A quantization proof of the uniform Yau-Tian-Donaldson conjecture
aa r X i v : . [ m a t h . DG ] F e b A QUANTIZATION PROOF OF THE UNIFORM YAU–TIAN–DONALDSONCONJECTURE
KEWEI ZHANG
Abstract.
Using quantization techniques, we show that the δ -invariant of Fujita–Odaka coincides withthe optimal exponent in certain Moser–Trudinger type inequality. Consequently we obtain a uniformYau–Tian–Donaldson theorem for the existence of twisted K¨ahler–Einstein metrics with arbitrary polar-izations. Our approach mainly uses pluripotential theory, which does not involve Cheeger–Colding–Tiantheory or the non-Archimedean language. A new computable criterion for the existence of constant scalarcurvature K¨ahler metrics is also given. Contents
1. Introduction 12. Setup and the main results 23. Existence of canonical metrics 44. Quantization 65. Proving δ = δ A Introduction
A fundamental problem in K¨ahler geometry is to find canonical metrics on a given manifold. A problemof this sort is often called the Yau–Tian–Donaldson (YTD) conjecture, which predicts that the existence ofcanonical metrics is equivalent to certain algebro-geometric stability notion. This article, as a continuationof the author’s recent joint work with Rubinstein–Tian [34], is mainly concerned with the existence oftwisted K¨ahler–Einstein (tKE) metrics on projective manifolds. We will present a quantization proofof a uniform version of the YTD conjecture, by directly relating Fujita–Odaka’s δ -invariant [26] (thatcharacterizes unform Ding stability [10, 12]) to the existence of tKE metrics.The key ingredient in our approach is the analytic δ -invariant defined as the optimal exponent of theMoser–Trudinger inequality, which we denote by δ A . This analytic threshold characterizes the coercivityof Ding functionals and hence governs the existence of tKE metrics. In the prequel [34] we set up aquantization approach whose goal is to show that δ and δ A are actually equal, a conjecture previouslymade by the author in [42]. If this works out then one would have a new proof for the uniform YTDconjecture. Although this goal was not achieved in [34], we were able to prove a quantized version sayingthat δ m = δ Am indeed holds at each level m , so that δ m characterizes the existence of certain balancedmetrics in the m -th Bergman space, making our conjectural picture about δ and δ A even more promising.In this article we completely solve our conjecture. Our result can be viewed as an analogue of Demailly’swork [13, Appendix] (see also Shi [35]) who showed that Tian’s analytically defined α -invariant is equalto the global log canonical threshold studied in algebraic geometry, the proof of which actually greatlyinfluenced this article and its prequel [34]. Main Theorem.
The equality δ ( L ) = δ A ( L ) holds for any ample line bundle L . Consequently we obtain a new proof of the uniform YTD conjecture, in a much simpler fashion than theother known approaches in the literature. More precisely, our approach only uses the following analyticingredients: • Tian’s seminal work [38] on the asymptotics of Bergman kernels; • the lower semi-continuity result of Demailly–Koll´ar [20]; • the existence of geodesics in the space of K¨ahler metrics going back to Chen [14]; • the variational approach of Berman, Boucksom, Eyssidieux, Guedj and Zeriahi [6, 5]; • a quantized maximum principle due to Berndtsson [9].While on the algebraic side, we only need • Fujita–Odaka’s basis divisor characterization of δ m [26]; • Blum–Jonsson’s valuative definition of δ [10].When the underlying manifold is Fano, a special case of our main theorem has already been obtainedby Berman–Boucksom–Jonsson [2], who showed that min { s, δ } = min { s, δ A } = the greatest Ricci lowerbound (here s denotes the nef threshold). Note that their approach crucially relies on the convexity oftwisted K-energy and the compactness of weak geodesic rays, which unfortunately cannot directly yield δ = δ A when these thresholds surpass s . In contrast, our quantization argument mainly takes place in thefinite dimensional Bergman space without involving any convexity of functionals. Hence as a consequence,we can treat arbitrary polarizations and establish the very much desired equality δ = δ A in the generalsetting. Somewhat surprisingly, our approach not only yields stronger results, but also comes with a quiteshort proof . Note that our approach extends easily to the case of klt currents as treated in [2] (whichwe will indeed adopt in what follows), and more generally also to the coupled soliton case considered in[34]. Our work even has applications in finding constant scalar curvature K¨ahler (cscK) metrics, since wewill give a new computable criterion for the coercivity of the K-energy. Organization.
The rest of this article is organized as follows. We will fix our setup and notation, andstate more precisely our main results in Section 2. In Section 3 we elaborate on how δ A is related to theexistence of canonical metrics. Then in Section 4 we recall some necessary quantization techniques on theBergman space and prove the key estimate, Proposition 4.2. Finally, our main results, Theorems 2.2, 2.3and 2.4, are proved in Section 5. 2. Setup and the main results
Notation and definitions.
Let X be a projective K¨ahler manifold of dimension n with an ample R -line bundle L over it. Fix a smooth Hermitian metric h on L such that ω := − dd c log h ∈ c ( L )is a K¨ahler form (here dd c = √− ∂ ¯ ∂ π ). Put V := R X ω n = L n . To make our result a bit more general, wewill also fix (following [2]) a positive (1 , θ with klt singularities,meaning that, when writing θ = dd c ψ locally, one has e − ψ ∈ L ploc for some p >
1. A case of particularinterest is when θ = [∆] is the integration current along some effective klt divisor ∆, which relates to theedge-cone metrics for log pairs. The reader may take θ = 0 for simplicity as it will make no essentialdifference.Now we recall the definition of δ -invariant, which was first introduced by Fujita–Odaka [26] using basistype divisors, and then reformulated by Blum–Jonsson [10] in a more valuative fashion. To incorporate θ , we will use the following definition of Berman–Boucksom–Jonsson [2]: δ ( L ; θ ) := inf E A θ ( E ) S L ( E ) . Here E runs through all the prime divisors over X , i.e., E is a divisor contained in some birational model Y π −→ X over X . Moreover, A θ ( E ) := 1 + ord E ( K Y − π ∗ K X ) − ord F ( θ )denotes the log discrepancy, where ord F ( θ ) is the Lelong number of π ∗ θ at a very generic point of F . And S L ( E ) := 1vol( L ) Z ∞ vol( π ∗ L − xE ) dx denotes the expected vanishing order of L along E .Historically, the case of the most interest is when L = − K X and θ = 0, i.e., the Fano case. Regardingthe existence of K¨ahler–Einstein metrics on such manifolds, a notion called K-stability was introducedby Tian [39] and later reformulated more algebraically by Donaldson [23]. This stability notion hasrecently been further polished by Fujita and Li’s valuative criterion [28, 25], and we now know (see[10, Theorem B]) that δ ( − K X ) > X, − K X ) being uniformly K-stable, a condition However we should emphasize that the non-Archimedean formalism in [2] indeed plays a key role when it comes to thecscK problem; see e.g. [30] for some recent breakthrough.
QUANTIZATION PROOF OF THE UNIFORM YAU–TIAN–DONALDSON CONJECTURE 3 stronger than K-stability (but actually these two are equivalent, at least in the smooth setting). It is alsoknown that uniform K-stability is equivalent to the uniform Ding stability of Berman [3]. More recentlyBoucksom–Jonsson [10] further extend the definition of uniform Ding stability to general polarizationsusing δ -invariants, which we will adopt in this article. Definition 2.1.
We say ( X, L, θ ) is uniformly Ding stable if δ ( L ; θ ) > . Under the YTD framework, it is expected that such a notion would imply the existence of tKE metrics.In the literature, the most examined case is when c ( L ) = c ( X ) − [ θ ], namely, the “log Fano” setting.By using continuity methods (cf. [40, 16, 19, 31, 41]) or the variational approach (cf. [2, 32, 29]), wenow have a fairly good understanding of the YTD conjecture in this scenario. The upshot is that one canindeed find a K¨ahler current ω tKE ∈ c ( L ) solvingRic( ω tKE ) = ω tKE + θ under the stability assumption. Here Ric( · ) := − dd c log det( · ) denotes the Ricci operator. The solution ω tKE is precisely what we mean by a twisted K¨ahler–Eisntein metric (cf. also [5, 2]).However, to the author’s knowledge, all the known approaches to the above statement does not workwell in the case where θ is merely quasi-positive, one main difficulty being that there is no convexityavailable for twisted K-energy in the non-Fano setting. In what follows we will present a quantizationapproach to circumvent this difficulty, which allows us to work even without the Fano condition.More precisely, given any (not necessarily semi-positive) smooth representative η ∈ c ( X ) − c ( L ) − [ θ ],we want to investigate the following tKE equation:(2.1) Ric( ω tKE ) = ω tKE + η + θ. To study this, a crucial input is taken from the work of Ding [22], who essentially showed that thesolvability of the above equation is governed by certain Moser–Trudinger type inequality. Inspired by thisviewpoint, the author introduced an analytic δ -invariant in [42], which we now turn to describe.Put H ( X, ω ) := (cid:8) φ ∈ C ∞ ( X, R ) (cid:12)(cid:12) ω φ := ω + dd c φ > (cid:9) . Let E : H ( X, ω ) → R denote the Monge–Amp`ere energy defined by E ( φ ) := 1( n + 1) V n X i =0 Z X φω n − i ∧ ω iφ for φ ∈ H ( X, ω ) . Also fix a smooth representative θ ∈ [ θ ], so we can write θ = θ + dd c ψ for some usc function ψ on X .We may rescale ψ such that(2.2) µ θ := e − ψ ω n defines a probability measure on X (i.e., R X dµ θ = 1). Note that θ being klt is equivalent to saying thatfor any p >
1, sufficiently close to 1, there exists A p > Z X e − pψ ω n < A p . The analytic δ -invariant of ( X, L, θ ) is then defined by(2.4) δ A ( L ; θ ) := sup (cid:26) λ > (cid:12)(cid:12)(cid:12)(cid:12) ∃ C λ > Z X e − λ ( φ − E ( φ )) dµ θ < C λ for any φ ∈ H ( X, ω ) (cid:27) , which does not depend on the choice of ω or θ . As explained in [42], δ A ( L ; θ ) > δ ( L ; θ ) = δ A ( L ; θ ) . Given this, then (2.1) can be solvedwhen δ ( L ; θ ) >
1, i.e., when (
X, L, θ ) is uniformly Ding stable.2.2.
Main results.
In this article we confirm the aforementioned conjecture.
Theorem 2.2 (Main Theorem) . For any ample R -line bundle L , one has δ ( L ; θ ) = δ A ( L ; θ ) . In particular uniform Ding stability implies the coercivity of twisted Ding functionals and as a con-sequence, we obtain a new proof of the uniform YTD conjecture and generalize the known results inthe log Fano case (e.g., [2, Theorem A]) to the following more general setting, with possibly irrationalpolarizations.
K. ZHANG
Theorem 2.3.
Assume that ( X, L, θ ) is uniformly Ding stable. Then for any smooth form η ∈ ( c ( X ) − c ( L ) − [ θ ]) , there exists a K¨ahler current ω tKE ∈ c ( L ) solving Ric( ω tKE ) = ω tKE + η + θ. As mentioned in Introduction, the proof of Theorem 2.2 uses the quantization approach initiated in[34], which already implies one direction: δ A ( L ; θ ) ≤ δ ( L ; θ ) when L is an ample Q -line bundle. Forcompleteness we will recall its proof in Section 5. For the other direction, δ A ( L ; θ ) ≥ δ ( L ; θ ), we willcrucially use a quantized maximum principle due to Berndtsson [9], which enables us to bound δ A frombelow using finite dimensional data, hence the result. The general case of R -line bundle then follows byinvoking the continuity of δ and δ A in the ample cone (cf. [42]). At the end of this article we will brieflyexplain how to generalize our approach to the coupled soliton case considered in [34].In fact we expect that our approach can be generalized to the case of big line bundles, yielding newexistence results for the general Monge–Amp`ere equations considered in [11], and answering some questionsproposed in [42, Section 6.3]. Another direction to pursue would be to consider the the case of singularvarieties (as in [36, 31, 32]) or the equivariant case (as in [29, 27]).Now take θ = 0, in which case we will drop θ from our notation. Then Theorem 2.2 has the followinginteresting application, yielding a new criterion for the existence of cscK metrics. This also answers [42,Question 6.14]. Theorem 2.4.
Let L be an ample R -line bundle. Assume that δ ( L ) > nµ ( L ) − ( n − s ( L ) , where µ ( L ) := − K X · L n − L n and s ( L ) := sup { s ∈ R | − K X − sL > } . Then X admits a unique constant scalarcurvature K¨ahler (cscK) metric in c ( L ) . Recent progress made by Ahmadinezhad–Zhuang [1] shows that one can effectively compute δ -invariantsby induction and inversion of adjunction. So we expect that Theorem 2.4 can be applied to find morenew examples of cscK manifolds. Also observe that the assumption in Theorem 2.4 is purely algebraic, sothe author wonders if one can show uniform K-stability for ( X, L ) under the same condition using onlyalgebraic argument; see [21] for related discussions.3.
Existence of canonical metrics
In this section we explain how is δ A related to the canonical metrics in K¨ahler geometry, following [42].The discussions below in fact hold for general K¨ahler classes as well.We begin by introducing a twisted version of the α -invariant of Tian [37]. Set(3.1) α ( L ; θ ) := sup (cid:26) α > (cid:12)(cid:12)(cid:12)(cid:12) ∃ C α > Z X e − α ( φ − sup φ ) dµ θ < C α for all φ ∈ H ( X, ω ) (cid:27) . Lemma 3.1.
One always has α ( L ; θ ) > .Proof. Using H¨older’s inequality, the assertion follows from [37, Proposition 2.1] and (2.3). (cid:3)
As a consequence, one also has δ A ( L ; θ ) > E ( φ ) ≤ sup φ . Note that α ( L ; θ ) will be used several times in this article, as it can effectively controlbad terms when doing integration.3.1. Twisted Ding functional.
In this part we relate δ A to tKE metrics. Pick any smooth representative η ∈ c ( X ) − c ( L ) − [ θ ]. Then we can find f ∈ C ∞ ( X, R ) satisfyingRic( ω ) = ω + η + θ + dd c f, where we recall that θ ∈ [ θ ] is the smooth representative we have fixed. Then the twisted Ding functionalis defined by D θ + η ( φ ) := − log Z X e f − φ dµ θ − E ( φ ) for φ ∈ H ( X, ω ) . Actually one can extend D θ + η ( · ) to the larger space E ( X, ω ) (see [6] for the definition). Using variationalargument, a critical point φ ∈ E ( X, ω ) of D θ + η ( · ) will give rise to a solution to (2.1) (see [5, Section4]). A sufficient condition to guarantee the existence of such a critical point is called coercivity , which werecall as follows. QUANTIZATION PROOF OF THE UNIFORM YAU–TIAN–DONALDSON CONJECTURE 5
Definition 3.2.
The twisted Ding functional D θ + η ( · ) is called coercive if there exist ε > and C > such that D θ + η ( φ ) ≥ ε (sup φ − E ( φ )) − C for all φ ∈ H ( X, ω ) . Using Demailly’s regularization, the above definition is equivalent the coercivity investigated in [5] andhence D θ + η being coercive implies the existence of a solution to (2.1) by [5, Section 4]. Proposition 3.3. If δ A ( L ; θ ) > , then D θ + η ( · ) is coercive for any smooth representative η ∈ c ( X ) − c ( L ) − [ θ ] .Proof. This is already contained in [42, Proposition 3.6] (which in fact says that the reverse direction isalso true). It suffices to show that, for some ε >
C > − log Z X e − φ dµ θ − E ( φ ) ≥ ε (sup φ − E ( φ )) − C for any φ ∈ H ( X, ω ) . To see this, fix λ ∈ (1 , δ A ( L ; θ )) and α ∈ (0 , min { , α ( L ; θ ) } ). Then by H¨older’s inequality, − log Z X e − φ dµ θ − E ( φ ) ≥ − − αλ − α log Z X e − λφ dµ θ − λ − λ − α Z X e − αφ dµ θ − E ( φ )= − − αλ − α log Z X e − λ ( φ − E ( φ )) dµ θ − λ − λ − α Z X e − α ( φ − sup φ ) dµ θ + α ( λ − λ − α (sup φ − E ( φ )) . Then the assertion follows from (2.4) and (3.1). (cid:3)
Corollary 3.4. If δ A ( L ; θ ) > , then there exists a solution to (2.1) for any smooth representative η ∈ c ( X ) − c ( L ) − [ θ ] . K-energy and constant scalar curvature metric.
In this part we relate δ A to cscK metrics.For simplicity assume θ = 0, and hence θ will be abbreviated in our notation. Let us first recall severalfunctionals. For φ ∈ H ( X, ω ), define • I -functional: I ( φ ) := V R X φ ( ω n − ω nφ ); • J -functional: J ( φ ) := V R X φω n − E ( φ ); • Entropy: H ( φ ) := V R X log ω nφ ω n ω nφ ; • J -Energy: J ( φ ) := n ( − K X ) · L n − L n E ( φ ) − V R X φ Ric( ω ) ∧ P n − i =0 ω i ∧ ω n − − iφ ; • K-energy: K ( φ ) := H ( φ ) + J ( φ ) . A K¨ahler metric ω φ ∈ c ( L ) is a cscK metric if and only if φ is a critical point of the K-energy (cf. [33]).The following result says that δ A ( L ) is the coercivity threshold of H ( φ ). Proposition 3.5. [42, Proposition 3.5]
We have δ A ( L ) = sup (cid:26) λ > (cid:12)(cid:12)(cid:12)(cid:12) ∃ C λ > s.t. H ( φ ) ≥ λ ( I − J )( φ ) − C λ for all φ ∈ H ( X, ω ) (cid:27) . Now let µ ( L ) := − K X · L n − L n denote the slope and s ( L ) := sup { s ∈ R | − K X − sL > } the nef threshold.As explained in [42, Section 6.2], if δ A ( L ) + ( n − s ( L ) − nµ ( L ) >
0, then for some ε > C ε > K ( φ ) ≥ ε ( I − J )( φ ) − C ε for all φ ∈ H ( X, ω ) , meaning that the K-energy is coercive. So by Chen–Cheng [15, Theorem 4.1], there exists a cscK metricin c ( L ). Moreover by [4, Theorem 1.3] such a metric is unique as in this case the automorphism groupmust be discrete. As a consequence, we have the following Corollary 3.6. [42, Corollary 6.13]
Assume that δ A ( L ) > nµ ( L ) − ( n − s ( L ) , then there exists a unique cscK metric in c ( L ) . K. ZHANG Quantization
We collect some necessary quantization techniques for the proof of our main theorem. In this sectionwe assume L to be an ample line bundle over X . By rescaling L we will assume further that mL is veryample for any m ∈ N > .Put R m := H ( X, mL ) and d m := dim R m . As in Section 2, fix a smooth positively curved Hermitian metric h on L with ω := − dd c log h .4.1. Bergman space.
Note that there is a natural Hermitian inner product H m := Z X h m ( · , · ) ω n on R m induced by h . More generally, for any bounded function φ on X , we may consider H φm := Z X ( he − φ ) m ( · , · ) ω n . So in particular, H m = H m .Now put P m ( X, L ) := (cid:26) positive Hermitian inner product on R m (cid:27) . and B m ( X, ω ) := (cid:26) φ = 1 m log d m X i =1 | σ i | h m (cid:12)(cid:12)(cid:12)(cid:12) { σ i } is a basis of R m (cid:27) . The classical Fubini–Study map
F S : P m ( X, L ) → B m ( X, ω ) is a bijection, where
F S is defined by
F S ( H ) := 1 m log d m X i =1 | σ i | h m for H ∈ P m where { σ i } is any H -orthonormal basis . In particular B m ( X, ω ) ⊆ H ( X, ω ) is a finite dimensional subspace (when identified with P m ( X, L ) ∼ = GL ( d m , C ) /U ( d m )).For any φ ∈ H ( X, ω ), we set for simplicity φ ( m ) := F S ( H φm ) . It then follows from the definition that(4.1) Z X e m ( φ ( m ) − φ ) ω n = d m for any φ ∈ H ( X, ω ) . This simple identity will be used in the proof of Theorem 2.2.Note that any two Hermitian inner products can be joint by the (unique)
Bergman geodesic . Morespecifically, given any two H m, , H m, ∈ P m ( X, L ), one can find an H m, -orthonormal basis under which H m, = diag( e µ , ..., e µ dm ) is diagonalized. Then the Bergman geodesic H t takes the form H m,t := diag( e µ t , ..., e µ dm t ) . Quantized δ -invariant. Now as in [34], we consider the following quantized Monge–Amp`ere energy : E m ( φ ) := 1 md m log det H m det F S − ( φ ) for φ ∈ B m ( X, ω ) . In the literature this is also known as (up to a sign) Donaldson’s L m -functional (cf. [24]). Observe that E m ( F S ( · )) is linear along any Bergman geodesics emanating from H m . So in particular(4.2) E m ( F S ( H m, )) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E m ( F S ( H m,t ))for any Bergman geodesic [0 , ∋ t H m,t with H m, = H m . Put(4.3) δ m ( L ; θ ) := sup (cid:26) λ > (cid:12)(cid:12)(cid:12)(cid:12) ∃ C λ > Z X e − λ ( φ − E m ( φ )) dµ θ < C λ for any φ ∈ B m (cid:27) . QUANTIZATION PROOF OF THE UNIFORM YAU–TIAN–DONALDSON CONJECTURE 7
By our previous work [34, Theorem B.3], this coincides with the original basis divisor formulation ofFujita–Odaka [26]. Moreover, by [10, Theorem A] and [2, Theorem 7.3] the limit of δ m ( L ; θ ) exists andone has(4.4) δ ( L ; θ ) = lim m →∞ δ m ( L ; θ ) . Note that δ m ( L ; θ ) characterizes the coercivity of certain quantized Ding functional, whose critical pointscorrespond to “balance metrics”; see [34, Theorem B.7] for a quantized version of Theorem 2.3.4.3. Comparing E with E m . Given any φ ∈ H ( X, ω ), it has been known since the work of Donaldsonthat E ( φ ) = lim m →∞ E m ( φ ( m ) ) . But this convergence is not uniform when φ varies in H ( X, ω ), which isthe main stumbling block in the quantization approach. To overcome this, we recall a quantized maximumprinciple due to Berndtsson [9].The setup goes as follows. For any ample line bundle E over X , let g be a smooth positively curvedmetric on E with η := − dd c log g > φ , φ ∈ H ( X, η ).It was shown by Chen [14] and more recently by Chu–Tosatti–Weinkove [17] that there always exists a C , -geodesic φ t joining φ and φ . For the reader’s convenience, we briefly recall the definition. Let[0 , ∋ t φ t be a family of functions on [0 , × X with C , regularity up to the boundary. Let S := { < Re s < } ⊂ C be the unit strip and let π : S × X → X denote the projection to the secondcomponent. Then we say φ t is a C , -subgeodesic if it satisfies π ∗ η + dd cS × X φ Re s ≥ . We say it is a C , -geodesic if it further satisfies the homogenous Monge–Amp`ere equation: (cid:0) π ∗ η + dd cS × X φ Re s (cid:1) n +1 = 0 . Now given any C , subgeodesic joining φ and φ , one may consider Hilb φ t := Z X g ( · , · ) e − φ t , which is a family of Hermitian inner products on H ( X, E + K X ) joining Hilb φ and Hilb φ . Note thatwe do not need any volume form in the above integral. Then Berndtsson’s quantized maximum principlesays the following, which in fact holds for subgeodesics with much less regularity; see [18, Proposition2.12]. Proposition 4.1. [9, Proposition 3.1]
Let [0 , ∋ t H t be the Bergman geodesic connecting Hilb φ and Hilb φ . Then one has H t ≤ Hilb φ t for t ∈ [0 , . We will now apply this result to the setting where E := mL − K X and g := h m ⊗ ω n . As a consequence,we obtain the following key estimate, which can be viewed as a weak version of the “partial C estimate”. Proposition 4.2.
For any ε ∈ (0 , , there exist m = m ( X, L, ω, ε ) ∈ N such that E ( φ ) ≤ E m (cid:0) ((1 − ε ) φ ) ( m ) (cid:1) + ε sup φ for any m ≥ m and any φ ∈ H ( X, ω ) .Proof. Since the statement is translation invariant, we assume that sup φ = 0. Let [0 , ∋ t φ t be a C , geodesic connecting 0 and φ , with φ = 0 and φ = φ . The geodesic condition implies that φ t isconvex in t so we have ˙ φ := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 φ t ≤ φ ≤
0. Put ˜ φ t := (1 − ε ) φ t . Observe that ( he − ˜ φ t ) m ⊗ ω n gives rise to a family of Hermitian metricson mL − K X , which is in fact a C , subgeodesic whenever m satisfies mεω ≥ − Ric( ω ). Indeed, let S := { < Re s < } ⊂ C be the unit strip and let π : S × X → X denote the projection to the secondcomponent. Then ( he − ˜ φ Re s ) m ⊗ ω n induces a Hermitian metric on π ∗ ( mL − K X ) over S × X whosecurvature form satisfies π ∗ ( mω + Ric( ω )) + m (1 − ε ) dd cS × X φ Re s ≥ mεω ≥ − Ric( ω ). It then follows from Proposition 4.1 that H m,t ≤ H ˜ φ t m for t ∈ [0 , , where [0 , ∋ t H m,t is the Bergman geodesic in P m ( X, L ) joining H m and H (1 − ε ) φm with H m, = H m and H m, = H (1 − ε ) φm . So we obtain that E m ( F S ( H m,t )) ≥ E m ( F S ( H ˜ φ t m )) for t ∈ [0 , , K. ZHANG with equality at t = 0 ,
1. Fixing an H m -orthonormal basis { s i } of R m , then by (4.2) we obtain that E m (cid:0) ((1 − ε ) φ ) ( m ) (cid:1) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E m ( F S ( H m,t )) ≥ ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E m ( F S ( H ˜ φ t m )) = 1 − εd m Z X ˙ φ (cid:18) d m X i =1 | s i | h m (cid:19) ω n , where the last equality is from a direct calculation using the definition of E m . Now by the first orderexpansion of Bergman kernels going back to Tian [38] (with respect to the background metric ω ), one has P d m i =1 | s i | h m d m ≤ − ε ) V for all m ≫
1. So we arrive at (recall ˙ φ ≤ E m (cid:0) ((1 − ε ) φ ) ( m ) (cid:1) ≥ V Z X ˙ φ ω n = E ( φ ) , where the last equality follows from the well-known fact that E is linear along the geodesic φ t . Thiscompletes proof. (cid:3) Remark 4.3.
After the appearance of this work on arXiv, the author was informed by Berndtssonthat Proposition 4.2 also follows from the fact that E m ( F S ( H ˜ φ t m )) is convex in t. And Berman kindlycommunicated to the author that, using Berndtsson’s convexity, our estimate is essentially contained in[7]; see in particular (3.4) in loc. cit. The author is very grateful to them for communications! But weneed to emphasize that our proof here is slightly different, with a small advantage that it can be directlygeneralized to the weighted setting to treat soliton type metrics; see also Remark 5.3.One can also bound E from below in terms of E m on the Bergman space B m ( X, ω ). This direction isalready known; see [6, Lemma 7.7] or [34, Lemma 5.2]. We record it here for completeness.
Proposition 4.4.
For any ε > , there exists m = m ( X, L, ω, ε ) ∈ N such that E m ( φ ) ≤ (1 − ε ) E ( φ ) + ε sup φ + ε. for any m ≥ m and φ ∈ B m ( X, ω ) . Proving δ = δ A In this section we prove our main results. Firstly, we prove Theorem 2.2 in the case where L is a bonafide ample line bundle, so that we can apply quantization techniques. Theorem 5.1.
Let L be an ample line bundle, then one has δ A ( L ; θ ) = δ ( L ; θ ) Proof.
The proof splits into two steps.
Step 1: δ A ( L ; θ ) ≤ δ ( L ; θ ).In the view of (4.4), it suffices to show that, for any λ ∈ (0 , δ A ( L ; θ )) one has δ m ( L ; θ ) > λ for all m ≫
1. In other words, for any m ≫
1, we need to find some constant C m,λ > Z X e − λ ( φ − E m ( φ )) dµ θ < C m,λ for all φ ∈ B m ( X, ω ) . Assume that sup φ = 0. For any small ε >
0, by Proposition 4.4 and H¨older’s inequality, Z X e − λ ( φ − E m ( φ )) dµ θ ≤ Z X e − λ ( φ − (1 − ε ) E ( φ ))+ λε dµ θ = e λε · Z X e − λ (1 − ε )( φ − E ( φ )) · e − λεφ dµ θ ≤ e λε (cid:18) Z X e − λ (1 − ε )1 − λεα ( φ − E ( φ )) dµ θ (cid:19) − λεα (cid:18) Z X e − αφ dµ θ (cid:19) λεα holds for all m ≥ m ( X, L, ω, ε ), where α ∈ (0 , α ( L ; θ )) is some fixed number. We may fix ε ≪ λ (1 − ε )1 − λεα < δ A ( L ; θ ) . Then by (2.4) and (3.1), there exist C λ > C α > Z X e − λ ( φ − E m ( φ )) dµ θ < e λε ( C λ ) − λεα ( C α ) λεα for all φ ∈ B m ( X, ω ) whenever m is large enough. This proves the assertion. Step 2: δ A ( L ; θ ) ≥ δ ( L ; θ ) . QUANTIZATION PROOF OF THE UNIFORM YAU–TIAN–DONALDSON CONJECTURE 9
It suffices to show that, for any λ ∈ (0 , δ ( L )), there exists C λ > Z X e − λ ( φ − E ( φ )) dµ θ < C λ for any φ ∈ H ( X, ω ).Again assume that sup φ = 0. Fix any number α ∈ (0 , α ( L ; θ )). Fix p > p ∈ (1 , p ). Let also ε > φ := (1 − ε ) φ. Then byProposition 4.2 and the generalized H¨older inequality, for any m ≥ m ( X, L, ω, ε ), we can write Z X e − λ (cid:0) φ − E ( φ ) (cid:1) dµ θ ≤ Z X e − λ (cid:0) φ − E m ( ˜ φ ( m ) ) (cid:1) dµ θ = Z X e λ (cid:0) ˜ φ ( m ) − ˜ φ (cid:1) · e − λ (cid:0) ˜ φ ( m ) − E m ( ˜ φ ( m ) ) (cid:1) · e − λεφ dµ θ ≤ (cid:18) Z X e √ m ( ˜ φ ( m ) − ˜ φ ) dµ θ (cid:19) λ √ m (cid:18) Z X e − λ (cid:0) ˜ φ ( m ) − Em (˜ φ ( m )) (cid:1) − λ √ m − λεα dµ θ (cid:19) − λ √ m − λεα (cid:18) Z X e − αφ dµ θ (cid:19) λεα ≤ ( d m ) λm (cid:18) Z X e − √ mψ √ m − ω n (cid:19) λ √ m − λm (cid:18) Z X e − λ (cid:0) ˜ φ ( m ) − Em (˜ φ ( m )) (cid:1) − λ √ m − λεα dµ θ (cid:19) − λ √ m − λεα (cid:18) Z X e − αφ dµ θ (cid:19) λεα , where we used (2.2) and (4.1) in the last inequality. We now fix ε ≪ m ≫ m ( X, L, ω, ε ) such that √ m √ m − < p and λ − λ √ m − λεα < δ m ( L ; θ ) . Then by (2.3), (4.3) and (3.1) there exist A m > C m,λ > C α > φ = 0) such that Z X e − λ ( φ − E ( φ )) ω n < ( d m ) λm · ( A m ) λ √ m − λm · ( C m,λ ) − λ √ m − λεα · ( C α ) λεα . Note that all the constants are uniform, independent of φ . So we finally arrive at R X e − λ ( φ − E ( φ )) ω n < C λ for some uniform C λ >
0, as desired. (cid:3)
Proof of Theorem 2.2.
Since δ ( L ; θ ) = δ A ( L ; θ ) holds for any ample line bundle, by rescaling, it holdsfor any ample Q -line bundle. Now by the continuity of δ and δ A in the ample cone (cf. [42]), the sameassertion holds for any ample R -line bundle. (cid:3) Proof of Theorem 2.3.
The result follows from Theorem 2.2 and Corollary 3.4. (cid:3)
Proof of Theorem 2.4.
The result follows from Theorem 2.2 and Corollary 3.6. (cid:3)
By Proposition 3.5 we also obtain an algebraic characterization of the coercivity threshold of theentropy. One should compare this with the non-Archimedean formulation [12, (2.9)] proposed by Berman.
Corollary 5.2.
For any ample R -line bundle L one has δ ( L ) = sup (cid:26) λ > (cid:12)(cid:12)(cid:12)(cid:12) ∃ C λ > s.t. H ( φ ) ≥ λ ( I − J )( φ ) − C λ for all φ ∈ H ( X, ω ) (cid:27) . Remark 5.3.
Finally we explain how to generalize our approach to the coupled KE/soliton case consid-ered in [34], which then yields a uniform YTD theorem for the existence of coupled KE/soliton metrics.The extension to the coupled KE case is straightforward: one only needs to replace φ and E ( φ ) by P i φ i and P i E ω i ( φ i ) respectively, and then slightly adjust the proof of Theorem 5.1. For the more generalcoupled soliton case, essentially one only needs to replace E by its “ g -weighted” version, E g , and thenadjust Propositions 4.2 and 4.4 accordingly, which can be done with the help of [8, Proposition 4.4], theasymptotics for weighted Bergman kernels. Then the argument goes through almost verbatim. See ourprevious work [34] for more explanations. The details are left to the interested reader. Acknowledgments.
The author is grateful to Gang Tian for inspiring conversations during thisproject. He also thanks Chi Li, Yanir Rubinstein, Yalong Shi, Feng Wang, Mingchen Xia and XiaohuaZhu for reading the first draft and for many valuable comments. Special thanks go to Bo Berndtssonand Robert Berman for letting me know of an alternative proof of Proposition 4.2 and also to S´ebastienBoucksom for clarifying some points in [2]. The author is supported by the China post-doctoral grantBX20190014.
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