What is a Kählerian variety? I know what a Kähler manifold is, and I (roughly) know what a variety is. However, I don't know what a Kählerian variety is. Is it just a variety which is also a Kähler manifold, or is it a separate concept? Is there any distinction between affine and projective varieties in the definition of a Kählerian variety?
 A: As far as I know, the definition of Kähler variety (in a non-smooth sense) introduced by Moishezon for the first time as follows. (Warining: there is another definition of Kähler variety due to Grauert)
B. MOISHEZON, Singular Kählerian spaces, Proc. Int. Conf. on manifolds and related topics in topology, Univ. of Tokyo Press, 1974, 343-351
Definition:  $X$ is Kähler variety i.e., $X$ admits a coveting by open sets $X_A$ and a plurisubharmonic function $\phi_\alpha$ on each $X_\alpha$ ($\alpha\in A$) such that $\phi_\alpha-\phi_\beta$ is pluriharmonic on $X_\alpha\cap X_\beta$.
Example(Varouchas): Let $X$ be a compact complex surface. Let $X^{[m]}$ be the Hilbert scheme or Douady space of zero-dimensional subspaces $(Z,\mathcal O_Z)$ of $X$, of the length $\dim_{\mathbb C}\mathcal O_Z=m$, then $X^{[m]}$ is Kähler variety whenever $X$ is Kähler variety.
Fujiki's Example: Let $X$ be a Kähler manifold and $G$ a finite group of
biholomorphic automorphisms of $X$. Suppose that the set of those points
which are fixed by some elements of $G$ is isolated. Then $X/G$ is a Kähler
variety.
Let me give some well-known theorems about Kähler variety.
Fujiki's Theorem: Any compact smooth analytic surface $X$ in Fujiki class $\mathcal C$ is Kähler variety.
Fujiki's Theorem: Let $X$ be a compact complex manifold of dimension
2. Suppose that $X$ is in Fujiki class $\mathcal C$. Then $X$ is Kähler variety.
We have the following important Moishezon stability theorem for Kähler variety. For smooth version, it is due to Kodaira.
Theorem: Let $X_0$ be a normal compact Kähler variety and we take it as the central fiber of $\pi:X\to\Delta$ where $\Delta$ is a disc. If the natural map $H^2(X_0,\mathbb R)\to H^2(X_0,\mathcal O_{X_0})$ is surjective. Then every small deformation $X_t$ of $X$ is again Kähler variety. 
J. Varouchas also gave the same definition several years later
J. Varouchas, Stabilité de la classe des variétés Kählériennes par certains morphismes propres, Inventiones mathematicae, February 1984, Volume 77, Issue 1, pp 117–127
After that, some other people introduced the relative Kähler form with the same "nature". See this comments
Let $ f:X\to S $ be a holomorphic map. A Kähler metric  $\varphi$ for $f$ is given by a collection ($\varphi_i$) of real differentiable functions on an open covering ($U_i$) of $X$ such that the differences $\varphi_i-\varphi_j$ on the intersections $U_j$ are locally the real part of holomorphic functions, and the Levi forms $L(\varphi_i)=i\partial\bar\partial \varphi_i$ are strictly 
positive on the complement of the zero section of the tangent bundle $T_{X/S}|_{U_i}$. If $\varphi_i-\varphi_j$ is only locally the real part of a differentiable function being holomorphic on the fibres of $f$, we call $\varphi$ a weak Kähler metric for $f$. We say that $f$ is Kähler (resp. weakly Kähler), if such a metric exists. In the case that $S=Spec(\mathbb{C})$ is a point, we obtain the notion of a Kähler space. Obviously the fibres of weakly Kähler maps are Kähler spaces. For example projective maps are Kähler.
See https://link.springer.com/article/10.1007%2FBF01389138#page-1
Example: Let $\pi:X\to S$ be a holomorphic fibre space over a disc. Let the fibers $X_s$ admits Kähler-Einstein metrics of negative Ricci curvature. Then we have the relative Kähler-Einstein metric
$$
Ric_{X/S}(\omega)=-\lambda(s)\omega
$$
where $\omega$ here is the relative Kähler metric and $\lambda$ is the fiberwise constant which when $s\to 0$, we have the violation of stability and hence existence of relative Kähler-Einstein metric corresponds to stability of relative tangent sheaf $T_{X/S}$ In the sense of Mumford.
In this case the right flow is the following Hyperbolic Relative Kähler Ricci flow
$$
\frac{\partial^2}{\partial s^{\prime}\partial t}=-Ric_{X/S}(s^{\prime},t)-\lambda(s^{\prime})\omega_{s^{\prime}}(t)
$$
where here $s^{\prime}=\frac{1}{s}$ and $s\to 0$
Note that there is another definition of Kähler variety due to  H. Grauert,
See  H. Grauert, Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368.
Grauert's definition of Kähler variety(1962):
Let $Y$ be an arbitrary reduced holomorphic space with a Hermitian metric $ds^2=\sum g_{i\bar k}dz_i\wedge dz_{\bar k}$ in $Y\setminus Y_{sing}$ (with continuous coefficients). Then it is called a Kähler metric in $Y$ if and only if for every point $a\in Y$ , there is an open neighborhood $Y$ of a and a strictly plurisubharmonic function $p$ on $Y$ such that $g_{i\bar k}=\frac{\partial^2p}{\partial z_i\partial z_{\bar k}}$ in $Y\setminus Y_{sing}$.
We have the notion of stratified Kähler variety. The following result show the more effectiveness of this notion
Definition: A complex space $Z$ is called a stratified Kähler space if there exists a complex stratification $S=(S_{\gamma})_{\gamma\in\Gamma}$ on $Z$ which is finer than the stratification of $Z$ as a complex space and there exists a Kähler structure $\omega=(U_{\alpha},\rho_{\alpha})_{\alpha\in I}$ on $Z$ such that $\rho_{\alpha|_{S_{\gamma}\cap U_{\alpha}}}$ is smooth.
For a proper Hamiltonian $G$-action on a normal Kähler space $X$ the quotient $X_0/G$ is a stratified normal Kählerian space with structure sheaf $\mathcal O_{X_0/G} $. Here $X_0$ is the zero level of the moment map $μ:X\to Lie(G)^∗$, which is assumed to be non-empty).
Hassan JOLANY
A: In my experience, the term means "a (complex) variety admitting a Kähler metric" (i.e., the holomorphic structure is fixed without specifying a metric, or even a polarization), whereas a "Kähler manifold" comes equipped with a holomorphic structure and a metric, and (anomalously) a "polarized Kähler manifold" comes with a positive $(1, 1)$ class but no (distinguished) metric.
A: I am not sure what the context for this is, but Kahlerian variety seems like franglais: the french term is "variété kählérienne", which is an exact translation of "Kahler manifold". (probably the French studied them first, so the translation is actually in the other direction).
