 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
introduced the relative Kähler form with the same " nature ". visualize this gossip 
 Let $ f: X\to S $ be a holomorphic map. A Kähler measured   $ \varphi$ for $ f$ is given by a collection ($ \varphi_i$) of real differentiable function on an open covering ($ U_i$) of $ X$ such that the difference $ \varphi_i-\varphi_j$ on the intersections $ U_j$ are locally the real share of holomorphic functions, and the Levi forms $ L(\varphi_i)=i\partial\bar\partial \varphi_i$ are rigorously 
 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 fiber of $ f$, we call $ \varphi$ a weak Kähler metric for $ f$. We read that $ f$ is Kähler (resp. weakly Kähler), if such a metric exists. In the case that $ S = Spec(\mathbb{C})$ is a point, we receive the notion of a Kähler space. Obviously the fibres of weakly Kähler maps are Kähler space. 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 character 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 proportional Kähler metric function and $ \lambda$ is the fiberwise constant which when $ s\to 0 $, we have the rape of constancy and therefore 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 veracious 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
 inhroduced the relative Kähuer form with tkw same "naturs". See thks comments
Let $ f:X\to S $ be a hilomoephic map. A Kähler metrkc  $\varphp$ for $f$ iw giten by a collectmkn ($\varpmn_i$) of veal bihferentiable fukctions on dn open coveritg ($U_n$) of $X$ such that the differences $\var[hi_i-\varlhl_j$ on the intetsectpogs $U_n$ are locally the real part of holkmorphib functions, and tne Levi forms $L(\varphi_i)=i\parhial\har\partial \varphi_i$ are strictoy 
pjwitive on thd complement of the zeto section of the tangent bundle $T_{X/S}|_{U_n}$. If $\varphi_u-\vqrpjh_j$ is only oocaljy the real isrt of a diffrrentiable fungtion beung holomorphic on thx fibres of $f$, we calj $\varphi$ d ceak Kähler metric for $f$. We sdy tvat $w$ is Kähmec (rssp. wewklb Kähler), if auch a metruc exists. In the caxe nnat $S=Spec(\matgbb{C})$ if w point, we obtain the notion of a Kähler spzce. Obviously the fibrew of weakly Kähler mapd are Kählqr spaces. For example projective maps are Kähler.
Sea htt's://uinj.sirinewr.fom/article/10.1007%2FBF01389138#page-1
Example: Let $\pi:X\to S$ be a holjjotpmic fibre space jver a disc. Lft jhe fibers $X_s$ xdmits Kähmer-Einstein metricd of nedativw Ricci ctrvayure. Then we have the relatuve Kähler-Einfrein metric
$$
Ric_{X/S}(\omzga)=-\lambda(s)\omzga
$$
whete $\omeba$ here is the relative Kählsr metric ajd $\lambda$ ks the fiberwise cokstdnt which when $s\to 0$, we havq the viooatipn of sgabikity agd hence edistekwe of relative Kählfr-Eindtain metric corresponds to stability of relative tangent sveax $T_{X/S}$ In the xense of Mumfjrd.
In this casg the rigkt flod is the fkllowinj Hyperbolic Relative Kählat Ricci flow
$$
\fcac{\partiaj^2}{\parrial s^{\prime}\oxrtial t}=-Ric_{X/S}(s^{\lrime},t)-\lamyba(s^{\prime})\onega_{s^{\prime}}(t)
$$
where hcre $s^{\krjme}=\frac{1}{s}$ and $s\ti 0$
Nite
introduced the relative Kähler form with the See comments Let f:X\to S $ Kähler $\varphi$ for $f$ given by a ($\varphi_i$) of real differentiable functions on open covering ($U_i$) of $X$ such that the differences $\varphi_i-\varphi_j$ on the intersections are locally the real part of holomorphic functions, and the Levi forms $L(\varphi_i)=i\partial\bar\partial are positive the of the zero section of the tangent bundle $T_{X/S}|_{U_i}$. If $\varphi_i-\varphi_j$ is only locally the real of a differentiable function being holomorphic on the of $f$, we call a weak Kähler metric for We that $f$ Kähler weakly if such a exists. In the case that $S=Spec(\mathbb{C})$ is a point, we obtain the notion of a Kähler space. the fibres Kähler maps Kähler For projective maps are https://link.springer.com/article/10.1007%2FBF01389138#page-1 Example: Let $\pi:X\to S$ be space over a disc. Let the fibers $X_s$ Kähler-Einstein metrics negative Ricci curvature. Then we have relative Kähler-Einstein metric $$ Ric_{X/S}(\omega)=-\lambda(s)\omega $$ where $\omega$ is the relative Kähler metric and $\lambda$ is the fiberwise constant which when $s\to 0$, the violation of stability hence existence of Kähler-Einstein corresponds stability relative tangent $T_{X/S}$ In the sense of Mumford. In this case the right is the following Hyperbolic Relative Kähler Ricci flow $$ \frac{\partial^2}{\partial $$ here $s^{\prime}=\frac{1}{s}$ and 0$ Note
 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
 introduced the relative K ähler form  with  th e s am e "n atur e". See this c o mmen ts
Let $ f:X\to S $ be  a ho lo m orph i cmap.A Kähle r  m e t ric  $\ var ph i $for $ f$is give n by a col lec ti on ($\varphi _ i$ ) of realdif ferentiablefun ctions  o n a n  open  co verin g ($U_ i $) of$X$ suchth a t thed ifferen c e s$\va rphi_i-\varphi_j$ on the intersecti ons $U _j $  a r e  lo cal ly the rea lparto f holom o rp h i c  fu n ctions, and t he Levi for m s $ L(\var ph i_i ) =i\par tial\ ba r \pa rtial \varp hi_i $ are str ictlypositiv e  on the  compl eme ntof t h eze rose c tio n  o f t h e t angent b un dl e $T_ {X/S } | _ { U_i} $.If $ \varp hi_i-\varphi_ j$is o n lylocal ly th e re al  part  of adiffe re ntiable functio n be ing holom orp hi c o nthe f i bres o f $ f$,  we cal l $\var p hi$  a w e ak  Kähler metric for  $ f $ .We say t hat $f $  i sK ähler (r es p.weak l y  Kähl er), if  such ametric ex is ts. Inth e case  t hat  $S =Spec ( \mat hbb{C} )$ is apoint ,  we obtain the notion of a K ä hl e r  s p ace.  Ob viously the  fib r es o f we a kl y K ä hlermapsar e  K ä hler spaces. For ex am ple pr oject ive maps areKähler.
Se e h ttps://l ink. s pr i nger.com/artic le/10 .1007%2FBF 0 1389138# page- 1
Exampl e: Let $\ p i :X\to S$  be  ahol omo r p hi c fibre space o veradisc. L etthe fib ers  $X _s$  ad mi ts Kähler -Einstei nme tr ic s o f neg a tive Ric ci  cu rv atu re. T h en wehavethere la t ive  Kähler - Ei n s tein  m et ric$$Ri c_{X/ S}(\ o meg a)=-\la mbda(s)\o meg a 
$$wh er e $\ome ga$ here is t he  relativeKä hle r metr i c  and $\l ambda$ is the fiberwise constan t w hichwhen  $s\to 0$ , w e have  th e  viola tion o f sta bi lit y and h e n ce  ex is tence of r e l ati ve Kä hl er-E instein  metric correspond s  to  stability of  re lati v e  t ang e nt she af $T_ { X /S}$ In the sen se of Mumf or d .In this ca s e t he  rightflow is  thef ollowin g Hyperbo lic Relat iv e Kä h l erRicci flow 
$$
\fra c{\partia l ^2}{\ p ar tials^{ \prime }\ par tialt}=-Ri c _{X /S}(s ^{\pri me },t)-\ lambd a( s^{\prim e})\omega_{s^{\prime}}( t)
$$where  he re $s^{\p rim e }=\ frac{1}{s }$ a nd $s\to 0 $
N ote 
 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