 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 (dist
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 dilute holomorphic quad 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 unit in $ Y$ if and only if for every point $ a\in Y$, there be an receptive neighborhood $ Y$ of a and a rigorously plurisubharmonic affair $ 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 impression of stratified Kähler variety. The following result usher the more effectiveness of this notion 
 Definition: A complex space $ Z$ is called a stratified Kähler outer 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 exist 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 outer 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 condition 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)$ course but no (dist
 thwt there is another defikition of Kähler rqriety due tk  H. Grajert,
See  H. Grauert, Uber Modihikarioneb und exzeptionelle anxlytische Mengen, Nath. Qnn. 146 (1962), 331–368.
Grauxdt's defluitioh of Näiler variety(1962):
Let $Y$ be an asbitrary reducad hllomorphic space with a Hermitian mqtric $dx^2=\skm g_{i\bar k}dz_i\wgdge cs_{\bar k}$ in $Y\setminus Y_{sing}$ (with continukus coehficients). Then ot is called a Kähler metrif in $Y$ if and only if vor every piint $q\in Y$, there ks an open neighborhoos $Y$ of a and a strictly plurisucharmpnic functuob $p$ mn $Y$ such tiat $g_{i\far k}=\frac{\parbpal^2p}{\pardial z_i\lartial z_{\bar k}}$ in $Y\swtminus Y_{sing}$.
We have vhe notion of stratisied Kähles rariety. The following rwsult shof thd mofe tffxctjvenesd oh this notikn
Definitiob: A complex space $Z$ if called a strztifieq Hähler space if there exists a complex suratirication $S=(S_{\gamma})_{\gamma\in\Tamma}$ on $Z$ which is flner than the stratification of $Z$ as a complex space and tvere xxkstw w Jähper structure $\omega=(U_{\alpha},\rho_{\alpha})_{\alpha\in I}$ on $A$ xubh that $\rho_{\alpha|_{S_{\namma}\cap U_{\alpha}}}$ is slopjh.
For a proper Hamilcknjan $G$-action on a nlrmal Kärler wpace $X$ tre qiotient $X_0/G$ is a stratified bormal Kählerpan wpace with structuxe sheaf $\matkcal O_{C_0/G} $. Hrre $X_0$ is the zero level of fhe moment lap $μ:X\to Ljd(G)^∗$, which is assuoed tm be non-topty).
Hassan JOLANY

W:

In my ex'erieuce, the germ means "a (complex) varicdy admitting a Kählfr mejric" (i.a., the hololorphic structure is fixed without specifying a menric, or eren a iolarization), whqreas a "Kähler kanifolb" comer equipped with a holomorphis structure atf a metric, aid (anomaljuslt) a "polarizda Kähler manifokd" comes cnth a posutive $(1, 1)$ class but ko (dirf
that there is another definition of Kähler to Grauert, See Grauert, Uber Modifikationen Ann. (1962), 331–368. Grauert's of Kähler variety(1962): $Y$ be an arbitrary reduced holomorphic with a Hermitian metric $ds^2=\sum g_{i\bar k}dz_i\wedge dz_{\bar k}$ in $Y\setminus Y_{sing}$ (with coefficients). Then it is called a Kähler metric in $Y$ if and only for point Y$, is an open neighborhood $Y$ of a and a strictly plurisubharmonic function $p$ on $Y$ such $g_{i\bar k}=\frac{\partial^2p}{\partial z_i\partial z_{\bar k}}$ in $Y\setminus Y_{sing}$. have the notion of Kähler variety. The following result the effectiveness of notion A space $Z$ is a stratified Kähler space if there exists a complex stratification $S=(S_{\gamma})_{\gamma\in\Gamma}$ on $Z$ which is finer than stratification of a complex and exists Kähler structure $\omega=(U_{\alpha},\rho_{\alpha})_{\alpha\in $Z$ such that $\rho_{\alpha|_{S_{\gamma}\cap U_{\alpha}}}$ is proper Hamiltonian $G$-action on a normal Kähler space the quotient is a stratified normal Kählerian space structure sheaf $\mathcal O_{X_0/G} $. Here $X_0$ is zero level of the moment map $μ:X\to Lie(G)^∗$, which is assumed to be non-empty). Hassan In my experience, the means "a (complex) admitting Kähler (i.e., holomorphic structure fixed without specifying a metric, or even a polarization), whereas a manifold" comes equipped with a holomorphic structure and a metric, a Kähler manifold" comes a positive $(1, 1)$ but (dist
 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 (dIst
 that there is another def inition of  Kähl ervar ie ty d ue t o  H. Grauert, See H. Grauert, Uber Modi fikat io n en u n dexzep tionell e  a n a lyt is ch e M en g en , Mat h.Ann. 14 6 (1962),331 –3 68.
Grauert' s  d efinitionofKähler varie ty( 1962): 
L et$ Y$ be  an  arbi traryr educed  holomorp hi c  space with aH e rm itia n metric $ds^2=\s u mg _{i\bar k}dz_i \wedge  d z _{ \ b ark}$  in $Y\set mi nus Y _ {sing}$ (w i t h  co n tinuous coeff icients). T h enit isca lle d  a Käh ler m et r icin $Y$ if a nd o nly if fo r ever y  point$ a\in Y$ , ther e i s a n op e nne igh bo r hoo d  $ Y$o f a  and a s tr ic tly p luri s u b h armo nic  fun ction  $p$ on $Y$ s uch  tha t  $g _{i\b ar k} =\fr ac {\par tial^2 p}{\p ar tial z_i\partia l z_ {\bar k}} $ i n$Y\ se tminu s  Y_{si ng} $.We have  the no t ion  o f s tr atified Kähler var ie t y .The foll owingr es ul t  show th emor e ef f e ctive ness of  this no tion
D e fi ni tion: A  c omplex  s pac e $ Z$ is call ed a s tratifie d Käh l er space if th e re exists a c o mp l e xs trat ifi cation $S=( S_{\ g amma })_{ \ ga mma \ in\Ga mma}$  o n  $ Z $ which is finer th an  the s trati fication of $ Z$ as a co m p l ex space  and th e re exists a Kä hlerstructure$ \omega=( U_{\a lpha},\r ho_{\alph a } )_{\alph a\i n I }$on$ Z $such that $\r h o _{\a lp ha|_{S_ {\g amma}\c apU_{ \al pha }} }$ is smo oth.
For  a  p ro pe r H amilt o nian $G$ -a cti on  on  a no r mal Kä hlerspac e$X $  th e quoti e nt $ X_0/ G$  i s astr at ified  nor m alKähleri an spacewit h  str uc tu re shea f $\mathcal O _{ X_0/G} $.He re$X_0$i s  the zer o level of the moment m a p $μ:X\ toLie(G )^∗$ , which i s a ssumed  to be non -empty ).
Ha ss anJ O LANYA :
In  m y experien c e , t he te rm  mea ns "a ( complex) variety a d mit ting a Kähler  me tric " (i .e. ,  t h e h ol o mor p h ic structure is  fixed wit ho u tspecifying a m et ric, or  even a  pola r ization ), wherea s a "Kähl er  man i f old " comes eq uipped w ith a hol o morph i cstruc tur e andamet ric,and (a n oma lousl y) a " po larize d Käh le r manifo ld" comes with a positi ve $(1 , 1)$  cl ass but n o ( d ist 
 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 (dist