  We present a detailed account and extension of our claim in arXiv:1610.01490.
We test the gauge/gravity duality between the ${\cal N} = 6$ mass-deformed ABJM
theory with U$_k(N)\times$U$_{-k}(N)$ gauge symmetry and the 11-dimensional
supergravity on LLM geometries with SO(4)/${\mathbb Z}_k$
$\times$SO(4)/${\mathbb Z}_k$ isometry, in the large $N$ limit. Our analysis is
based on the evaluation of vacuum expectation values of chiral primary
operators from the supersymmetric vacua of mass-deformed ABJM theory and from
the implementation of Kaluza-Klein holography to the LLM geometries. We focus
on the chiral primary operator with conformal dimension $\Delta = 1$. We show
that $\langle {\cal O}^{(\Delta=1)}\rangle= N^{\frac32} \, f_{(\Delta=1)}$ for
all supersymmetric vacuum solutions and LLM geometries with $k=1$, where the
factor $f_{(\Delta)}$ is independent of $N$. We also confirm that the vacuum
expectation value of the the energy momentum tensor is vanishing as expected by
the supersymmetry. We extend our results to the case of $k\ne 1$ for LLM
geometries represented by rectangular-shaped Young-diagrams. In analogy with
the Coulomb branch of the ${\cal N} = 4$ super Yang-Mills theory, we argue that
the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM
geometries are parametrized by the vevs of the chiral primary operators.
