[proofplan] We prove that a bounded domain $\Omega \subset \mathbb{C}^n$ with $C^2$ boundary is a domain of holomorphy if and only if the Levi form of its defining function, restricted to the complex tangent space at each boundary point, is positive semidefinite. The proof combines two equivalences: the [Solution of the Levi Problem](/theorems/3416) (a domain is a domain of holomorphy iff it is pseudoconvex) and the [Equivalence of Pseudoconvexity Definitions](/theorems/3407) (for bounded domains with smooth boundary, pseudoconvexity is equivalent to the Levi semidefiniteness condition). [/proofplan] [step:Reduce the domain-of-holomorphy condition to pseudoconvexity via the Levi problem equivalence] By the [Solution of the Levi Problem](/theorems/3416), for any domain $\Omega \subset \mathbb{C}^n$, the following are equivalent: - $\Omega$ is a domain of holomorphy. - $\Omega$ is pseudoconvex (admits a continuous psh exhaustion function $\phi: \Omega \to \mathbb{R}$ with $\{\phi < c\} \Subset \Omega$ for all $c$). This equivalence holds for general domains in $\mathbb{C}^n$ without any [boundary regularity](/theorems/99) assumption. In particular, it applies to our bounded domain $\Omega$ with $C^2$ boundary. [/step] [step:Reduce pseudoconvexity to the Levi form condition for domains with $C^2$ boundary] Let $\rho: \mathbb{C}^n \to \mathbb{R}$ be the $C^2$ defining function with $\Omega = \{\rho < 0\}$ and $\nabla\rho \neq 0$ on $\partial\Omega$. At each boundary point $p \in \partial\Omega$, the complex tangent space to $\partial\Omega$ is \begin{align*} T^{1,0}_p(\partial\Omega) = \left\{w \in \mathbb{C}^n : \sum_{j=1}^n \frac{\partial\rho}{\partial z_j}(p)\, w_j = 0\right\} = \ker(\partial\rho|_p). \end{align*} The Levi form of $\rho$ at $p$, restricted to this complex tangent space, is the Hermitian form \begin{align*} \mathcal{L}_\rho(p; w) = \sum_{j,k=1}^n \frac{\partial^2 \rho}{\partial z_j \partial \bar{z}_k}(p)\, w_j \overline{w_k}, \quad w \in T^{1,0}_p(\partial\Omega). \end{align*} By the [Equivalence of Pseudoconvexity Definitions](/theorems/3407), the following are equivalent for a bounded domain with $C^2$ boundary: - $\Omega$ is pseudoconvex (admits a continuous psh exhaustion). - The Levi form $\mathcal{L}_\rho(p; w) \geq 0$ for all $p \in \partial\Omega$ and all $w \in T^{1,0}_p(\partial\Omega)$. The direction "Levi semidefiniteness implies pseudoconvexity" proceeds by constructing a psh exhaustion from the defining function. Near $\partial\Omega$, the function $-\log(-\rho)$ is a candidate: its complex Hessian on $\Omega$ near $\partial\Omega$ involves the Levi form of $\rho$ on the level sets $\{\rho = c\}$, and the Levi semidefiniteness condition ensures positive semidefiniteness of this Hessian on the complex tangent directions. Since $-\log(-\rho) \to +\infty$ as $z \to \partial\Omega$ (because $\rho \to 0^-$), and $\Omega$ is bounded, the function provides the desired exhaustion after patching with a smooth psh function in the interior. The direction "pseudoconvexity implies Levi semidefiniteness" shows that the sublevel sets $\Omega_c = \{\phi < c\}$ of a psh exhaustion $\phi$ are Levi pseudoconvex for regular values $c$. Since $\phi$ is psh, the complex Hessian of $\phi$ is positive semidefinite, which forces the Levi form of each level set to be positive semidefinite. Taking $c \to \sup \phi$ recovers the Levi condition on $\partial\Omega$. [guided] The Levi form condition is a pointwise differential condition at the boundary, while pseudoconvexity is a global condition involving an exhaustion function defined on all of $\Omega$. The equivalence between these two conditions -- one local and boundary-based, the other global and interior-based -- is the content of the [Equivalence of Pseudoconvexity Definitions](/theorems/3407). Why is the Levi form the correct boundary quantity? Consider a complex line $\ell = \{p + \zeta w : \zeta \in \mathbb{C}\}$ tangent to $\partial\Omega$ at $p$ in the complex sense ($w \in T^{1,0}_p(\partial\Omega)$). The restriction $\rho|_\ell$ is a real-valued function of the complex variable $\zeta$, and its Laplacian at $\zeta = 0$ is \begin{align*} \Delta_\zeta (\rho \circ \ell)(0) = 4\sum_{j,k} \frac{\partial^2 \rho}{\partial z_j \partial \bar{z}_k}(p)\, w_j \overline{w_k} = 4\,\mathcal{L}_\rho(p; w). \end{align*} If $\mathcal{L}_\rho(p; w) < 0$ for some complex tangent direction $w$, then $\rho|_\ell$ is superharmonic at $\zeta = 0$, meaning $\ell$ curves into $\Omega$ near $p$ -- the boundary is concave in the complex direction $w$. Such concavity forces holomorphic extension past $p$ (as in the Hartogs extension phenomenon), contradicting the domain-of-holomorphy property. The Levi semidefiniteness condition $\mathcal{L}_\rho(p; w) \geq 0$ ensures that the boundary is "convex" in every complex tangent direction, preventing forced extension. This is precisely the answer to Levi's 1910 question: the semidefiniteness condition he identified as necessary for a domain to resist holomorphic extension is also sufficient, as confirmed by the full [solution of the Levi problem](/theorems/3416). [/guided] [/step] [step:Combine the two equivalences to obtain the theorem] Chaining the two equivalences: \begin{align*} &\Omega \text{ is a domain of holomorphy} \\ &\quad \Longleftrightarrow \quad \Omega \text{ is pseudoconvex} && \text{(Solution of the Levi Problem)} \\ &\quad \Longleftrightarrow \quad \mathcal{L}_\rho(p; w) \geq 0 \text{ for all } p \in \partial\Omega,\; w \in T^{1,0}_p(\partial\Omega) && \text{(Equivalence of Pseudoconvexity Definitions)} \end{align*} This gives the desired characterisation: $\Omega$ is a domain of holomorphy if and only if the Levi form of $\rho$, restricted to the complex tangent space $T^{1,0}_p(\partial\Omega) = \ker(\partial\rho|_p)$, is positive semidefinite at every boundary point $p \in \partial\Omega$. [/step]