Relative Kähler-Einstein metric on Kähler varieties of positive Kodaira dimension

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1 Relative Kähler-Einstein etric on Kähler varieties of positive Kodaira diension Hassan Jolany Abstract In this paper I have introduced a new notion of canonical etric. The notion of generalized Kähler-Einstein etric on the Kähler varieties with an interediate Kodaira diension is not suitable and we need to replace Kähler-Einstein etric (KE) to new notion of Relative Kähler- Einstein etric (RKE) for such varieties and its connection with Song-Tian-Yau-Vafa foliation 1. Relative Kähler-Einstein etric Let X 0 be a projective variety with canonical line bundle K X 0 of Kodaira diension κ(x 0 ) = li sup log di H0 (X 0, K l ) log l This can be shown to coincide with the axial coplex diension of the iage of X 0 under pluri-canonical aps to coplex projective space, so that κ(x 0 ) {, 0, 1,..., }. Lelong nuber: Let W C n be a doain, and Θ a positive current of degree (q, q) on W. For a point p W one defines 1 v(θ, p, r) = Θ(z) (dd c z 2 ) n q r 2(n q) z p <r The Lelong nuber of Θ at p is defined as v(θ, p) = li r 0 v(θ, p, r) Let Θ be the curvature of singular heritian etric h = e u, one has v(θ, p) = sup{λ 0 : u λ log( z p 2 ) + O(1)} Definition 1. Let π : X Y be a holoorphic ap of coplex anifolds. A real d-closed (1, 1)-for ω on X is said to be a relative Kähler for for π, if for every point y Y, there exists an open neighbourhood W of y and a sooth plurisubharonic function Ψ on W such that ω + π ( 1 Ψ) is a Kähler for on π 1 (W ). A orphis π is said to be Kähler, if there exists a relative Kähler for for π, and π : X Y is said to be a Kähler fiber space, if π is proper, Kähler, and surjective with connected fibers. We consider an effective holoorphic faily of coplex anifolds. This eans we have a holoorphic ap π : X Y between coplex anifolds such that. Hassan Jolany, 2015

2 Page 2 of 12 HASSAN JOLANY 1.The rank of the Jacobian of π is equal to the diension of Y everywhere. 2.The fiber X t = π 1 (t) is connected for each t Y 3.X t is not biholoorphic to X t for distinct points t; t B. It is worth to ention that Kodaira showed that all fibers are dieoorphic to each other. The relative Kähler for is denoted by ω X/Y = 1g α, β(z, s)dz α d z β Moreover take ω X = 1 log det g α, β(z, y) on the total space X. The fact is ω X in general is not Kähler on total space and ω X Xy = ω Xy. More precisely ω X = ω F + ω H where ω F is a for along fiber direction and ω H is a for along horizontal direction. ω H ay not be Kähler etric in general, but ω F is Kähler etric. Now let ω be a relative Kähler for on X and := di X di Y, We define the relative Ricci for Ric X/Y,ω of ω by Ric X/Y,ω = 1 log(ω π dy 1 dy 2... dy k 2 ) where (y 1,..., y k ) is a local coordinate of Y, where Y is a curve. Let for faily π : X Y ρ y0 : T y0 Y H 1 (X, T X) = H 0,1 σ (T X) be the Kodaira Spencer ap for the corresponding deforation of X over Y at the point y 0 Y where X y0 = X If v T y0 Y is a tangent vector, say v = y y 0 and s + bα z is any lift to X along X, then α ( s + ) bα z α = bα (z) β z β dz zα is a -closed for on X, which represents ρ y0 ( / y). The Kodaira-Spencer ap is induced as edge hooorphis by the short exact sequence 0 T X/Y T X π T Y 0 Weil-Petersson etric when fibers are Calabi-Yau anifolds can be defined as follows[6]. Definition 2. Calabi-Yau anifold is a copact Kähler anifold with trivial canonical bundle. The local Kuranishi faily of polarized Calabi-Yau anifolds X Y is sooth (unobstructed) by the Bogoolov-Tian-Todorov theore. Let each fibers is a Calabi-Yau anifold. One can assign the unique (Ricci-flat) Yau etric g(y) on X y. The etric g(y) induces a etric on 0,1 (T X). For v, w T y (Y ), one then defines the Weil-Petersson etric on the base Y by g W P (v, w) = 1.1. Vafa-Yau s sei Ricci-flat etric X < ρ(v), ρ(w) > g(y) The volue of fibers π 1 (y) = X y is a hoological constant independent of y, and we assue that it is equal to 1. Since fibers are Calabi-Yau anifolds so c 1 (X y ) = 0, hence there is a sooth function F y such that Ric(ω y ) = 1 F y and X y (e Fy 1)ωy n = 0. The function F y vary soothly in y. By Yau s theore there is a unique Ricci-flat Kähler etric ω SRF,y on X y cohoologous to ω 0. So there is a sooth function ρ y on π 1 (y) = X y such that ω 0 Xy + 1 ρ y = ω SRF,y is the unique Ricci-flat Kähler etric on X y. If we noralize

3 RELATIVE KÄHLER-EINSTEIN METRIC Page 3 of 12 by X y ρ y ω n 0 Xy = 0 then ρ y varies soothly in y and defines a sooth function ρ on X and we let ω SRF = ω ρ which is called as Sei-Ricci Flat etric. Such Sei-Flat Calabi-Yau etrics were first constructed by Greene-Shapere-Vafa-Yau on surfaces [4]. More precisely, a closed real (1, 1)- for ω SRF on open set U X \ S, (where S is proper analytic subvariety contains singular points of X) will be called sei-ricci flat if its restriction to each fiber X y U with y f(u) be Ricci-flat. Notice that ω SRF is sei-positive For the log-calabi-yau fibration f : (X, D) Y, such that (X t, D t ) are log Calabi-Yau varieties. If (X, ω) be a Kähler variety with Poincaré singularities then the sei-ricci flat etric has ω SRF Xt is quasi-isoetric with the following odel which we call it fibrewise Poincaré singularities. ( 1 n n ) dz k d z k 1 π z k 2 (log z k 2 ) π k=1 (log t 2 dz k n d z k n k=1 log z k 2 ) 2 z k z k k=1 k=1 We can define the sae fibrewise conical singularities. and the sei-ricci flat etric has ω SRF Xt is quasi-isoetric with the following odel 1 π ( n dz k d z k 1 1 n z k 2 + π (log t 2 n k=1 log z k 2 ) 2 k=1 In fact the previous reark will tell us that the sei Ricci flat etric ω SRF singularities with Poincare growth. k=1 dz k z k n k=1 d z k z k ) has pole Definition 3. Let X be a sooth projective variety with κ(x) 0. Then for a sufficiently large > 0, the coplete linear syste!k X gives a rational fibration with connected fibers f : X Y. We call f : X Y the Iitaka fibration ofx. Iitaka fibration is unique in the sense of birational equivalence. We ay assue that f is a orphis and Y is sooth. For Iitaka fibration f we have 1. For a general fiber F, κ(f ) = 0 holds. 2. di Y = κ(y ). Let X be a Kähler variety with an interediate Kodaira diension κ(x) > 0 then we have an Iitaka fibration π : X Y = ProjR(X, K X ) = X can such that fibers are Calabi-Yau varieties. We set K X/Y = K X π K 1 Y and call it the relative canonical bundle of π : X Y Definition 4. Let X be a Kähler variety with κ(x) > 0 then the relative Kähler- Einstein etric is defined as follows where Ric hω X/Y X/Y (ω) = ω Ric hω X/Y X/Y (ω) = 1 log( ωn π ωcan π ωcan ) and ω can is a canonical etric on Y = X can. Ric hω SRF X/Y X/Y (ω) = 1 log( ωn SRF π ωcan π ωcan ) = ω W P

4 Page 4 of 12 HASSAN JOLANY here ω W P is a Weil-Petersson etric[6]. Note that if κ(x) = then along Mori fibre space f : X Y we can define Relative Kähler-Einstein etric as when fibers and base are K-poly-stable, see[5] Ric hω X/Y X/Y (ω) = ω Note that, if X be a Calabi-Yau variety and we have a holoorphic fibre space π : X Y, which fibres are Calabi-Yau varieties, then we have the relative Ricci flat etric Ric X/Y (ω) = 0, which turns out to be Ric(ω) = 0 + π (ω Y ) where ω Y = ω W P, see[3] Then by the definition of Relative Kähler etric Ric(ω) = ω + π (ω Y ) which ω Y = ω W P is Weil-Petersson etric(we can define Weil-Petersson etric copletely on the base of Iitaka fibration) by using higher canonical bundle forula of Fujino-Mori For the existence of Kähler-Einstein etric when our variety is of general type, we need to the nice deforation of Kähler-Ricci flow and for interidiate Kodaira diension we need to work on relative version of Kähler Ricci flow. i.e ω t = Ric X/Y (ω) ω take the reference etric as ω t = e t ω 0 + (1 e t )Ric( ωn SRF π ω can π ω ) then the version of Kähler can Ricci flow is equivalent to the following relative Monge-Apere equation φ t t = ( ω t + 1 φt ) n π ωcan ωsrf n π ωcan φ t Take the relative canonical volue for Ω X/Y = ωn SRF π ω can π ω can ω t t = ω t t + 1 φ t t By taking ω = Ric(Ω X/Y ) + 1 φ we obtain after using estiates log By taking 1 of both sides we get ωn Ω X/Y φ = 0 and ω t = ω t + 1 φ t, then Ric X/Y (ω ) = ω hence by the definition of relative Kähler-etric and higher canonical bundle forula we have the Song-Tian etric[1, 2] Ric(ω ) = ω + π (ω W P ) More explicitly on pair (X, D) where D is a snc divisor, we can write Ric(ω can ) = ω can + ω D W P + P (b(1 t D P ))[π (P )] + [B D ] where B D is Q-divisor on X such that π O X ([ib D + ]) = O B ( i > 0). Here s D P := b(1 td P ) where t D P is the log-canonical threshold of π P with respect to (X, D B D /b) over the generic point η P of P. i.e., t D P := ax{t R ( X, D B D /b + tπ (P ) ) is sub log canonical over η P }

5 RELATIVE KÄHLER-EINSTEIN METRIC Page 5 of 12 and ω can has zero Lelong nuber Reark:Note that the log sei-ricci flat etric ωsrf D is not continuous in general. But if the central fiber has at worst canonical singularities and the central fiber (X 0, D 0 ) be itself as Calabi-Yau pair, then by open condition property of Kahler-Einstein etrics, sei-ricci flat etric is sooth in an open Zariski subset. Reark:So by applying the previous reark, the relative volue for Ω (X,D)/Y = (ωd SRF )n π ω can π ω can S 2 is not sooth in general, where S H 0 (X, L N ) and N is a divisor which coe fro canonical bundle forula of Fujino-Mori. Note that Song-Tian easure is invariant under birational change Now we try to extend the Relative Ricci flow to the fiberwise conical relative Ricci flow. We define the conical Relative Ricci flow on pair π : (X, D) Y where D is a siple noral crossing divisor as follows ω t = Ric (X,D)/Y (ω) ω + [N] where N is a divisor which coe fro canonical bundle forula of Fujino-Mori. Take the reference etric as ω t = e t ω 0 + (1 e t )Ric( ωn SRF π ω can π ω ) then the conical can relative Kähler Ricci flow is equivalent to the following relative Monge-Apere equation φ t t = log ( ω t + Ric(h N ) + 1 φt ) n π ωcan S N 2 (ωsrf D φ )n π ωcan t Now we prove the C 0 -estiate for this relative Monge-Apere equation. We use the following iportant lea fro Schuacher and also Cheeger-Yau, Lea 1.1. Suppose that the Ricci curvature of ω is bounded fro below by negative constant 1. Then there exists a strictly positive function P n (dia(x, D)), depending on the diension n of X and the diaeter dia(x, D) with the following property: Let 0 < ɛ 1. If g is a continuous function and f is a solution of then ( ω + ɛ)f = g, f(z) P n (dia(x, D)). gdv ω X So along relative Kähler-Ricci flow we have Ric(ω) 2ω where ω is the solution of Kähler- Ricci flow. But if we restrict our relative Monge-Apere equation to each fiber (X s, D s ), then we need diaeter bound on the fibers, i.e., dia(x s \ D s, ω s ) C But fro recent result of Takayaa(On Moderate Degenerations of Polarized Ricci-Flat Kähler Manifolds,J. Math. Sci. Univ. Tokyo, 22 (2015), ) we know that we have dia(x s \ D s, ω s ) 2 + D ( 1) n 2 /2 Ω s Ω s X s\d s S s 2 if and only if we have 1) central fiber X 0 \ D 0 has at worst canonical singularities and K X0 + D 0 = O X0 (D 0 ) which eans the central fiber itself be log Calabi-Yau variety.

6 Page 6 of 12 HASSAN JOLANY So this eans that we have C 0 -estiate for relative Kähler-Ricci flow if and only if the central fiber be Calabi-Yau variety with at worst canonical singularities. Note that to get C - estiate we need just check that our reference etric is bounded. So it just reain to see that ω W P is bounded. But when fibers are not sooth in general, Weil-Petersson etric is not bounded and Yoshikawa in Proposition 5.1 in [?] showed that under the soe additional condition when central fiber X 0 is reduced and irreducible and has only canonical singularities we have 1 s 2r ds d s 0 ω W P C s 2 ( log s ) 2 So we can get easily by ancient ethod! the C -solution. Note that the ain difficulty of the solution of C for the solution of relative Kähler-Einstein etric is that the null direction Vafa-Yau sei Ricci flat etric ω SRF gives a foliation along Iitaka fibration π : X Y and we call it Song-Tian-Yau-Vafa foliation (shortly we call it STYV foliation) and can be defined as follows F = {θ T X ω SRF (θ, θ) = 0} and along log Iitaka fibration π : (X, D) Y, we can define the following foliation F = {θ T X ω D SRF (θ, θ) = 0} where X = X \ D. In fact the ethod of Song-Tian (only in fiber direction and they couldn t prove the estiates in horizontal direction which is the ain part of coputation) works when ω SRF > 0. More precisely, in null direction, the function ϕ satisfies in the coplex Monge- Apere equation (e t ω D W P + (1 e t )ω 0 + Ric(h N ) + 1σ σϕ) κ = 0 gives rise to a foliation by X by coplex sub-anifolds. For the null direction we need to an extension of Monge-Apere foliation ethod of Gang Tian in [18]. It will appear in y new paper [19] A coplex analytic space is a topological space such that each point has an open neighborhood hoeoorphic to soe zero set V (f 1,..., f k ) of finitely any holoorphic functions in C n, in a way such that the transition aps (restricted to their appropriate doains) are biholoorphic functions. Lea: Fiberwise Calabi-Yau foliation F = {θ T X ω D SRF (θ, θ) = 0} is coplex analytic space and its leaves are also coplex analytic spaces. Moreover the direct iage π F is coplex analytic spaces on the log canonical odel X D can Lea: Let L be a leaf of f F, then L is a closed coplex subanifold and the leaf L can be seen as fiber on the oduli ap η : Y M D can where M D can is the oduli space of log calabi-yau fibers with at worst canonical singularites and Y = {y Y reg (X y, D y ) is Kawaata log terinal pair} Definition 5. Let π : X B be a faily of Kähler-Einstein varieties, then we introduce the new notion of stability and call it fiberwise KE-stability, if the Weil-Petersson distance

7 RELATIVE KÄHLER-EINSTEIN METRIC Page 7 of 12 d W P <. Note when fibers are Calabi-Yau varities, Takayaa, by using Tian s Kählerpotential for Weil-Petersson etric for oduli space of Calabi-Yau varieties showed that Fiberwise KE-Stability is as sae as when the central fiber is Calabi-Yau variety with at worst canonical singularities So along canonical odel π : X X can for ildly singular variety X, we have Ric(ω) = ω + ω W P if and only if our faily of fibers be fiberwise KE-stable Let π : (X, D) B is a holoorphic subersion onto a copact Kähler anifold B with c 1 (K B ) < 0 where the fibers are log Calabi-Yau anifolds and D is a siple noral crossing divisor in X. Let our faily of fibers is fiberwise KE-stable. Then (X, D) adits a unique twisted Kähler-Einstein etric ω B solving Ric(ω B ) = ω B + ω D W P + (1 β)[n] where ω W P is the logarithic Weil-Petersson for on the oduli space of log Calabi-Yau fibers and [D] is the current of integration over D. More precisely, we have Ric(ω can ) = ω can + ω D W P + P (b(1 t D P ))[π (P )] + [B D ] where B D is Q-divisor on X such that π O X ([ib D + ]) = O B ( i > 0). Here s D P := b(1 td P ) where t D P is the log-canonical threshold of π P with respect to (X, D B D /b) over the generic point η P of P. i.e., t D P := ax{t R ( X, D B D /b + tπ (P ) ) is sub log canonical over η P } and ω can has zero Lelong nuber. With cone angle 2πβ, (0 < β < 1) along the divisor D, where h is an Heritian etric on line bundle corresponding to divisor N, i.e., L N. This equation can be solved. Take, ω = ω(t) = ω B + (1 β)ric(h) + 1 v where ω B = e t ω 0 + (1 e t )Ric( (ωd SRF )n π ω can π ω ), by can using Poincare-Lelong equation, 1 log sn 2 h = c 1 (L N, h) + [N] we have Ric(ω) = = 1 log ω = 1 log π Ω (X,D)/Y 1 v (1 β)c 1 ([N], h) + (1 β){n} and 1 log π Ω (X,D)/Y + 1 v = = 1 log π Ω (X,D)/Y + ω ω B Ric(h)

8 Page 8 of 12 HASSAN JOLANY Hence, by using we get ω D W P = 1 log( (ωd SRF )n π ω can π ω can S 2 ) 1 log π Ω (X,D)/Y + 1 v = = ω ω D W P (1 β)c 1 (N) So, which is equivalent with Ric(ω) = ω + ω D W P + (1 β)[n] Ric (X,D)/Y (ω) = ω + [N] Uniqueness result of Relative Kähler-Einstein etric: Uniqueness of the solutions of relative Kahler Ricci flow along Iitaka fibration or π : X X can or along log canonical odel π : (X, D) X D can. Let φ 0 and ψ 0 be be ω-plurisubharonic functions such that v(φ 0, x) = 0 for all x X, let φ t, and ψ t be the solutions of relative Kähler Ricci flow starting fro φ 0 and ψ 0, respectively. Then in [16] it has been proven that if φ 0 < ψ 0 then φ t < ψ t for all t. In particular, the flow is unique. So fro the deep result of Tsuji-Schuacher[15], it has been showen that Weil-Petersson etric has zero Lelong nuber on oduli space of Calabi-Yau varieties, and by the sae ethod we can show that logarithic Weil-Petersson etric has zero Lelong nuber on oduli space of log Calabi-Yau varieties, hence by taking the initial etric to be Weil-Petersson etric or logarithic Weil-Petersson etric and since Weil-Petersson etric or logarithic Weil-Petersson etric are Kahler and sei-positive hence we get the uniquenessof the solutions of relative Kähler Ricci flow. The relation between the Existence of Zariski Decoposition and the Existence of Initial Kähler etric along relative Kähler Ricci flow: Finding an initial Kähler etric ω 0 to run the Kähler Ricci flow is iportant. Along holoorphic fibration with Calabi-Yau fibres, finding such initial etric is a little bit ysterious. In fact, we show that how the existence of initial Kähler etric is related to finite generation of canonical ring along singularities. Let π : X Y be an Iitaka fibration of projective varieties X, Y,(possibily singular) then is there always the following decoposition K Y + 1! π O X (!K X/Y ) = P + N where P is seiaple and N is effective. The reason is that, If X is sooth projective variety, then as we entioned before, the canonical ring R(X, K X ) is finitely generated. We ay thus assue that R(X, kk X ) is generated in degree 1 for soe k > 0. Passing to a log resolution of kk X we ay assue that kk X = M + F where F is the fixed divisor and M is base point free and so M defines a orphis f : X Y which is the Iitaka fibration. Thus M = f O Y (1) is seiaple and F is effective. In singular case, if X is log terinal. By using Fujino-Mori s higher canonical bundle forula, after resolving X, we get a orphis X Y and a klt pair K Y + B Y. The Y described above is the log canonical odel of K Y + B Y and so in fact (assuing as above that Y Y is a orphis), then K Y + B Y Q P + N where P is the pull-back of a rational ultiple of O Y (1) and N is effective (the stable fixed divisor). If Y Y is not a orphis, then P will have a base locus corresponding to the indeterinacy locus of this ap.

9 RELATIVE KÄHLER-EINSTEIN METRIC Page 9 of 12 So the existence of Zariski decoposition is related to the finite generation of canonical ring (when X is sooth or log terinal). Now if such Zariski decoposition exists then, there exists a singular heritian etric h, with sei-positive Ricci curvature 1Θ h on P, and it is enough to take the initial etric ω 0 = 1Θ h + [N] or ω 0 = 1Θ h + 1δ S N 2β along relative Kähler Ricci flow ω(t) = Ric X/Y (ω(t)) ω(t) t with log terinal singularities. So when X, Y have at worst log terinal singularities(hence canonical ring is f.g and we have initial Kähler etric to run Kähler Ricci flow with starting etric ω 0 ) and central fibre is Calabi-Yau variety, and K Y < 0, then all the fibres are Calabi-Yau varieties and the relative Kähler-Ricci flow converges to ω which satisfies in Ric(ω) = ω + f ω W P Reark: The fact is that the solutions of relative Kähler-Einstein etric or Song-Tian etric Ric(ω) = ω + f ω W P ay not be C. In fact we have C of solutions if and only if the Song-Tian easure or Tian s Kähler potential be C. Now we explain that under soe following algebraic condition we have C -solutions for Ric(ω) = ω + f ω W P along Iitaka fibration. We recall the following Kawaata s theore [17]. Theore 1.2. Let f : X B be a surjective orphis of sooth projective varieties with connected fibers. Let P = j P j, Q = l Q l, be noral crossing divisors on X and B, respectively, such that f 1 (Q) P and f is sooth over B \ Q. Let D = j d jp j be a Q- divisor on X, where d j ay be positive, zero or negative, which satisfies the following conditions A,B,C: A) D = D h + D v such that any irreducible coponent of D h is apped surjectively onto B by f, f : Supp(D h ) B is relatively noral crossing over B \ Q, and f(supp(dv)) Q. An irreducible coponent of D h (resp. D v ) is called horizontal (resp. vertical) B)d j < 1 for all j C) The natural hooorphis O B f O X ( D ) is surjective at the generic point of B. D) K X + D Q f (K B + L) for soe Q-divisor L on B. Let f Q l = j w lj P j d j = d j + w lj 1, if f(p j ) = Q l w lj δ l = ax{ d j ; f(p j ) = Q l }. = δ l Q l. l M = L. Then M is nef. The following theore is straightforward fro Kawaata s theore

10 Page 10 of 12 HASSAN JOLANY Theore 1.3. Let d j < 1 for all j be as above in Theore 0.11, and fibers be log Calabi- Yau pairs, then ( 1) n 2 /2 Ω s Ω s X s\d s S s 2 is continuous on a nonepty Zariski open subset of B. Since the inverse of volue gives a singular heritian line bundle, we have the following theore fro Theore 0.11 Theore 1.4. Let K X + D Q f (K B + L) for soe Q-divisor L on B and f Q l = j w lj P j d j = d j + w lj 1, if f(p j ) = Q l w lj δ l = ax{ d j ; f(p j ) = Q l }. = δ l Q l. l M = L. Then ( ( 1) n 2 /2 Ω s Ω s X s\d s S s 2 is a continuous heritian etric on the Q-line bundle K B + when fibers are log Calabi-Yau pairs. ) 1 Conjecture: The twisted Kähler-Einstein etric Ric(ω) = ω + α where α is a seipositive current has unique solution if and only if α has zero Lelong nuber Theore 1.5. The axial tie existence T for the solutions of relative Kähler Ricci flow is T = sup{t e t [ω 0 ] + (1 e t )c 1 (K X/Y + D) K((X, D)/Y )} where K ((X, D)/Y ) denote the relative Kähler cone of f : (X, D) Y Now take we have holoorphic fibre space f : X Y such that fibers and base are Fano K-poly stable, then we have the relative Kähler-Einstein etric Ric X/Y (ω) = ω we need to work on relative version of Kähler Ricci flow. i.e ω t = Ric X/Y (ω) + ω

11 RELATIVE KÄHLER-EINSTEIN METRIC Page 11 of 12 take the reference etric as ω t = e t ω 0 + (1 + e t )Ric( ( X/Y ωn+1 SKE ) π ω Y π ω ) then the version of Y Kähler Ricci flow is equivalent to the following relative Monge-Apere equation φ t t = ( ω t + 1 φt ) n π ωy ( X/Y ωn+1 SKE ) π ωy where ω Y is the Kähler-Einstein etric corresponding to Ric(ω Y ) = ω Y and ω SKE is the fiberwise Fano Kähler-Einstein etric. In fact the relative volue for is Ω X/Y = ( X/Y ωn+1 SKE ) π ω Y π ω and we have the following Y relative Monge-Apere equation φ t t = ( ω t + 1 φt ) n + φ t Ω X/Y Hence fro Ric X/Y (ω) = ω by using the definition of relative Kähler etric we obtain Ric(ω) = ω + f (ω Y ), for soe ω Y on the base which this etric is correspond to canonical etric on oduli part of faily of fibers, which is Weil-Petersson etric ω W P = X/Y c 1(K X/Y, h) n+1. Moreover I think the equation of Ric(ω) = ω + f (ω W P ) work when we have 0 < X/Y ωn+1 SKE < C which in general ω SKE ay not be sei-positive and sei-positivity of fiberwise Fano Kähler-Einstein etric is correspond to K-poly stability of total space X(this is y conjecture). The sae assuption ust holds when fibers are of general types. See [5] Reark: Note that we still don t know canonical bundle type forula along Mori-fiber space. So finding explicit Song-Tian type etric on pair (X, D) along Mori fiber space when base and fibers are K-poly stable is not known yet. + φ t Conjecture:Let π : X B is sooth, and every X t is K-poly stable. Then the plurigenera P (X t ) = di H 0 (X t, K Xt ) is independent of t B for any. Idea of proof. We can apply the relative Kähler Ricci flow ethod for it. In fact if we prove that ω(t) = Ric X/Y (ω(t)) + ω(t) t has long tie solution along Fano fibration such that the fibers are K-poly stable then we can get the invariance of plurigenera in the case of K-poly stability References 1. Jian Song; Gang Tian, The Kähler-Ricci flow on surfaces of positive Kodaira diension, Inventiones atheaticae. 170 (2007), no. 3, Jian Song; Gang Tian, Canonical easures and Kähler-Ricci flow, J. Aer. Math. Soc. 25 (2012), no. 2, , 3. Valentino Tosatti, Adiabatic liits of Ricci-flat Kähler etrics, J. Differential Geo. 84 (2010), no.2, B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosic strings and noncopact Calabi-Yau anifolds. Nuclear Physics B, 337(1):1-36, Hassan Jolany, Generalized Kähler-Einstein etric along Q-Fano bration, preprint 6. Hassan Jolany, A short proof of Tian s forula for logarithic Weil-Petersson etric, Preprint 7. Robert J. Beran, Relative Kähler-Ricci flows and their quantization, Analysis and PDE, Vol. 6 (2013), No. 1, Georg Schuacher, Moduli of fraed anifolds,invent. ath. 134, (1998) 9. P. Eyssidieux, V. Guedj, A. Zeriahi: Singular Kähler-Einstein etrics, J. Aer. Math. Soc. 22 (2009), no. 3, Hajie Tsuji, Canonical easures and the dynaical systes of Bergan kernels, arxiv: Georg Schuacher and Hajie Tsuji, Quasi-projectivity of oduli spaces of polarized varieties, Annals of Matheatics,159(2004),

12 Page 12 of 12 RELATIVE KÄHLER-EINSTEIN METRIC 12. H. Skoda. Sous-ensebles analytiques d ordre fini ou infini dans C n. Bulletin de la Societe Matheatique de France, 100: Jean-Pierre Deailly, Coplex Analytic and Differential Geoetry, preprint, Shigeharu Takayaa, On Moderate Degenerations of Polarized Ricci-Flat Kähler Manifolds, J. Math. Sci. Univ. Tokyo, 22 (2015), Georg Schuacher, Hajie Tsuji, Quasi-projectivity of oduli spaces of polarized varieties,annals of Matheatics, Pages fro Volue 159 (2004), Issue Eleonora Di Nezza, Uniqueness and short tie regularity of the weak Kähler-Ricci flow, arxiv: Yujiro Kawaata, Subadjunction of log canonical divisors, II, Aer. J. Math. 120 (1998), Chen, X. X.; Tian, G. Geoetry of Kähler etrics and foliations by holoorphic discs. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 19. Hassan Jolany, Coplex Monge-Apere foliation via Song-Tian-Yau-Vafa foliation. In preparation