Generalized Kähler-Einstein metric along Q-Fano fibration

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HAL Id: hal-01413759

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Preprint submitted on 16 Sep 2017

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Generalized Kähler-Einstein metric along Q-Fano fibration

Hassan Jolany

To cite this version:

Hassan Jolany. Generalized Kähler-Einstein metric along Q-Fano fibration. 2017. �hal-01413759v3�

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Generalized K¨ ahler-Einstein metric along Q -Fano fibration

Hassan Jolany

University of Lille1, France

Correspondence to be sent to: [email protected]

In this paper, we show that along

Q

-Fano fibration, when general fibres, base and central fiber (with at worst

Kawamata log terminal singularities)are K-poly stable then there exists a relative K¨ ahler-Einstein metric. We introduce the fiberwise K¨ ahler-Einstein foliation and we mention that the main difficulty to obtain higher estimates is to solve relative CMA equation along such foliation. We propose a program such that for finding a pair of canonical metric (ω

X, ωB

), which satisfies in

Ric(ωX

) =

π^∗ωB

+

π^∗

(ω

W P

) + [N ] on K-poly stable degeneration

π

:

X →B, where Ric(ωB

) =

ωB

, we need to have Canonical bundle formula.

1 Introduction

Let (X, L) be a polarized projective variety. Given an ample line bundle L → X , then a test configuration for the pair (X, L) consists of:

- a scheme X with a C

^∗

^-action

- a flat C

^∗

-equivariant map π : X → C with fibres X

_t

; - an equivariant line bundle L → X , ample on all fibres;

- for some r > 0, an isomorphism of the pair ( X

1

, L

1

) with the original pair (X, L

^r

).

Let U

_k

= H

⁰

( X

0

, L

^k₀

|

X₁

) be vector spaces with C

^∗

-action, and let A

_k

: U

_k

→ U

_k

be the endomorphisms generating those actions. Then

dim U

k

= a

0

k

ⁿ

+ a

1

k

ⁿ⁻¹

+ . . .

Tr (A

k

) = b

0

k

ⁿ⁺¹

+ b

1

k

ⁿ

+ . . . Then the Donaldson-Futaki invariant of a test configuration ( X , L ) is

F ut( X , L ) = 2(a

1

b

0

− a

0

b

1

) a

0

.

A Fano variety X is K-stable (respectively K-poly stable) if F ut( X , L ) ≥ 0 for all normal test configurations X such that X

0

6= X and equality holds if (X , L) is trivial (respectively, X = X × A

¹

).See [5, 6, 7, 8, 9] and references therein.

Let X be a projective variety with canonical line bundle K

X

→ X of Kodaira dimension κ(X ) = lim sup log dim H

⁰

(X, K

_X^⊗`

)

log `

This can be shown to coincide with the maximal complex dimension of the image of X under pluri-canonical maps to complex projective space, so that κ(X ) ∈ {−∞, 0, 1, ..., m}.

A K¨ ahler current ω is called a conical K¨ ahler metric (or Hilbert Modular type) with angle 2πβ, (0 < β < 1) along the divisor D, if ω is smooth away from D and asymptotically equivalent along D to the model conic metric

Received 1 Month 20XX; Revised 11 Month 20XX; Accepted 21 Month 20XX Communicated by A. Editor

c

Hassan Jolany 2015

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ω

_β

= √

−1 dz

1

∧ d z ¯

1

|z

1

|

^2(1−β)

+

n

X

i=2

dz

_i

∧ d z ¯

_i

!

here (z

₁

, z

₂

, ..., z

_n

) are local holomorphic coordinates and D = {z

₁

= 0} locally. See[18]

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 K¨ ahler variety with Poincar singularities then the semi-Ricci flat metric has ω

SRF

|

X_t

is quasi-isometric with the following model which we call it fibrewise Poincar singularities.

√ −1 π

n

X

k=1

dz

k

∧ d z ¯

k

|z

k

|

²

(log |z

k

|

²

)

²

+

√ −1 π

1 log |t|

²

− P

n

k=1

log |z

k

|

²

n

X

k=1

dz

k

z

_k

∧

n

X

k=1

d z ¯

k

¯ z

_k

!

We can define the same fibrewise conical singularities. and the semi-Ricci flat metric has ω

SRF

|

Xt

is quasi-isometric with the following model

√ −1 π

n

X

k=1

dz

k

∧ d z ¯

k

|z

k

|

²

+

√ −1 π

1 log |t|

²

− P

n

k=1

log |z

k

|

²

2 n

X

k=1

dz

k

z

k

∧

n

X

k=1

d z ¯

k

¯ z

k

!

In fact the previous remark will tell us that the semi Ricci flat metric ω

SRF

has pole singularities.

A pair (X, D) is log Q -Fano if the Q -Cartier divisor −(K

X

+ D) is ample. For a klt pair (X, D) with κ(K

X

+ D) = −∞, according to the log minimal model program, there exists a birational map φ : X → Y and a morphism Y → Z such that for D

⁰

= φ

_∗

D , the pair (Y

z

, D

_z⁰

) is log Q -Fano with Picard number ρ(Y

z

) = 1 for general z ∈ Z. In particular, log Q -Fano pairs are the building blocks for pairs with negative Kodaira dimension.

Now, we consider fibrations with K¨ ahler Einstein metrics with positive Ricci curvature and CM-line bundle.

Let π : (X, D) → B be a holomorphic submersion between compact K¨ ahler manifolds of any dimensions, whose log fibers (X

_s

, D

_s

) and base have no no-zero holomorphic holomorphic vector fields and whose log fibers admit conical K¨ ahler Einstein metrics with positive Ricci curvature on (X, D). We give a sufficient topological condition that involves the CM-line bundle on B.

Let B be a normal variety such that K

B

is Q -Cartier, and f : X → B a resolution of singularities. Then, K

X

= f

^∗

(K

B

) + X

i

a

i

E

i

where a

i

∈ Q ^{and the} ^E

ⁱ

are the irreducible exceptional divisors. Then the singularities of B are terminal, canonical, log terminal or log canonical if a

i

> 0, ≥ 0, > −1 or ≥ −1, respectively.

Firstly, we introduce the log CM-line bundle. Let π : (X, D) → B be a holomorphic submersion between compact K¨ ahler manifolds and let L + D be a relatively ample line bundle. Let Y = (X

s

, D

s

) be a log fiber n = dim X

s

and let η denote the constant

η = nc

₁

(Y )c

₁

(L + D) |

ⁿ⁻¹_Y

c

1

(L + D) |

ⁿ_Y

Let K

_X/B

denote the relative canonical bundle. The log CM-line bundle is the virtual bundle and introduced by Tian [39] as follows

L

^D_CM

= n + 1 (K

_X/B

+ D)

^∗

− (K

_X/B

+ D)

⊗ ((L + D) − (L + D)

^∗

)

ⁿ

− η ((L + D) − (L + D)

^∗

)

ⁿ⁺¹

For finding the first Chern classof log CM-line bundle, we need to the following Grothendieck-Riemann-Roch theorem.

Let X be a smooth quasi-projective scheme over a field and K

0

(X) be the Grothendieck group of bounded complexes of coherent sheaves. Consider the Chern character as a functorial transformation

ch : K

₀

(X) → A(X, Q ⁾

, where A

d

(X, Q ) is the Chow group of cycles on X of dimension d modulo rational equivalence, tensored with the rational numbers. Now consider a proper morphism

f : X → Y

between smooth quasi-projective schemes and a bounded complex of sheaves F

^•

on X . Let R be the right

derived functor, then we have the following theorem of Grothendieck-Riemann-Roch.

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Theorem 1.1. The Grothendieck-Riemann-Roch theorem relates the pushforward map f ! =

X (−1)

ⁱ

R

ⁱ

f

_∗

: K

0

(X) → K

0

(Y ) and the pushforward f

_∗

: A(X) → A(Y ) by the formula

ch(f ! F

^•

)td(Y ) = f

_∗

(ch(F

^•

)td(X))

Here td(X ) is the Todd genus of (the tangent bundle of) X. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck-Riemann-Roch formula as

ch(f ! F

^•

) = f

_∗

(ch(F

^•

)td(T

f

))

where T

f

is the relative tangent sheaf of f , defined as the element T X − f

^∗

T Y in K

⁰

(X ). For example, when f is a smooth morphism, T

f

is simply a vector bundle, known as the tangent bundle along the fibers of f .

So by applying Grothendieck-Riemann-Roch theorem, the first Chern class of log CM-line bundle is c

₁

(L

CM

) = 2

ⁿ⁺¹

π

_∗

(n + 1)c

₁

(K

_X/B

+ D) + ηc

₁

(L + D)

c

₁

(L + D)

ⁿ

We need to introduce log Weil-Petersson metric on the moduli space of conical K¨ ahler-Einstein Fano varieties. We can introduce it by applying Deligne pairing. First we recall Deligne pairing here.[48]

Let π : X → S be a projective morphism. Let F be a coherent sheaf on X . Let det Rπ

_∗

F be the line bundle on S. If R

ⁱ

π

_∗

F is locally free for any i, then

det Rπ

_∗

F = O

i

det R

ⁱ

π

_∗

F

(−1)ⁱ

For any flat morphism π : X → S of relative dimension one between normal integral schemes X and S, we denote the Deligne pairing by

h−, −i : Pic(X) × Pic(X ) → Pic(S)

In this case, the Deligne’s pairing can be defined as follows. Let L

1

; L

2

∈ Pic(X ), then hL

₁

, L

₂

i : det Rπ

_∗

(L

₁

⊗ L

₂

) ⊗ det Rπ

_∗

(L

⁻¹₁

) ⊗ Rπ

_∗

(L

⁻¹₂

) ⊗ Rπ

_∗

O

_X

In fact, Deligne pairings provide a method to construct a line bundle over a base S from line bundles over a fiber space X .

Suppose π : X → S is a flat morphism of schemes of relative dimension n, i.e., n = dimX

_s

, and suppose L

_i

are line bundles over the fiber space X . Then the Deligne pairing hL

0

, ..., L

n

i

_X/S

is a line bundle over S. The sections are given formally by hs

0

, ..., s

n

i, where each s

i

is a rational section of L

i

and whose intersection of divisors is empty.

The transition functions between sections are defined

hs

₀

, ..., f

_j

s

_j

, ..., s

_n

i = N

_f

[∩

_i6=j

div(s

_i

)] hs

₀

, ..., s

_j

, ..., s

_n

i where X

_p

∩

_i6=j

div(s

_i

) = P _n

k

p

_k

, the formal sum of zeros and poles, N

_f

[∩

_i6=j

div(s

_i

)] is the product Q _f(p

k

)

ⁿ^k

, where the p

_i

are the zeros and poles in the common intersection, and n

_k

the multiplicities.

Let π : (X, D) → S be a projective family of canonically polarized varieties. Equip the relative canonical bundle K

_X/S

+ D with the hermitian metric that is induced by the fiberwise K¨ ahler-Einstein metrics. The log Weil-Petersson form is equal, up to a numerical factor, to the fiber integral

ω

_{W P}^D

= Z

X_s\Ds

c

₁

K

_X0/S

ⁿ⁺¹

= Z

X_s\Ds

|A|

²_ω

s

ds ∧ d¯ s A represents the Kodaira-Spencer class of the deformation. See [20]

In fact if we take π : (X, D) → S be a projective family of canonically polarized varieties. Since, the Chern form of the metric on hL

0

, ..., L

n

i equals the fiber integral [20],

Z

X/S

c

₁

(L

₀

, h

₀

) ∧ ... ∧ c

₁

(L

_n

, h

_n

)

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The curvature of the metric on the Deligne pairing hK

_X/S

+ D, ..., K

_X/S

+ Di given by the fiberwise K¨ ahler- Einstein metric coincides with the generalized log Weil-Petersson form ω

^D_{W P}

on S.

Take

α = c

1

(L

^D_CM

)

2

ⁿ⁺¹

(n + 1)π

_∗

(c

₁

(L + D)

ⁿ

) ∈ H

^1,1

(S)

In Song-Tian program when the log-Kodaira dimension κ(X, D) is −∞ then we don’t have canonical model and by using Mori fibre space we can find canonical metric(with additional K-poly stability assumption on fibres ), the following theorem give an answer for finding canonical metric by using Song-Tian program in the case when the log-Kodaira dimension κ(X, D) is −∞. We can introduce the logarithmic Weil-Petersson metric on moduli space of log Fano varieties.

Let M

ⁿ

be the space of all n-dimensional K¨ ahler-Einstein Fano manifolds, normalized so that the K¨ ahler form ω is in the class 2πc

1

(X ), modulo biholomorphic isometries.It is pre-compact in the Gromov-Hausdorff topology, see [40]. Donaldson and Sun [23], introduced the refined Gromov-Hausdorff compactification M

ⁿ

of M

ⁿ

such that every point in the boundary M

ⁿ

\ M

ⁿ

is naturally a Q -Gorenstein smoothable Q -Fano variety which admits a K¨ ahler-Einstein metric. Recently a proper algebraic compactification M of M was constructed by Odaka, Chi Li, et al. [28],[30] for proving the projectivity of M.

There is a belief due to Tian saying that a moduli space with canonical metric is likely to be quasi-projective Let W ⊂ C

ⁿ

be a domain, and Θ a positive current of degree (q, q) on W . For a point p ∈ W one defines

v (Θ, p, r) = 1 r

^2(n−q)

Z

|z−p|<r

Θ(z) ∧ (dd

^c

|z|

²

)

^n−q

The Lelong number of Θ at p is defined as

v (Θ, p) = lim

r→0

v (Θ, p, r) Let Θ be the curvature of singular hermitian metric h = e

^−u

, one has

v (Θ, p) = sup{λ ≥ 0 : u ≤ λ log(|z − p|

²

) + O(1)}

see [16]

We set K

_X/Y

= K

X

⊗ π

^∗

K

_Y⁻¹

and call it the relative canonical bundle of π : X → Y

Definition 1.2. Let X be a K¨ ahler variety with κ(X ) > 0 then the relative K¨ ahler-Einstein metric is defined as follows

Ric

^h

ω X/Y

X/Y

(ω) = −Φω where

Ric

^h

ω X/Y

X/Y

(ω) = √

−1∂ ∂ ¯ log( ω

ⁿ

∧ π

^∗

ω

^m_can

π

^∗

ω

^m_can

)

and ω

can

is a canonical metric on Y = X

can

. Where Φ is the fiberwise constant function.

Ric

^h

ωSRF X/Y

X/Y

(ω) = √

−1∂ ∂ ¯ log( ω

ⁿ_SRF

∧ π

^∗

ω

^m_can

π

^∗

ω

_can^m

) = ω

W P

here ω

_{W P}

is a Weil-Petersson metric.

Note that if κ(X ) = −∞ then along Mori fibre space f : X → Y we can define Relative K¨ ahler-Einstein metric as

Ric

^h

ω X/Y

X/Y

(ω) = Φω

when fibers and base are K-poly-stable, and we have fiberwise KE-stability

Remark 1:From [28], by the same method we can show that, the log CM line bundle L

^D_CM

descends to a

line bundle Λ

^D_CM

on the proper moduli space M

^D

. Moreover, there is a canonically defined continuous Hermitian

metric h

^D_DP

, called as Deligne Hermitian metric on Λ

^D_CM

whose curvature form is a positive currentω

^D_{W P}

and

called logarithmic Weil-Petersson metric on compactified moduli space of K¨ ahler-Einstein log Fano manifolds

and it can be extended to canonical Weil-Petersson current ˙ ω

^D_{W P}

on M

^D

. Viehweg [29], showed that the moduli

space of polarized manifolds with nef canonical line bundle is quasi-projective and by the same method of Chi

Li [28], Λ

^D_CM

on the proper moduli space M

^D

is nef and big. Hence M

^D

is quasi-projective. We can prove it

by using Lelong number method also which is more simpler.

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In fact, if we prove that the Lelong number of singular hermitian metric (L

CM

, h

W P

) corresounding to Weil-Petersson current on virtual CM-bundle has vanishing Lelong number, then L

CM

is nef and if we know the nefness of L

CM

then due to a result of E.Viehweg, we can prove that the moduli space of Kahler-Einstein Fano manifolds M is quasi-projective and its compactification M is projective. From Demailly’s theorem, We know if (L, h) be a positive , singular hermitian line bundle , whose Lelong numbers vanish everywhere. Then L is nef.

Gang Tian recently proved that, the CM-bundle L

CM

is positive, So by the same theorem 2 and Poroposition 6, of the paper,[42] we can show that the Weil-Petersson current on mduli space of Fano Kahler-Einstein varieties has zero Lelong number(in smooth case it is easy, for singular case(for log terminal singularities ) we just need to show that the diameter of fibers are bounded ).

Moreover, by using the same method of Theorem 2, in [42], hermitian metric corresponding to L

^D_CM

has vanishing Lelong number, and since from the recent result of Tian, L

^D_CM

is positive, hence L

^D_CM

is nef. So by the theorem of Viehweg [29], M

^D

is quasi-projective. In fact this give a new analytical method to the recent result of [28].

We have

ω

^D_{W P}

:= −

√ −1

2π ∂ ∂ ¯ log h

DP

where localle h

_DP

= e

^−Ψ

and hence

ω

_{W P}^D

:=

√ −1 2π ∂ ∂Ψ ¯ which Ψ called Deligne potential.

Moreover, we have

ω

_{W P}

= Ric





ω

_SKE^κ

∧ kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ ¯ v)

^1/m

kv

^∗

k

^2/m

(i^∗hF S)m¹

(v ∧ ¯ v)

^1/m

| S |

²





Here semi-K¨ ahler-Einstein metric ω

_SKE

is a (1, 1)-current with log-pole singularities such that its restriction on each fiber is Kahler-Einstein metric. Note that from Kodaira-Spencer theory, we always have such semi- K¨ ahler-Einstein metric.

Definition 1.3. The null direction of fiberwise K¨ ahler-Einstein metric ω

SKE

gives a foliation along Mori-fibre space π : X → Y and we call it fiberwise K¨ ahler-Einstein foliation and can be defined as follows

F = {θ ∈ T

_X/Y

|ω

SKE

(θ, θ) = 0} ¯

and along Mori-fibre space π : (X, D) → Y , we can define the following foliation F

⁰

= {θ ∈ T

_X⁰_/Y

|ω

_SKE^D

(θ, θ) = 0} ¯

where X

⁰

= X \ D. In fact from Theorem 0,9. the Weil-Petersson metric ω

W P

vanishes everywhere if and only if F = T

_X/Y

. Note that such foliation may be fail to be as foliation in horizontal direction. But is is foliation in fiber direction.

We have the following result due to Bedford-Kalka [49]

Lemma: Let L be a leaf of f

_∗

F

⁰

, then L is a closed complex submanifold and the leaf L can be seen as fiber on the moduli map

η : Y → M

^D_KE

where M

^D_KE

is the moduli space of Kahler-Einstein fibers with at worst log terminal singularites and Y = {y ∈ Y

reg

|(X

y

, D

y

) is Kawamata log terminal pair}

2 Main Theorems

Now by applying Tian-Li [22] method on Orbifold regularity of weak K¨ ahler-Einstein metrics, we have the

following theorem. We show that if general fibers and base are K-poly stable then there exists a generalized

K¨ ahler-Einstein metric on the total space which twisted with logarithmic Wil-Petersson metric and current of

integration of special divisor, which is not exactly D. In fact along Minimal model π : X → X

can

, we can find

such divisor N by Fujino-Mori’s higher canonical bundle formula. But along Mori-fibre space I don’t know such

formula to apply it in next theorem.

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Let f : X → B be the surjective morphism of a normal projective variety X of dimension n = m + l to a nonsingular projective l-fold B such that:

i) (X, D) is sub klt pair.

ii) the generic fiber F of π is a geometrically irreducible variety with vanishing log Kodaira dimension. We fix the smallest b ∈ Z ^> 0 such that the

π

_∗

O

X

(b(K

_X

+ D)) 6= 0

The Fujino-Mori [25] log-canonical bundle formula for π : (X, D) → B is b(K

_X

+ D) = π

^∗

(bK

_B

+ L

^ss_(X,D)/B

) + X

P

s

^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 − t

^D_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

:= max{t ∈ R ^| ^{X, D} ⁻ ^B

^D

^/b ⁺ ^tπ

^∗

^(P ⁾

is sub log canonical over η

_P

} Now if fibres (X

_s

, D

_s

) are log Calabi-Yau varieties and the base c

₁

(B) < 0, then we have

Ric(ω

X

) = −ω

B

+ ω

^D_{W P}

+ X

P

(b(1 − t

^D_P

))[π

^∗

(P)] + [B

^D

] Note that such pair of canonical metric is not unique.

Moreover, Let π : (X, D) → B is a holomorphic submersion onto a compact K¨ ahler manifold B with B be a Calabi-Yau manifold, log fibers (X

s

, D

s

) are log Calabi-Yau, and D is a simple normal crossing divisor in X with conic singularities. Then (X, D) admits a unique smooth metric ω

B

solving

Ric(ω

can

) = ω

^D_{W P}

+ X

P

(b(1 − t

^D_P

))[π

^∗

(P )] + [B

^D

]

as current where ω

_can

has zero Lelong number and is good metric in the sense of Mumford.[45]

Theorem 2.1. Let π : (X, D) → B be a holomorphic submersion between compact K¨ ahler manifolds and B and central fiber is Q -Fano K-poly stable variety with klt singularities. Let L + D be a relatively ample line bundle on (X, D) such that the restriction of c

₁

(L) to each fiber (X

_s

, D

_s

) admits a K¨ ahler-Einstein metric.

Suppose that α − c

1

(B) ≥ 0. Then, for all sufficiently large r, the adiabatic class κ

r

= c

1

(L + D) + rπ

^∗

κ

B

contains a K¨ ahler Einstein metric which is twisted by log Weil-Petersson current and current of integration of special divisor N which is not D in general. In fact such current of integration must come from higher canonical bundle formula along Mori-fibre space(I don’t know such formula at the moment).

Ric(ω

_(X,D)

) = ω

B

+ ω

_{W P}^D

+ (1 − β)[N ]

If α = c

1

(B), κ

B

is any K¨ ahler class on the base, whilst if α − c

1

(B) > 0, then κ

B

= α − c

1

(B).

Proof . Let B be any Q -Fano variety with klt singularities.Assume p ∈ B

^orb

is a quotient singularity where B

^orb

is the orbifold locus of X. Then there exists a branched covering map ˜ U

p

→ U ˜

p

/G ∼ = U

p

where U

p

is the small neighborhood of p. Now since B is Q -Fano variety, we have an embedding i : B → P

ⁿ

of linear system | − mK

B

| for m ∈ N sufficiently large and divisible.Now if h

F S

be the Fubini Study hermitian metric, then

ω = √

−1∂ ∂ ¯ log (i

^∗

h

_{F S}

)

⁻^m¹

gives a smooth positive (1, 1)-form on regular part of B and we show it with B

^reg

. but the fact is that on the singular locus B

^sing

the form ω in general is not canonically related to the local structure of B. So we must define new canonical measure for it. The following measure introduced by Bo Berndtsson,

Ω = kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ ¯ v)

^1/m

Here v is any local generator of O(m(K

B

+ D)) and v

^∗

is the dual generator of O(−m(K

B

+ D)).

Now consider the following Monge-Ampere equation on B.

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(e

^t

ω

W P

+ (1 + e

^t

)ω

0

+ Rich + √

−1∂ ∂φ) ¯

ⁿ

=

e

^−φ+Ψ

kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ ¯ v)

^1/m

kS k

²

= e

^−φ

ω

^κ_SKE

∧ kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ v) ¯

^1/m

kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ v) ¯

^1/m

| S |

²

Where Ψ is Deligne potential on log Tian’s line bundle L

^D_CM

and ω

SKE

is called fiberwise K¨ ahler-Einstein metric. Note that diam(X

s

, ω

s

) ≤ C if and only of central fiber be K-poly stable with klt singularities , so we can get C

⁰

-estimate. Note that we are facing with three type Complex-Monge-Ampere equation, 1) MA equation on fiber direction, 2) MA equation on horizontal direction , and 3) MA foliation when fiber-wise Kahler-Einstein metric iz zero, which the main difficulty for solving estimates is about Monge-Ampere equation corresponding to fiberwise K¨ ahler-Einstein foliation. I don’t know it yet. It is supper difficult!

Take a resolution π : ( ˜ B, D) ˜ → (B, D) with a simple normal crossing exceptional divisor E = π

⁻¹

(B

^sing

) =

∪

^r_i=1

E

i

S

∪

^s_j=1

F

i

where by klt property we have a

i

> 0 and 0 < b

j

< 1. Then we have K

B˜

+ ˜ D = π

^∗

(K

B

+ D) +

r

X

i=1

a

i

E

i

−

s

X

j=1

b

j

F

j

So by choosing a smooth K¨ ahler metric ˜ ω on ˜ B, there exists f ∈ C

^∞

( ˜ B) such that π

^∗

( Ω

_X/B

kSk

²

) = e

^f

Q

r

i=1

kθ

i

k

^2aⁱ

k Sk ˜

²

Q

s

j=1

kσ

j

k

^2bⁱ

ω ˜

ⁿ

where θ

_i

, σ

_j

and ˜ S are defining sections of E

_i

, F

_j

and ˜ D respectively. Then we can pull back our Monge- Ampere equation to ( ˜ B, D) and by taking ˜ ω = e

^t

ω

W P

+ (1 + e

^t

)ω

0

(π

^∗

ω + Ric˜ h + √

−1∂ ∂ψ) ¯

ⁿ

= e

^f−ψ+Ψ

Q

r

i=1

kθ

i

k

^2aⁱ

k Sk ˜

²

Q

s

j=1

kσ

j

k

^2bⁱ

ω ˜

ⁿ

Now by applying Theorem 0.8, if we take ψ

₊

= f + P

i

a

_i

log |θ

i

|

²

and ψ

₋

= ψ − Ψ + P

j

b

_j

log |σ

j

|

²

+ P

l

log c

_l

|˜ s

_l

|

²

, where ˜ S = P

l

s

_l

S ˜

_l

then we have (π

^∗

ω + Ric˜ h + √

−1∂ ∂ψ) ¯

ⁿ

= e

^ψ⁺^−ψ⁻

ω ˜

ⁿ

Its easily can be checked they satisfy the quasi-plurisubharmonic condition:

√ −1∂ ∂ψ ¯

+

≥ −C ω ˜

ⁿ

, √

−1∂ ∂ψ ¯

₋

≥ −C ω ˜

ⁿ

for some uniform constant C > 0. By regularizing our Monge-Ampere equation away from singularities we have (ω

+ Ric˜ h + √

−1∂ ∂ψ ¯

)

ⁿ

= e

^ψ^+,^−ψ^−,

ω ˜

ⁿ

where ω = π

^∗

ω − √

−1∂ ∂|S ¯

E

|

²

is a K¨ ahler metric on ˜ B \ D. The Laplacian estimate for the solution ˜ ω away from singularities is due to M. Paun [21] and Demailly-Pali[26](a more simpler proof). So we have C

^1,α

estimate for our solution. So, we get

Ric(ω

_X

) = ω

_B

+ ω

_{W P}^D

+ (1 − β)[N ] .

Remark 2: A Q -Fano variety X is K-stable if and only if it be K-stable with respect to the central fibre of test configuration. Hence X is a Q -Fano K-stable if and only if

Z

Xy

kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ ¯ v)

^1/m

≤ C

Here v is any local generator of O(m(K

_X_y

)) and v

^∗

is the dual generator of O(−m(K

_X_y

)) and C is a constant and X

_t

is a general fibre with respect to test configuration.

We give the following conjecture about K-stability via Weil-Petersson geometry

Conjecture: A Fano variety X is K-stable if and only if 0 is at finite Weil-Petersson distance from C

^o

i.e.

d

W P

(C

⁰

, 0) < +∞

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where here C is just base of test configuration

Now we introduce relative Tian’s alpha invariant along Fano fibration and Mori fibre space which correspounds to twisted K-stability.

Tian in [5], obtained a suficient condition to get C

⁰

-estimates for Monge-Ampere equation related to K¨ ahler- Einstein metric for Fano varieties. It involves an invariant of the manifold and called the Tian’s α-invariant, that encodes the possible singularities of singular hermitian metrics with non negative curvature on K

_M

. The suficient condition for the existence of K¨ ahler-Einstein metric remain the same. One can also define the α-invariant for any ample line bundle on a complex manifold M .

Definition 2.2. Let M be an n-dimensional compact K¨ ahler manifold with an ample line bundle L. We fix ω a K¨ ahler metric in c

₁

(L), and define Tians α-invariant

α(L) = sup{α > 0| ∃C > 0 with Z

M

e

^{−α(ϕ−sup}^M^ϕ)

ω

ⁿ

≤ C}

where we have assumed ϕ ∈ C

^∞

and ω + √

−1∂ ∂ϕ > ¯ 0

Now we have the following remark as relative α-invariant on the degeneration of Fano K¨ ahler-Einstein metrics.

Remark 3: Let π : X → B be a holomorphic submersion between compact K¨ ahler manifolds whose fibers and the base admit no non-zero holomorphic vector fields and B is Fano variety. Let each Fano fiber X

s

admits a K¨ ahler-Einstein metric. Define the relative α-invariant

α(X/B) = sup{α > 0| ∃C > 0 with Z

X/B

e

^{−α(ϕ−sup}^M^ϕ)

ω

_SKE^κ

∧ kv

^∗

k

^2/m

(i^∗h_{F S})m¹

(v ∧ ¯ v)

^1/m

kv

^∗

k

^2/m

(i^∗hF S)m¹

(v ∧ ¯ v)

^1/m

| S |

²

≤ C}

and let

α(X/B) > n n + 1 then we have relative K¨ ahler Einstein metric

Ric

_X/B

(ω) = Φω where Φ is the fiberwise constant function.

2.1 Relative K¨ ahler Ricci soliton

In this subsection, we extend the notion of K¨ ahler Ricci soliton to relative K¨ ahler Ricci soliton along holomorphic fibre space π : X → B. We think that the notion of relative K¨ ahler Ricci soliton is more suitable for Song-Tian program along Mori fibre space or in general for Fano fibration.

A K¨ ahler-Ricci soliton is one of the generalization of a K¨ ahler-Einstein metric and closely related to the limiting behavior of the normalized K¨ ahler-Ricci flow.

A K¨ ahler metric h is called a K¨ ahler-Ricci soliton if its K¨ ahler form ω

h

satisfies equation Ric(ω

h

) − ω

h

= L

X

ω

h

= d(i

X

ω

h

)

where Ric(ω

h

) is the Ricci form of h and L

X

ω

h

denotes the Lie derivative of ω

h

along a holomorphic vector field X on M .

Since i

_X

ω

_h

is (0, 1) ¯ ∂-closed form, by using Hodge theory, i

_X

ω

_h

= α + ¯ ∂ϕ where α is a (0, 1)-harmonic form and ϕ ∈ C

^∞

(M, C ) hence, we have

Ric(ω

h

) − ω

h

= ∂ ∂ϕ ¯ we can find a smooth real-valued function θ

X

(ω) such that

i

X

ω = √

−1 ¯ ∂θ

X

(ω)

This means that if the K¨ ahler metric h be a K¨ ahler-Ricci soliton, then ω

h

∈ c

1

(M )

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Theorem: Let X and B are compact K¨ ahler manifolds with holomorphic submersion π : X → B, and let fibers π

⁻¹

(b) = X

b

are K-poly stable Fano varieties and the Fano variety B admit K¨ ahler Ricci soliton. Then there exists a generalized K¨ ahler Ricci soliton

Ric(ω

X

) = ω

B

+ π

^∗

ω

W P

+ L

X

ω

B

for some holomorphic vector field on X , where ω

W P

is a Weil-Petersson metric on moduli space of K-poly stable Fano fibres.

Idea of Proof. It is enough to solve the following Monge-Ampere equation (ω + √

−1∂ ∂ϕ) ¯

ⁿ

= e

^Ψ−ϕ−θ^X^−X(ϕ)

ω

ⁿ

where Ψ is the Deligne potential on Tian’s CM-line bundle L

CM

. We need to show properness of the relative K-energy. We postpone it for future work.

Moreover, we have the same result on pair (X, D) → B. i.e.

Ric(ω

_X

) = ω

_B

+ π

^∗

ω

_{W P}^D

+ L

_X

ω

_B

+ [N]

We define the relative Soliton-K¨ ahler Ricci flow as

∂ω(t)

∂t = −Ric

X/Y

(ω(t)) + Φω(t) + L

X

ω(t)

along holomorphic fiber space π : X → Y , where here the relative Ricci form Ric

_X/Y,ω

of ω is defined by Ric

_X/Y,ω

= − √

−1∂ ∂ ¯ log(ω

^m

∧ π

^∗

|dy

1

∧ dy

2

∧ ... ∧ dy

k

|

²

) where (y

₁

, ..., y

_k

) is a local coordinate of Y .

Note that if fibers X

y

be K-poly stable, base Y and central fiber admit K¨ ahler Ricci soliton then relative Soliton-K¨ ahler Ricci flow flow converges to

Ric(ω

_X

) = ω

_B

+ π

^∗

ω

_{W P}

+ L

_X

ω

A relative K¨ ahler metric g on holomorphic fibre space π : M → B where M and B are compact K¨ ahler manifolds is called a relative K¨ ahler-Ricci soliton if there is a relative holomorphic vector field X on the relative tangent bundle T

_M/B

such that the relative K¨ ahler form ω

_g

of g satisfies

Ric

_M/B

ω

g

− Φω

g

= L

X

ω

g

So we can extend the result of Tian-Zhu for future work. i.e. If we have already relative K¨ ahler Ricci soliton ω

RKS

, then the solutions of the relative K¨ ahler Ricci flow

∂ω(t)

∂t = −Ric

X/Y

(ω(t)) + Φω(t)

converges to ω

RKS

in the sense of Cheeger-Gromov with additional assumption on the initial metric g

0

. We give a conjecture about invariance of plurigenera using K-stability and K¨ ahler Ricci flow.

Theorem 2.3. (Siu [32]) Assume π : X → B is smooth, and every X

_t

is of general type. Then the plurigenera P

_m

(X

_t

) = dim H

⁰

(X

_t

, mK

_X_t

) is independent of t ∈ B for any m.

See [31, 33, 34] also for singular case.

Conjecture:Let π : X → B is smooth, and every X

_t

is K-poly stable. Then the plurigenera P

_m

(X

_t

) = dim H

⁰

(X

_t

, −mK

_X_t

) is independent of t ∈ B for any m.

Idea of proof. We can apply the relative K¨ ahler Ricci flow method for it. In fact if we prove that

∂ω(t)

∂t = −Ric

_X/Y

(ω(t)) + Φω(t)

has long time 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

Conjecture: The Fano variety X is K-poly stable if and only if for the proper holomorphic fibre space

π : X → D which the Fano fibers have unique K¨ ahler-Einstein metric with positive Ricci curvature, then the

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fiberwise K¨ ahler Einstein metric (Semi-K¨ ahler Einstein metric) ω

SKE

be smooth and semi-positive. Note that if fibers are K-poly stable then by Schumacher and Berman [35, 36] result we have

−∆

ω_t

c(ω

SKE

) − c(ω

SKE

) = |A|

²_ω_t

where A represents the Kodaira-Spencer class of the deformation and since ω

ⁿ⁺¹_SKE

= c(ω

SKE

)ω

_SKEⁿ

ds ∧ d¯ s so c(ω

SKE

) and ω

SKE

have the same sign. By the minimum principle inf ω

SKE

< 0. But our conjecture says that the fibrewise Fano K¨ ahler-Einstein metric ω

SKE

is smooth and semi-positive if and only if X be K-poly stable.

Chi Li in his thesis showed that an anti-canonically polarized Fano variety is K-stable if and only if it is K-stable with respect to test configurations with normal central fibre. So if the central fibre X

0

admit Kahler-Einstein metric with positrive Ricci curvature then along Mori-fibre space all the general fibres X

t

admit K¨ ahler-Einstein metric with positive Ricci curvature(we will prove it in this paper) and hence we can introduce fibrewise K¨ ahler-Einstein metric ω

SKE

. See [35, 36]

Let L → X be a holomorphic line bundle over a complex manifold X and fix an open cover X = ∪U

α

for which there exist local holomorphic frames e

α

: U

α

→ L. The transition functions g

αβ

= e

β

/e

α

∈ O

^∗_X

(U

α

∩ U

β

) determine the Cech 1-cocycle {(U

α

, g

αβ

)}. If h is a singular Hermitian metric on L then h(e

α

, e

α

) = e

^−2ϕ^α

, where the functions ϕ

α

∈ L

¹_loc

(U

α

) are called the local weights of the metric h. We have ϕ

α

= ϕ

β

+ log |g

αβ

| on U

α

∩ U

β

and the curvature of h,

c

₁

(L, h)|

Uα

= dd

^c

ϕ

_α

is a well defined closed (1, 1) current on X.

One of the important example of singular hermitian metric is singular hermitian metric with algebraic singularities. Let m be a positive integer and {S

i

} a finite number of global holomorphic sections of mL. Let ϕ be a C

^∞

-function on X. Then

h := e

^−ϕ

. 1 ( P

i

|S

i

|

²

)

^1/m

defines a singular hermitian metric on L. We call such a metric h a singular hermitian metric on L with algebraic singularities.

Remark 4:Fiberwise K¨ ahler-Einstein metric ω

SKE

along Mori fiber space(when fibers are K-poly stable) has non-algebraic singularities. In fact such metric ω

SKE

has pole singularites

Remark 5:Let f : X → Y be a surjective proper holomorphic fibre space such that X and Y are projective varieties and central fibre X

0

is Calabi-Yau variety with canonical singularities, then all the fibres X

t

are also Calabi-Yau varieties. Let for simplicity that Y is a smooth curve. Since X

0

has canonical, then, we may assume that X is canonical. Nearby fibers are then also canonical. O

X

(K

X

+ X

0

) → O

X₀

(K

X₀

) is surjective and so if K

X₀

is Cartier, then so is K

X

on a neighborhood of X

0

.We have that P

m

(X

y

) is deformation invariant for m ≥ 1 so in fact h

⁰

(K

X_y

) > 0 for y ∈ Y . Since K

X₀

≡ 0, then K

X_y

≡ 0 and so K

X_y

∼ 0

Note that if all of the fibres of the surjective proper holomorphic fibre space f : X → Y are Calabi-Yau varieties, then the central fiber X

0

may not be Calabi-Yau variety

Remark 6:Let f : X → Y be a surjective proper holomorphic fibre space such that X and Y are projective varieties and central fibre X

0

is of general type, then all the fibres X

t

are also of general type varieties. Note that if base and fibers are of general type then the total space is of general type

Remark 7:Let f : X → Y be a surjective proper holomorphic fibre space such that X and Y are projective varieties and central fibre X

0

is K-poly stable with discrete Automorphism group, then all the fibres X

t

are also K-poly stable. Note that if along Mori-fiber space, the general fibers and base space are of K-poly stable, then total space is also K-poly stable. Note that Mori fiber space is relative notion and we can assume that base is K-poly stable always.

Define

Scal(ω) = nRic(ω) ∧ ω

ⁿ⁻¹

ω

ⁿ

and take a holomorpic submerssion π : X → D over some disc D . It is enough to show smoothness of semi Ricci flat metric over a disc D . Define a map F : D ^× ^C

^∞

^(π

⁻¹

^(b)) ^→ ^C

^∞

^(π

⁻¹

^{(b)) by}

F(b, ρ) = Scal(ω

₀

|

_b

+ √

−1∂

_b

∂ ¯

_b

ρ)

We can extend F to a smooth map D ^× ^L

²k+4

(π

⁻¹

(b)) → L

²_k

(π

⁻¹

(b)). From the definition F (b, ρ

_b

) is constant.

Recall that, if L denotes the linearization of ρ → Scal(ω

ρ

), then

L(ρ) = D

^∗

D(ρ) + ∇Scal.∇ρ

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where D is defined by

D = ¯ ∂o∇ : C

^∞

→ Ω

^0,1

(T )

and D

^∗

is L

²

adjoint of D. The linearisation of F with respect to ρ at b is given by D

_b^∗

D

b

: C

^∞

π

⁻¹

(b)

→ C

^∞

π

⁻¹

(b)

Leading order term of L is ∆

²

. So L is elliptic and we can use the standard theory of elliptic partial differential equations. Since central fiber has discrite Automorphism, hence Kernel of such Laplace-Beltrami is zero, and hence D

_b^∗

D

b

is an isomorphism. By the implicit function theorem, the map b → ρ

b

is a smooth map

D ^→ ^L

²k

(π

⁻¹

(b)) ; ∀k.

By Sobolev embedding, it is a smooth map D → C

^r

^(π

⁻¹

(b)) for any r. Hence cscK or K¨ ahler-Einstein metric with positive Ricci curvature on the central fiber can be extended smoothly to general fibers. See [43, 44]

In final it is worth to mention that K-stability along special test configuration gives such foliation as we introduced and Tian , Chen-Sun-Donaldson didn’t consider such Monge-Ampere foliation for their solution of Yau-Tian-Donaldson, conjecture.

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