The Yau-Tian-Donaldson Conjectuture for general polarizations Toshiki Mabuchi, Osaka University 2015 Taipei Conference on Complex Geometry December 22, 2015
1. Introduction 2. Background materials Table of Contents 3. The invariant f 1 for sequences of test configurations 4. Strong K-stability Quantized version Stability Theorem Existence Problem 5. References
1. Introduction Setting-up: (X,L) : a polarized algebraic manifold; T : a maximal algebraic torus in Aut(X). As to the existence of extremal Kähler metrics, one can propose: Conjecture: If (X,L) is K-stable relative to T, then the polarization class c 1 (L) admits an extremal Kähler metric. If T is trivial, this is known as the Yau-Tian-Donaldson Conjecture: Conjecture: If (X,L) is K-stable, then the polarization class c 1 (L) admits a CSC Kähler metric. In the case L = (K X ) -1, the Yau-Tian-Donaldson Conjecture was affirmatively solved by Chen-Donaldson-Sun and Tian. However, if L is general, this is still open.
2. Background Materials
A test configuration (X,L,ϕ) for (X,L) Fix a Hermitian metric h for L such that ω = c 1 (L;h) is Kähler. Let X C P*(V) be a C*-invariant closed subset for the C*-action C* (C P*(V)) (t, (z,p)) (tz,ϕ(t)p), for a 1-PS ϕ: C* GL(V), where GL(V) acts naturally on the set P*(V) of all hyperplanes in V passing through the origin, and we usually take V = V l1 = Γ(X,Ll1 ) assuming that V has a natural metric structure induced by h and ω such that S 1 C*acts isometrically on V. A triple µ = (X,L,ϕ) is called a test configuration for (X,L) if (1) L is the restriction to X of the pullback pr 2 *O P*(V) (1) of the hyperplane bundle on P*(V) on which C* acts naturally; (2) (X t, L t ) (X, L l1 ), t, for some positive integer l 1 independent of the choice of t. This triple µ = (X,L,ϕ) is called the De Concini-Procesi family for ϕ, and l 1 is the exponent of the test configuration µ = (X,L,ϕ). Moreover, µ = (X,L,ϕ) is called trivial (resp. normal) if ϕ SL is trivial (resp. if X is normal).
The Chow norm on W j by Zhang d(j) := degree of X in P*(V j ) embedded by L l j, where V j := Γ(X,L l j ), lj = j l 1, n = dim X, W j := {S d(j) (V j )*} n+1. Let 0 CH j (X) W j be the Chow form for the irreducible reduced algebraic cycle X on P*(V j ), so that [CH j (X)] P(W j ) is the Chow point for the cycle X. Consider the Chow norm : W j R, 0 w R, w W j.
The Donaldson-Futaki invariant F 1 = F 1 (µ) for µ = (X,L,ϕ) N j := dim Γ(X,L l j ) = dim Γ(X0,L 0 j ), where l j = j l 1. w j := weight of the C*-action on det Γ(X 0,L 0 j ) w j / (l j N j ) (j >>1) = F 0 (µ) + F 1 (µ) l j -1 + F 2 (µ) l j -2 + F 1 = F 1 (µ) is called the Donaldson-Futaki invariant for the test configuration µ = (X,L,ϕ) of (X,L).
K-stability and Li-Xu s pathology (X,L) is called K-stable if the following conditions are satisfied: (1) F 1 (µ) 0 for all test configurations µ = (X,L,ϕ) for (X,L). (2) If F 1 (µ) = 0 and µ is normal, then µ is trivial. We here destinguish K-stability from K-polystability. In the original definition of K-stability by Donaldson, (2) is stated as If F 1 (µ) = 0, then µ is trivial. However, Li-Xu gave an example of a nontrivial test configuraton µ = (X,L,ϕ) for (P 1,O P1 (3)) such that F 1 (µ) vanishes, and that the normalization of µ is trivial. Hence the definition of K-stability is given in this form.
Characterization of F 1 in terms of the Chow norm For a test configuration µ = (X,L,ϕ) of exponent l 1 = 1, we consider the sequence of test configurations µ j = (X,L j,ϕ j ), j =1,2,, such that ϕ j : C* GL(V j ) is induced by ϕ, where by l j = j l 1 and l 1 = 1, we have l j = j, V j := Γ(X,L lj ) (= Γ(X,L j )) and W j := {S d(j) (V j )*} n+1. For the special linear form ϕ j SL : C* SL(V j ) (modulo finite group) of ϕ j, by using the Chow norm, we set ν(s) := l j log ϕ j SL (exp(s)) CH j (X), s R. Let q j be the (possibly rational) Chow weight of the C*-action by ϕ j SL on the line C CH j (X 0 ). Then by writing F i (µ) as F i shortly, we obtain l j q j = lim s - dν(s)/ds = (n +1)! c 1 (L) n [X] (F 1 + F 2 l j -1 + F 3 l j -2 + ).
3. The invariant f 1 for sequences of test configurations
Norms for the the infinitesimal generator of ϕ For a test configuration µ = (X,L,ϕ), we consider the infinitesimal generator u for ϕ SL, so that ϕ SL (exp(s)) = exp(su), s R. Then such a ϕ SL is called the 1-PS generated by u, and is written as ϕ u. Let b 1, b 2,, b N be the weights of the representation by ϕ, so that ϕ SL (t), t R +, is written as a diagonal matrix with the γ-th diagonal elements t bγ, γ=1,2,, N. Let n be the dimension of X, and l be the exponent of the test configuration µ = (X,L,ϕ). Put u 1 := l -1 ( b 1 + b 2 + + b N ), u := l -1 max{ b 1, b 2,, b N }.
Definition of f 1 : M R {- } Consider a sequence {µ j } of test configurations µ j = (X j,l j,ϕ j ) such that the exponent l j of µ j satisfies l j + as j. Let M be the set of all such sequences {µ j }. For s R, we define ν j (s) := ( u j / u j 1 ) l j log ϕ j SL (t) CH j (X) = ( u j / u j 1 ) l j log exp(su j / u j ) CH j (X), where t = exp(s/ u j ). We can then define (cf. [M1]) f 1 ({µ j }) := lim s - lim j dν j /ds. If the double limit commutes and if lim j u j 1 exists as a positive real number r > 0, then by the characterization of F 1 in terms of the Chow norm, we obtain (compare this with Szekelyhidi s approach) r f 1 ({µ j }) = (n+1)! c 1 (L) n [X] lim j F 1 (µ j ).
Some remark on f 1 For instance, for a test configuration µ = (X,L,ϕ) for (X,L) of exponent 1, let µ j = (X j,l j,ϕ j ), j =1,2,, be the test configurations such that X j = X, L j = L j and that ϕ j : C* GL(V j ) is induced by ϕ: C* GL(V), where V j := Γ(X 0,L j 0 ) and V := Γ(X 0,L 0 ). Replacing {µ j } by its subsequence if necessary, we may assume that r = lim j u j 1 0 exists. Then r f 1 ({µ j }) = (n+1)! c 1 (L) n [X] F 1 (µ). Note that, for the test configuration in Li-Xu s pathology, r = 0.
4. Strong K-stability
Definition of strong K-stability For a polarized algebraic manifold (X,L), let M be as above, i.e., the set of all sequences {µ j } of test configurations µ j = (X j,l j,ϕ j ), j =1,2,, for (X,L) such that the exponent l j of µ j satisfies l j + as j. Let us now introduce the following stronger concept: Definition: (1) (X,L) is called strongly K-semistable, if f 1 ({µ j }) 0 for all {µ j } M. (2) Let (X,L) be strongly K-semistable. Then (X,L) is called strongly K-stable, if the equality f 1 ({µ j }) = 0 for {µ j } M implies that µ j are trivial for all sufficiently large j.
A quantized version of the Conjecture Li-Xu s pathology doesn t occur in our new definition of f 1. Actually, for their example of a test configuration, f 1 = -. Hence we propose the following version of the Yau-Tian-Donaldson Conjecture: Conjecture: If (X,L) is strongly K-stable, then c 1 (L) admits a constant scalar curvature Kähler metric. By Donaldson, every constant scalar curvature Kähler metric can be approximated by a sequence of balanced metrics. In other words, a balanced metric is viewed as a quantized version of a constant scalar curvature Kähler metric. In this sense, the following quantized version of the above conjecture makes sense, and we can actually prove it (cf. [MN], Progr. Math. 308, Kobayashi memorial vol., Birkhäuser, 2015) Theorem (Nitta-M.): If (X,L) is strongly K-stable, then for a sufficiently large l, (X,L l ) admits a balanced metric, i.e., (X,L) is asymptotically Chow-stable.
Proof of the Theorem For contradiction, we assume that (X,L) is not asymptotically Chow-stable. Then we have an increasing sequence of exponents l j 1 such that (X,L l j ) are not Chow-stable. For Vj := Γ(X,L l j ) and the Kodaira embedding X P * (V j ), we have a destabilizing nontrivial one-parameter group ϕ j : C * SL(V j ) such that the Chow weight q j for X P * (V j ) satisfies q j 0. We now consider the De Concini-Procesi family µ j = (X j,l j,ϕ j ) for ϕ j. Then 0 l j q j = lim s - d{ l j log ϕ j SL (exp(s)) CH j (X) }/ds On the other hand, by setting exp(s/ u j ) = t = exp(s) we have s = s/ u j, and hence in view of ν j (s) = ( u j / u j 1 ) l j log ϕ j SL (t) CH j (X), we obtain dν j (s)/ds = ( u j / u j 1 ) d{ l j log ϕ j SL (t) CH j (X) }/ds = ( u j / u j 1 ) d{ l j log ϕ j SL (t) CH j (X) }/ds ds/ds = (1/ u j 1 ) d{ l j log ϕ j SL (t) CH j (X) }/ds (1/ u j 1 ) lim s - d{ l j log ϕ j SL (t) CH j (X) }/ds 0 for all s R. Hence lim j dν j (s)/ds 0 for all s R. Thus f 1 ({µ j }) = lim s - lim j dν j (s)/ds 0. By the strong K-stability of (X,L), we see that µ j are trivial for all sufficiently large j. Hence ϕ j are trivial for all sufficiently large j in contradiction.
Strong K-stability for CSC polarization Assume that c 1 (L) on X admits a CSC Kähler metric ω CSC. We then choose ω CSC as the reference metric ω = c 1 (L;h). Let {µ j } M, so that we write µ j = (X j,l j,ϕ j ). Recall that (1) dν j (s)/ds s = 0 = (-1/ u j 1 ) l j X (Σ α b α σ α 2 ) (Σ α σ α 2 ) -1 ω FS n where {σ 1, σ 2,, σ Nj} is the orthonormal basis for V j, and we put ω FS = ( - 1/2π) log (Σ α σ α 2 ). By the theorem of Yau-Tian-Zelditch-Catlin, the asymptotic Bergman kernel for ω = ω CSC has the following expansion: (2) Σ α σ α h 2 = (1/n!) {l j n + (S ω /2) l j n - 1 + O(l j n - 2 ) }, where O(l j n - 2 ) is a term having a C 2 orm bounded by a constant independent of j. Then by taking the ( - 1/2π) log of the both sides, we obtain ω FS - l j ω = O(l j - 2 ), i.e., (3) (l j - C 1 l j - 2 ) ω ω FS (l j + C 1 l j - 2 ) ω for some positive constant C 1 independent of j. Then by (1), (2) and (3), dν j (s)/ds s = 0 C 2 l j - 1 for some positive constant C 2 independent of j. Since the function lim j dν j (s)/ds is non-decreasing in s, f 1 ({µ j }) = lim s - lim j dν j (s)/ds lim j C 2 l j - 1 = 0. Hence (X,L) is strongly K-semistable (cf. [M1]). By a more delicate analysis, strong K-stability of (X,L) can also be proved.
An outline of our approach (work in progress) strong K-stability CSC Kähler metric Fix a Hermtian metric h for L such that ω = c 1 (L;h) is Kähler. Assume that (X,L) is strongly K-stable. Then by the Theorem, (X,L) is asymptotically Chow-stable, i.e., there exists an increasing sequence 1 < l 1 < l 2 < < l j < such that ω j = c 1 (L;h j ) is a balanced metric for (X,L l j ). Hence, we have an ONB {τ 1,τ 2,, τ Nj }for (V j,ρ j ) such that ( τ 1 2 + τ 2 2 + + τ Nj 2 ) hj is a constant, where ρ j is the Hermitian inner product for V j defined by ρ j (σ, σ ) = X (σ, σ ) hj ω j n, σ, σ V j, and we define another Hermitian inner product ρ 0 for V j by ρ 0 (σ, σ ) = X (σ, σ ) h ω n, σ, σ V j. By choosing a suitable ONB {σ 1, σ 2,, σ Nj } for (V j,ρ 0 ), we can write ρ 0 as an N j N j identity matrix, and ρ j is also written as an N j N j diagonal matrix in SL with the α-th daigonal element λ α > 0. Put b α := (1/2) log λ α. Then b 1 +b 2 + + b Nj = 0. Approximating each b α by a sequence of rational numbers, we may assume from the beginning that each b α is a rational number. By choosing a positive integer m j, we have that
b α := m j b α, α = 1,2,, N j, are all integers. Let u j (resp. u j ) be the diagonal matrices in sl(v j ) with the α-th diagonal element b α (resp. b α ). Let ϕ j : C* SL(V j ) be the algebraic group homomorphism with the fundamental generator u j, so that ϕ j (t) = t uj. In the definition ν j (s) := ( u j / u j 1 ) l j log exp(su j / u j ) CH j (X) of ν j (s), the function ν j (s) doesn t change even if we replace u j by u j, so that we can write ν j (s) := ( u j / u j 1 ) l j log exp(su j / u j ) CH j (X). On the other hand, u j can be viewed as the distance between ρ j and ρ 0. Let µ j = (X j,l j,ϕ j ) be the De Concini-Procesi family for ϕ j. Put d = sup j u j. Then the following two cases can occur. (Case 1) d = + (Case 2) d < + Suppose that Case 1 occurs. Then f 1 ({µ j }) = 0, and by the strong K-stability of (X,L), µ j would be trivial for j >> 1, in contradiction. I explain the details about this later. (cf. [M2], Springer Proc. in Math. And Stat.)
We therefore consider the Case 2. In this case, ρ j is not so far away from ρ 0 (and hence ω j is not so far away from ω). In view of the following theorem, Tian s peak sections method will allow us to obtain the convergence of the sequence of balanced metrics ω j to a CSC Kähler metric ω. Theorem [M3]: For a test configuration µ = (X,L,ϕ), of exponent l, for (X,L), we have the following: (1) a C*-equivariant desingularizations ι: X X and η: X Y, (2) a test configuration κ = (Y,Q,ψ), of exponent 1, for (X,L), (3) an effective divisor D on X sitting over the origin such that L:= ι*l, Q:= η*q and D := l -1 D satisfy (4) L = O X (D) Q l, and formally L = {O X (D) Q} l (5) l deg ι * D + deg η * D C 1 u (6) v C 2 u where u, v are fundamental generators for ϕ and ψ, respectively, and C 1 and C 2 are constants independent of the choice of µ and l. Remark: The pair (κ,d) is called an l-th root of µ = (X,L,ϕ).
The details for the Case 1 Recall that ϕ j (t) = exp(su j / u j ), t = exp(s / u j ), and that u j is a diagonal matrix with the α-th diagonal element b α. Note also that and b α := (1/2) log λ α. Hence ϕ j (t) ρ 0 s = 0 = ρ 0 ϕ j (t) ρ = exp(-u 0 s = - uj j ) ρ 0 Here exp(-u j ) is a diagonal matrix with the α-th diagonal element e -b α. However, exp(-u j ) acts on V j * by the contragradient representation. On the other hand, ρ 0 sitting in the space V j * V j * is written as an identity matrix I in terms of an ONB for ρ 0. Hence ϕ j (t) ρ 0 s = - uj can be viewed as a diagonal matrix t exp(u j ) I exp(u j ) whose α-th diagonal element is e 2b α = λα. Thus we obtain ϕ j (t) ρ = ρ 0 s = - uj j. Since the Chow norm takes the critical value at the balanced metric ρ j, the derivative of the function ν j (s) := ( u j / u j 1 ) l j log exp(su j / u j ) CH j (X) at s = - u j vanishes: dν j (s)/ds s = - uj = 0 for all j.
By the assumption of Case 1, d = +. Hence replacing u j by its subsequence if necessary, we may assume that u j, j = 1,2,, is a monotone increasing sequence diverging to +. We now fix an arbitrary j. Then by the monotonicity of the sequence, we have u j u j for all j j. Hence - u j - u j. Since dν j (s)/ds is a non-decreasing function, we obtain 0 = dν j (s)/ds dν s = - uj j (s)/ds s = - for all j j. uj By letting j, we obtain lim j dν j (s)/ds 0 for all s with - u j s +. Now let j. Then by u j +, it follows that lim j dν j (s)/ds 0 for all s with - s +. We here let s -. Then f 1 ({µ j }) = lim s - lim j dν j (s)/ds 0 By the strong K-stability of (X,L), f 1 ({µ j }) = 0, and µ j are trivial for all sufficiently large j. This is a contradiction, since in approximating b α by a sequence of rational numbers, we may choose them in such a way that u j (and hence u ) are nontrivial for all j, and hence µ j can be assumed to be nontrivial for all j in contradiction. Therefore, Case 1 cannot occur.
References [M1] M. : The Donaldson-Futaki invariants for sequences of test configurations, in Geometry and Analysis on Manifolds, Progr. Math. 308, Birkhäuser Boston, 2015, 395-403. [M2] M. : The Yau-Tian-Donaldson conjecture for general polarizations, I, to appear in Geometry and Topology of Manifolds, 10th Geom. Conf. for the friendship of China and Japan 2014, Springer Proc. in Math. and Stat. ed. by A. Futaki et al. [M3] M. : An l-th root of a test configuration of exponent l, preprint. [MN] Nitta and M. : Strong K-stability and asymptotic Chow stability, in Geometry and Analysis on Manifolds, Progr. Math. 308, Birkhäuser Boston, 2015, 405-411.
Thank you.