Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

Remark A.1.

By [27, p. 50, Corollaire 6.6] the non-decreasing function

is convex (generalized three-circle theorem), and we have

This implies in particular that the supremum in (A.1) is attained.

Theorem A.2.

Let π:X~→X be any resolution of singularities and let E⊂X~ be a prime divisor above x∈X. Then there exists C>0 such that

for all psh functions φ on X.

Corollary A.3.

If φ is a psh function with s⁢(φ,x)=0 for some x∈X, then

for every resolution of singularities and every p∈π-1⁢(x).

Proof.

Let b:X′→X~ be the blow-up of X~ at point p∈X~. Then π′=π∘b:X′→X is yet another resolution of singularities. Set E=β-1⁢(p). This is a prime divisor to which we can apply Theorem A.2. The conclusion follows then by recalling the following classical interpretation of Lelong number: ν⁢(φ∘π∘b,E)=ν⁢(φ∘π,p). ∎

Proof of Theorem A.2.

By Hironaka’s theorem we may assume that π:X~→X dominates the blow-up of X at x, so that the scheme-theoretic fiber π-1⁢(x) is an effective divisor ∑iai⁢Ei. Note that ∑iEi is connected by Zariski’s “main theorem”.

Set bi:=ν⁢(π*⁢φ,Ei). Using the Siu decomposition of the positive current T:=d⁢dc⁢π*⁢φ we may write T=R+B, where B=∑ibi⁢Ei is an effective ℝ-divisor and R is a positive current such that ν⁢(R,Ei)=0 for all i. We first claim that

Indeed, if we write 𝔪x=(fi) as above, then π*⁢log⁢∑i|fi| has analytic singularities described by the divisor π-1⁢(x), i.e. locally on X we have

where zi is a local equation of Ei. We thus see that

locally on X~, and the positivity of R=T-B shows that this holds if and only if

hence the claim.

In view of (A.2), the desired statement amounts to an estimate maxi⁡bi≤C⁢mini⁡bi for some C>0 independent of φ. Thanks to Lemma A.4 below, this will hold if we can show that -B|Ei is pseudoeffective for all i. Since the restriction to each Ei of the cohomology class of T=π*⁢d⁢dc⁢φ is trivial, we are reduced to showing that {R}|Ei is pseudoeffective. Since ν⁢(R,Ei)=0, this follows from Demailly’s regularization theorem. Let us recall the standard argument: by [28], after perhaps shrinking X slightly about 0 we may write R as a weak limit of closed positive (1,1)-currents Rk with analytic singularities such that {Rk}={R}, Rk≥-εk⁢ω for some εk→0 and Rk is less singular than R. In particular, ν⁢(Rk,Ei)=0 for all i, which means that the local potentials of Rk are not entirely singular along Ei, so that Rk|Ei is a well-defined closed (1,1)-current. We thus see that ({R}+εk⁢{ω})|Ei is pseudoeffective for all k, and the claim follows. ∎

Lemma A.4.

Let E=∑iEi be a reduced compact connected divisor on a Kähler manifold M. Let B=∑bi⁢Ei be an effective R-divisor supported in E, and assume that -B|Ei is pseudoeffective for all i. Then there exists a constant C>0 only depending on E such that maxi⁡bi≤C⁢mini⁡bi.

Proof.

Let ω be a Kähler form on M. Thanks to the connectedness of E, we may index the Ei such that B=∑i=1Nbi⁢Ei with b1=mini⁡bi, br=maxi⁡bi for 1≤r≤N, and Ei∩Ei+1≠∅ for all i=1,…,r-1. For each i we have

with

hence

Now ci,j≥0 if j≠i, and ci,i+1>0 for all i since Ei meets Ei+1. It follows that

for all i, hence maxi⁡bi=br≤C⁢b1=mini⁡bi with C:=∏i=1r-1|ci,i|ci,i+1 ∎

Remark A.5.

Besides the slope s⁢(φ,x) considered above, Demailly introduced in [27] a different generalization of Lelong numbers on normal complex spaces, defined as the intersection multiplicity

where (fi) are generators of 𝔪x, the definition being independent of that choice. When 𝔞=(g1,…,gr) is an 𝔪x-primary ideal and φ=log⁢∑i|gi|, then ν⁢(φ,x) computes the mixed (Hilbert–Samuel) multiplicity 〈𝔞,𝔪x,…,𝔪x〉. In particular, for φ=ψ we have ν⁢(ψ,x)=m⁢(X,x), the multiplicity of X at x. By Demailly’s comparison theorem we have

and the inequality is strict in general. Using the notation of the proof of Theorem A.2 and recalling that -π-1⁢(x) is π-nef, we conjecture by analogy with the algebraic case that

By Theorem A.2 this would imply in particular that conversely ν⁢(φ,x)≤C⁢s⁢(φ,x) for some C>0 independent of φ.

Theorem B.1.

Let μ be a positive measure on X of the form μ=eψ+-ψ-⁢d⁢V with ψ± quasi-psh and e-ψ-∈Lp for some p>1. Assume that φ is a bounded θ-psh function such that (θ+d⁢dc⁢φ)n=μ. Then we have Δ⁢φ=O⁢(e-ψ-) locally in Amp⁡(θ).

More precisely, assume given a constant C>0 such that

1. d⁢dc⁢ψ+≥-C⁢ω and supX⁡ψ+≤C,

2. d⁢dc⁢ψ-≥-C⁢ω and ∥e-ψ-∥Lp≤C.

Let also U⋐Amp⁡(θ) be a relatively compact open subset. Then there exists A>0 only depending on θ, p, C and U such that

on U.

Proof.

We may of course assume that φ is normalized. During the proof A,A1,… will denote positive constants that may vary from line to line, but are under control in the sense that they only depend on θ, p, C and U. Since U is contained in Amp⁡(θ), we may choose a Zariski open set Ω⊃U and a θ-psh function ψ such that (θ+d⁢dc⁢ψ)|Ω is the restriction of a Kähler form ω~ on a higher compactification X~ of Ω, so that

The proof of Theorem B.1 is divided into two steps. In the first and main one, an a priori estimate for smooth solutions of non-degenerate perturbations of the equation is established. In the second step we conclude using a regularization argument.

Step 1: A priori estimates. For 0<ε≤1 we set ωε:=ω~+ε⁢ω, viewed as a Kähler form on Ω. Note that ωε≥δ⁢ω, so that

for every positive (1,1)-form α. Assume that ψ+ and ψ- are smooth functions satisfying (i) and (ii) of Theorem B.1, and assume given a smooth normalized θε-psh function φε such that

The goal of Step 1 is to establish that |Δ⁢φε|≤A⁢e-ψ- on U with A>0 under control. Since we have ωε≤A⁢ω over U with A under control, it will be enough to prove that

satisfies trωε⁡(ωε′)≤A⁢e-ψ- on U.

We first recall the Laplacian inequality obtained in [57, pp. 98–99]: if τ,τ′ are two Kähler forms on a complex manifold, then there exists a constant B>0 only depending on a lower bound for the holomorphic bisectional curvature of τ such that

We remark that Siu’s argument uses the fact that τ and τ′ are d⁢dc-cohomologous. But the general case is valid as well since Siu’s computations are purely local and any Kähler form is even locally d⁢dc-exact. This being said, let us apply this inequality to the two Kähler forms ωε and ωε′ on Ω.

Since ω~ extends to a Kähler form on a higher compactification X~ of Ω, the holomorphic bisectional curvature of ωε=ω~+ε⁢ω is obviously bounded over Ω by a constant B>0 under control, and (B.3) yields

On the other hand, applying d⁢dc⁢log to (ωε′)n=eψ+-ψ-⁢ωn yields

where A is under control thanks to (i). Using trωε⁡(ω)≤n⁢δ-1 and the trivial inequality

we thus infer from (B.4) that

with A under control.

We next argue along the lines of [50, Lemma 3.2] to take care of the term Δωε⁢ψ-. By (ii) we have A⁢ωε+d⁢dc⁢ψ-≥0 with A under control. Applying trωε to the trivial inequality

yields

Plugging this into (B.6) and using again (B.5), we thus obtain

where A is under control. Now set

so that ωε′=ωε+d⁢dc⁢ρε. We then have n=trωε′⁡(ωε)+Δωε′⁢ρε, and we finally deduce from (B.7) that

on Ω, with A1,A2 under control.

We are now in a position to apply the maximum principle. On the one hand, ρε=φε-ψ tends to +∞ near ∂⁡Ω. On the other hand, trωε⁡(ωε′)≤δ-1⁢trω⁡(ωε′) is bounded above on Ω since ωε′ is smooth over X. The function

therefore achieves its maximum at some x0∈Ω, and (B.8) yields trωε′⁡(ωε)⁢(x0)≤A2. On the other hand, trivial eigenvalue considerations show that

for any two Kähler forms τ1,τ2, whence

by (B.2). Using ω≤δ-1⁢ωε it follows that

where A3,A4 are under control, and we obtain

with A5 under control, since ρε=φε-ψ and ψ≤0. By the L∞-estimate provided by [34], we now obtain

on Ω for some constant A under control. Since φε is normalized, we conversely have

over U⋐Ω, and we finally infer as desired trωε⁡(ωε′)≤A⁢e-ψ- on U.

Step 2: Regularization. We now consider the set-up of Theorem B.1. By Demailly’s regularization theorem [28], there exist two decreasing sequences of smooth functions ψj± such that

1. limj→∞⁡ψj±=ψ± on X,

2. d⁢dc⁢ψj±≥-A⁢ω for some A>0 under control.

In fact, the constant A>0 depends in principle on the Lelong numbers of the quasi-psh functions ψ± according to Demailly’s result, but these Lelong numbers can be uniformly bounded in terms of the lower bound -C⁢ω for d⁢dc⁢ψ± by a standard argument, see for instance [12, Lemma 2.5].

For each 0<ε≤1 the closed (1,1)-form θ+ε⁢ω is Kähler, and Yau’s theorem [70] yields smooth normalized θε-psh functions φε,j such that

where cε,j∈ℝ is a normalizing constant. Since eψj+-ψj-≤eC-ψ- is uniformly bounded in Lp, cε,j is under control and Step 1 of the proof shows that

over U, with A>0 under control.

Now for each fixed j it follows from [13, Lemma 5.3] that φε,j converges weakly as ε→0 to the normalized solution φj of

which therefore satisfies as well |Δ⁢φj|≤A⁢e-ψj- on U. But we also have eψj+-ψj-→eψ+-ψ- in Lp by dominated convergence, and it follows that φj→φ weakly on X by [34, Theorem A], which concludes the proof of Theorem B.1. ∎

Theorem C.1.

Let X be a compact Kähler manifold and L a line bundle on X such that

1. h0⁢(X,KX+L)=1 and h1⁢(X,KX+L)=0,

2. L=M+Δ, where M is a semipositive ℚ-line bundle, Δ=∑iai⁢Di is an effective ℚ-divisor with SNC support and ai∈(0,1).

Set S:={t∈C∣0<Re⁡t<1} and consider a psh metric ϕ on the pull-back of L to X×S of the form ϕ=τ+ϕΔ, where

1. τ is a bounded psh metric on the pull-back of M to X×S, with t↦τt only depending on Re⁡t and Lipschitz continuous,

2. ϕΔ=∑iai⁢log⁡|si|2 with si the canonical section of 𝒪⁢(Di), so that d⁢dc⁢ϕΔ=[Δ].

For each generator u of H0⁢(X,KX+L), viewed as an L-valued holomorphic n-form on X, the function

is then convex on (0,1). If it is further affine, then there exists a holomorphic vector field V on {u≠0}⊂X such that

on X×S, with LV the Lie derivative along V.

Lemma C.2.

For each L-valued (n,0)-form η on X, there exists a unique L-valued (n-1,0)-form v such that

1. ∂ϕ⁡v=P⁢η, the projection of η orthogonal to the kernel of ∂¯,

2. ω∧∂¯⁢v=0.

Proof.

The image of ∂¯ is closed, since it has finite codimension in Ker⁡∂¯. As a result, P⁢η∈(Ker⁡∂¯)⊥=Im⁡∂¯ϕ* may be uniquely written as P⁢η=∂¯ϕ*⁢β for an L-valued (n,1)-form β∈(Ker⁡∂¯ϕ*)⊥=Im⁡∂¯, and β=ω∧v for a unique L-valued (n-1,0)-form v.

Since we are assuming that Hn,1⁢(X,ℂ)=H1⁢(X,KX+L)=0, β above is in fact unique in Ker⁡∂¯, which concludes the proof. ∎

Remark C.3.

For later use, note that any L-valued (n-1,0)-form v for which (ii) holds satisfies

for all L-valued (n,1)-form α, by (C.3).

Lemma C.4.

There exists a constant C>0 such that for each t,ν and each L-valued (n,0)-form η, the unique L-valued (n-1,0)-form v solving

1. ∂zϕtν⁡v=Ptν⁢η,

2. ω∧∂¯z⁢v=0.

satisfies ∥v∥ϕtν≤C⁢∥η∥ϕtν.

Lemma C.5.

For each δ>0, there exists a neighborhood Uδ⊂X of supp⁡Δ such that

for all L-valued (n-1,0)-forms v on X, all ν and t∈Sν.

Lemma C.6.

For each ν, there exists a unique smooth family vν=(vtν)t∈Sν of L-valued (n-1,0)-forms such that

1. ∂zϕtν⁡vtν=Ptν⁢(ϕ˙tν⁢u),

2. ω∧∂¯z⁢vtν=0.

The L2-norm ∥vtν∥ϕtν is bounded independently of t and ν. After perhaps passing to a subsequence, we can further find a sequence of smooth cut-off functions 0≤χν≤1 on X (with χν≡0 on some neighborhood of supp⁡Δ) such that

1. χν⁢d⁢dc⁢ϕtν≥-εν⁢ω,

2. ∫X(1-χν)⁢|vtν|ϕtν2≤εν⁢(1+∥∂¯z⁢vtν∥ϕtν2),

for some sequence εν>0 converging to 0.

Lemma C.7.

The following fiber integrals converge weakly to zero on S as ν→∞:

1. ∫X|∂¯z⁢vtν|ϕtν2,

2. ∫Xχν⁢Θν,

3. ∫X(1-χν)⁢Θν,

4. ∫X(1-χν)⁢|vtν|ϕtν2.

Proof.

Part (i) follows directly from (C.9). Another application of the curvature formula (C.4) then yields ∫XΘν→0. Now let f∈Cc∞⁢(S) be a non-negative test function. Injecting (i) in (C.8) yields ∫X×Sf⁢(1-χν)⁢Θν≥-εKν, while (C.7) gives ∫X×Sf⁢χν⁢Θν≥-εKν. Since the sum converges to zero by what we just saw, we get (ii) and (iii). Finally, (iv) is a consequence of (i) and point (iv) of Lemma C.6. ∎