  We study the variational structure of the complex $k$-Hessian equation on
bounded domain $X\subset \mathbb C^n$ with boundary $M=\partial X$. We prove
that the Dirichlet problem $\sigma_k (\partial \bar{\partial} u) =0$ in $X$,
and $u=f$ on $M$ is variational and we give an explicit construction of the
associated functional $\mathcal{E}_k(u)$. Moreover we prove $\mathcal{E}_k(u)$
satisfies the Dirichlet principle. In a special case when $k=2$, our
constructed functional $\mathcal{E}_2(u)$ involves the Hermitian mean curvature
of the boundary, the notion first introduced and studied by X. Wang. Earlier
work of J. Case and and the first author of this article introduced a boundary
operator for the (real) $k$-Hessian functional which satisfies the Dirichlet
principle. The present paper shows that there is a parallel picture in the
complex setting.
