \( \varepsilon \to 0\).

Moreover, for \( u\in \mathcal{D}'(\Omega) \)

Positive measure.: \( u\in Sh(\Omega) \) iff \( \Delta u \geq 0 \) is a positive measure.

Locally integrable.: To see that \( u\in L^1_{loc}(\Omega) \) if \( \Omega \) is connected and \( u\not\equiv -\infty \), let \( x \) be a point in the boundary of \( W=\{y\in \Omega:\ u\text{ integrable in neighborhood of } y\} \), then apply mean value property in \( a\in W\) such that \( x\in B(a,r) \).
Pointwise decreasing limit.: Infimum of a family of u.s.c functions is still u.s.c. The mean value property comes from monotone convergence.
Regularisation.: Check first for \( C^2 \) functions, then regularise. One uses the following Gauss formula: \[ \mu_S(u,a,r) = u(a) + \frac{1}{n}\int_0^r\mu_B(\Delta u, a, t)tdt \] to see that \( \mu_S \) is increasing in \( r \) and \[ \mu_B(u,a,r) = m\int_0^1 t^{m-1}\mu_S(u,a,rt)dt \] to see that \( \mu_B \) is increasing. For the convolution, use \[ u*\rho_\varepsilon = \vol(S^{n-1})\int_0^1 \mu_S(u,a,\varepsilon t)\rho(t) t^{m-1}dt. \]
Positive measure.: \( \Delta u * \rho_\varepsilon \geq 0\) as function, therefore the limit \( \geq 0 \) as measure (dominated convergence).

Let \( u_k \in Sh(\Omega) \) then

If \( \{u_k\} \) decrease to \( u \) then \( u\in Sh(\Omega) \).
Let \( \chi \) be a convex function, non-decreasing in each variable then \( \chi(u_1,\dots,u_p) \in Sh(\Omega)\). Therefore, \( \sum u_i\) and \( \max\{u_i\} \) are subharmonic.

Let \( u \) be a real function on \( \Omega \) then \( u^*(x) = \lim_{\varepsilon\to 0} \sup_{x+\varepsilon B} u\), called the upper envelope of \( u \) is u.s.c and is in fact the smallest u.s.c function greater than \( u \).
Choquet lemma. Let \( \{u_\alpha\} \) be a family of real function, one defines the upper regularization of \(\{ u_\alpha\} \) by \( u^*\) where \(u=\sup_\alpha u_\alpha \). Then from every such family, on can always find a countable subfamily \( \{v_i\} \) such that \( u^* = v^* \).
If \( \{u_{\alpha}\} \subset Sh(\Omega)\) then \( u^* = u\) a.e. and \( u^*\in Sh(\Omega) \).

Obvious.
Let \( B_i \) be a countable base of the topology and \( x_{i,j} \) be a sequence such that \( u(x_{ij}) \to \sup_{B_i}u \). Let \(\{ u_{i,j,k} \}\) be a countable subfamily such that \( u_{ijk}(x_i) \to u(x_i) \) then it is a suitable subfamily.
WLOG, suppose that \( \{u_\alpha\} = \{u_i\} \) countable then \( u \) satisfies the submean value property: \( u(z)\leq \mu_B(u,z,r) \). By the continuity of \( \mu_B(u,z,r) \) one has \( u^*(z)\leq \mu_B(u,z,r)\leq \mu(u^*,z,r) \) therefore \( u^*\in Sh(\Omega) \) and \( u^*(z) = \lim_{r\to 0} \mu_B(u^*,z,r) = \lim_{r\to 0}\mu_B(u,z,r) \), from which \( u=u^* \)

2.2 Plurisubharmonic functions

The analog of harmonic functions over \( \mathbb{C} \) in multidimensional case \( \Omega\subset \mathbb{C}^n \) is in fact pluriharmonic functions which is defined through the notion of plurisubharmonic functions

A real function \( u \) is said to be plurisubharmonic if and only if its restriction to any complex line is subharmonic. One denotes by \( Psh(\Omega) \) the space of plurisubharmonic function on \( \Omega \).
In case \( u\in C^2 \) on \( \Omega\subset \mathbb{C}^n \), this is equivalent to \[ H(u)(\zeta) = \sum \frac{\partial^2 u}{\partial z^j \partial \bar z^k } \zeta^j \bar\zeta^k \geq 0\quad \forall \zeta\in \mathbb{C}^n \] where the notation \( H(u)(\zeta) \) is invariant, i.e. if \( f:\ M_1 \longrightarrow M_2\) is holomorphic then \( H(u\circ f)(\zeta) = H(u) df(\zeta) \).
In the general case, this is equivalent to \( H(u)(\zeta) \geq 0\quad\forall \zeta\in \mathbb{C}^n \) as a measure.

The invariance can be noticed using \( \zeta^j = \zeta^j d\zeta^j + \bar\zeta^j d\bar\zeta^j\) where LHS is intepreted as a vector in \( T \mathbb{C} \). This allows us to extend the notion of \( Psh(M) \) to any complex manifold \( M \).
By consequence, \( f^*u \in Psh(M_1) \) for all \( u\in Psh(M_2) \) and \( f:\ M_1 \longrightarrow M_2 \) holomorphic.

The construction of new plurisubharmonic function is the same as that of subharmonic function. Let \( u_k \in Psh(\Omega) \) then

If \( \{u_k\} \) decrease to \( u \) then \( u\in Psh(\Omega) \).
Let \( \chi \) be a convex function, non-decreasing in each variable then \( \chi(u_1,\dots,u_p) \in Psh(\Omega)\). Therefore, \( \sum u_i\) and \( \max\{u_i\} \) are plurisubharmonic.
The upper regularization \( u^* \) where \( u = \sup_\alpha u_\alpha \) is also plurisubharmonic and \( u=u^* \) almost everywhere.

The only nontrivial proof is the third one where upper envelop in \( \mathbb{C^n} \) and in a line can be different. To fix this, use Choquet lemma prop:upper-regularization and dominated convergence, \( u*\rho_\varepsilon \) satisfies the submean property on every complex line and decrease to \( u \) a.e.

2.3 Pluriharmonic functions

A function \( u\) is said to be pluriharmonic on \( \Omega \), denoted \( u\in Ph(\Omega)\) if \( u\in Psh(\Omega) and \( -u\in Psh(\Omega) \) \) where \( \Psh(\Omega) \).

This is obviously equivalent to \( H(u) = 0 \), i.e. \( \frac{\partial^2 u}{\partial z^j \bar\partial z^k} =0 \quad \forall j,k\), i.e. \( \partial \bar \partial u = 0 \).

By mean value property, \( Ph(\Omega)\subset Harm(\Omega) \).
If \( f\in \mathcal{O}(M)\) then \( \Re f, \Im f \in \( Ph(M) \)

If \( M \) is a complex manifold such that \( H^1_{dR}(X, \mathbb{R}) = 0 \) then every pluriharmonic function \( u \) is a real part of a holomorphic function \( f\in \mathcal{O}(M) \)

Since \( d (\bar \partial u) = 0 \), and \( H^{1}_{dR} = 0 \), one has \( \bar \partial u = dg \). Therefore \( d(u - 2\Re g) = (\bar\partial u - dg) + (\partial u - d\bar g) = 0 \), hence on chooses \( f=2g +C \) on each connected component.

3 Resolution of \( \bar\partial \), Dolbeault-Grothendieck lemma

The generalized Cauchy formula for several variables is the following (the formula in wikipedia is \( K^{0,0}_{BM} \))

The Bochner-Martinelli kernel is the following \( (n,n-1) \)-form on \( \mathbb{C}^n \) \[ k_{BM} = (-1)^{n(n-1)/2}\frac{(n-1)!}{(2\pi i)^n}\sum_{1\leq j\leq n} (-1)^j \frac{\bar z_j dz_1\wedge \dots\wedge dz_n\wedge d\bar z_1\wedge\dots \wedge \widehat{d\bar z_j} \wedge \dots \wedge d\bar z_n}{|z|^{2n}} \] then \( \bar \partial k_{BM} = \delta_0 \) on \( \mathbb{C}^n \).

Let \( K_{BM} = \pi^* k_{BM} \) where \( \pi: (z,\zeta)\mapsto z-\zeta \) so that \( \bar \partial K_{BM} = [\Delta] \), then: For any domain \( \Omega\subset \mathbb{C}^n \) bounded with piecewise \( C^1 \) boundary and \( v \) a \( (p,q) \)-form of class \( C^1 \) on \( \bar \Omega \) then \[ v(z) = \int_{\partial\Omega}K^{p,q}_{BM}(z,\zeta)\wedge v(\zeta) + \bar \partial \int_{\Omega} K^{p,q-1}_{BM}(z,\zeta)\wedge v(\zeta) + \int_\Omega K^{p,q}_{BM}(z,\zeta)\wedge \bar \partial v(\zeta) \] where \( K^{p,q}_{BM} \) denotes the component of \( K_{BM} \) type \( (p,q) \) in \( z \) and type \((n-p, n-q-1)\) in \( \zeta \)

Another consequence of thm:koppelman is the global resolution of \( \bar \partial \) in case of compact support.

If \( v\) is a \( (p,q) \)-form with \( q\geq 1 \) on \( \mathbb{C}^n \), compactly supported, with regularity of class \( C^s \) such that \( \bar \partial v = 0 \) then there exists an \( (p,q-1) \)-form \( u \) on \( \mathbb{C}^n \) with the same regularity as \( u \) such that \( \bar \partial u =v \). In fact one can take \[ u(z) = \int_{\mathbb{C}^n} K_{BM}^{p,q-1}(z,\zeta)\wedge v(\zeta) \] In case \( (p,q)=(0,1) \) then \( u \) is compactly support. This means that the compact support \( (0,1) \)-Dolbeault cohomology \( H_c^{0,1}(\mathbb{C}^n) = 0 \).

Since \( K_{BM} = O(|z|^{1-2n}) \), one has \( |u(z)| = O(|z|^{1-2n}) \) at infinity. Therefore the compact support of \( u \) in case \( (p,q)=(0,1) \) is explained by Liouville theorem.

The Dolbeault-Grothendieck lemma solves the equation \( \bar \partial u = v \) in a local scale if the compact support condition is dropped and gives regular result if \( v \) is a \((p,0)\)-form.

Let \( v \in \mathcal{D}'(p,q)(\Omega)\) such that \( \bar \partial v = 0 \).

If \( q=0 \) then \( v = \sum v_I dz^I \) where \( v_I\in \mathcal{O}(\Omega) \).
If \( q \geq 1 \) then there exists \( \omega\subset \Omega \) and \( u\in \mathcal{D}'^(p,q-1)(\Omega) \) such that \( \bar \partial u =v \). Moreover, if \(v\in \mathcal{E}^{p,q}(\Omega)\) then \( u\in \mathcal{E}^{p,q-1}(\Omega) \)

\( \bar \partial \) is hypoellipticity in bidegree \( (p,0) \), i.e. if \( \bar \partial u = v\), v of bidegree \( (p,1) \) and \(v \) is \( C^\infty \) then \( u \) is also \( C^\infty \) on the entire domain \( \Omega \).

4 Extension theorems, Domain of holomorphy

Let \( \Omega \subset \mathbb{C}^n \) be a domain and \( K \Subset \Omega \) such that \( \Omega\setminus K \) is connected. Then \( \restr{\mathcal{O}(\omega)}{\Omega\setminus K} = \mathcal{O}(\Omega\setminus K) \) every holomorphic function on \( \Omega\setminus K \) extends to \( \Omega \)

Let \( f\in \mathcal{O}(\Omega \setminus K \) be the function we want to extend. Let \( \varphi \) be a function with support in a neighborhood of \( K \) and is identically 1 on \( K \) and \( g = (1-\varphi)f \) which coincides with \( f \) outside of \( \supp \varphi \). Then \( v = \bar \partial g \in \mathcal{D}^{0,1} \) satisfies \( \bar \partial v = 0 \), therefore there exists \( u\in C_c^\infty(\mathbb{C}^n \) with \( \supp u \subset \supp \varphi\) such that \( \bar \partial u = v = \bar \partial g \), the holomorphic function \( g-u \) is well-defined on \( \Omega \) and coincides with \( f \) (and \( g \)) on \( \Omega\setminus \supp\varphi \), therefore coincides with \( f \) on \( \Omega\setminus K \).

Note that although we do not need \( \Omega \) to be small, this theorem counts as a local result due to the hypothesis that we are in \( \mathbb{C}^n \).

A global result can be obtained using the Hartog figure, that is the union of an anulus \( \{ (z_1,z'):\ r < |z_1| < R\} \) and an open set in other dimension \( \{ (z_1,z'):\ z'\in\omega \text{ open}\} \). and use the interpolation \( (z_1,z') \mapsto \int_{C_{R}} \frac{f(\zeta_1,z')}{\zeta_1-z_1}d\zeta_1 \) to extend \( f \). The open set in \( z' \)-dimension is to show that the interpolation and \( f \) coincide on it. With one dimension \( z_1 \) to form the annulus an another dimension (says \( z_2 \) to form the open set, one can extend any holomorphic function to a submanifold of (complex) codimension at least 2.

Let \( M \) be a complex manifold and \( N \) a sub complex manifold of codimension \( \geq 2 \) then any holomorphic function on \( M\setminus N \) extends uniquely to \( M \).

4.1 Generalities

An approach to the extension problem on complex manifolds is through the notion of holomorphic hull and holomorphic convexity.

Let \( K \) be a compact in a complex manifold \( M \). Then the holomorphic hull \( \hat K_{\mathcal{O}(M)} \) is the set \( \{ z\in M:\ f(z) \leq \sup_K |f| \quad\forall f\in \mathcal{O}(M) \} \).
A complex manifold \( M \) is said to be holomorphically convex if \( \hat K_{\mathcal{O}(X)} \) is compact for all compact \( K\subset M \).

The following statements are obvious

\( \hat K \) is a closed subset containing \( K \) and \( \hat{\hat K} = \hat K \).
If \( f:\ M_1 \longrightarrow M_2 \) is holomorphic then \( f(\hat K)\subset \widehat{f(K)} \). (Think inclusion)
Hole filling. In particular, if \( f:\ \bar B \longrightarrow X \) and \( f(\partial B) \subset K \) then \( f(\bar B)\subset \hat K \).

Let \( M \) be a holomorphically convex complex manifold then

\( M \) admits a exhaustive sequence of compact \( K_\nu \), i.e. \( K_\nu \Subset K_{\nu+1} \) and \( \widehat {K_\nu} = K_\nu \).
\( M \) is weakly pseudoconvex, i.e. there exists \( \psi\in Psh(M)\cap C^\infty(M) \) such that \( \{\psi

4.2 Case \( \Omega\subset \mathbb{C}^n \)

Domain of holomorphy

Let \( \Omega\subset \mathbb{C}^n \) be a domain then:

If \( \Omega \) is a domain of holomorphy then \( \hat K_{\mathcal{O}(\Omega)} \) is compact and \( d(K, \partial \Omega) = d(\hat K,\partial\Omega) \).
THe followings are equivalent:
1. \( \Omega \) is a domain of holomorphy.
2. \( \Omega \) is holomorphically convex.
3. Let \( \{z_k\} \) be a sequence in \( \Omega \) without accumulation in \( \Omega \) and \( c_k\in \mathbb{C} \). There exists a function \( f\in \mathcal{O}(\Omega) \) such that \( f(z_k)=c_k \).
4. There exists a function \( F\in \mathcal{O}(\Omega) \) that is unbounded locally in any point on \( \partial\Omega \).

#+END_theorem