Uniform K-stability of pairs Gang Tian Peking University
Let G = SL(N + 1, C) with two representations V, W over Q. For any v V\{0} and any one-parameter subgroup λ of G, we can associate a weight w λ (v) to be the unique integer defined by lim t 0 t wλ(v) λ(t)(v) = v 0 0.
Let v V\{0} and w W\{0}. (S. Paul) (v, w) is called K-semistable if for any oneparameter subgroup λ, w λ (w) w λ (v).
Paul/Boucksom-Hisamoto-Jonsson showed: The K-semistability of (v, w) is equivalent to G[v, w] G[v, 0] =. If V = C is trivial, then (1, w) is K-semistable if and only if 0 is not in the closure of the orbit Gw, i.e., w is semistable in the Geometric Invariant Theory.
To introduce the K-stability, we recall a few facts: First. If T be a maximal algebraic torus of G and M Z be its character lattice, then for any v 0 in V, we can associate a weight polytope N (v) in M R, where M R = M Z Z R.
Secondly, there is a natural representation gl = gl(n +1, C): G gl(n + 1, C) gl(n + 1, C) : (σ, B) σb. For I gl, N (I) is the standard N-simplex which contains the origin.
Define the degree q of V to be min{ k Z k > 0, N (v) k N (I), v V\{0} }. Clearly, we have q w λ (I) w λ (v).
The following definition is due to S. Paul. (v, w) is called K-stable if it is K-semistable and w λ (w) < w λ (v) whenever q w λ (I) < w λ (v).
Here is our main theorem: Theorem 0.1 If (v, w) is K-stable, then there is an integer m > 0 such that for any one-parameter subgroup λ, m (w λ (v) w λ (w)) w λ (v) q w λ (I). That is, (v, w) is uniformly K-stable.
We denote by a Hermitian norm on either V or W and define p v,w (σ) = log σ(w) 2 log σ(v) 2. As a corollary, we have Theorem 0.2 If (v, w) is K-stable, then there is an integer m > 0 such that m p v,w (σ) q log σ 2 log σ(v) 2.
Let us explain motivations for above theorems. Let (M, L) be a polarized projective manifold. A Kähler metric is locally given by ω = 1 n i,j=1 g i j dz i d z j, where z 1,, z n are holomorphic coordinates. Also, being Kähler, dω = 0, so ω represents a cohomology represented, referred as the Kähler class [ω].
A fundamental problem in complex geometry is the existence of Kähler metric ω on (M, L) such that [ω] = 2π c 1 (L) and s(ω) is constant, where s(ω) denotes the scalar curvature. In general, such a metric may not exist. A geometric condition is needed, that is, the K-stability of (M, L).
Fix a Kähler metric ω 0 with Kähler class 2πc 1 (L), other metrics with the same Kähler class are of the form: ω ϕ = ω 0 + 1 ϕ. A Kähler metric ω ϕ has constant scalar curvature if and only if ϕ is a critical point of Mabuchi s K-energy: M ω0 (ϕ) = 1 V 1 0 M ϕ (Ric(ω tϕ ) µ ω tϕ ) ω n 1 tϕ dt.
By the Kodaira embedding theorem, for l sufficiently large, a basis of H 0 (M, L l ) gives an embedding: φ l : M CP N, where N = dim C H 0 (M, L l ) 1. Any other basis gives an embedding of the form σ φ l, where σ G = SL(N + 1, C).
Given an embedding M CP N by L l, we have an induced function on G: F(σ) = M ω0 (ψ σ ), where ψ σ is defined by 1 l σ ω F S = ω 0 + 1 ψ σ.
Similarly, we can define J on G by where J ω0 (ϕ) = n 1 i=0 i + 1 n + 1 J(σ) = J ω0 (ψ σ ), M 1 ϕ ϕ ω i 0 ω n i 1 ϕ. Note that both F and J are well-defined.
Assuming a conjectured analytic estimate, we expect that the existence of constant scalar curvature metrics on (M, L) can be reduced to the properness of F modulo J with respect to L l for a sufficiently large l. The properness means: F is bounded from below and F(σ i ) whenever J(σ i ), {σ i } G. In my 1997 paper, it was referred as CM-stability of M with respect to L l.
As an application of our main result, we have Theorem 0.3 Let (M, L) be a polarized projective manifold which is K-stable with respect to L l. Then there are positive constants δ and C, which may depend on l, such that F(σ) δ J(σ) C on G. Clearly, this implies the properness of F modulo J with respect to L l.
Let me explain how to prove this theorem. First we recall a previous result of mine which relates the K-stability to the asymptotic behavior of the K-energy: If (M, L) is K-stable with respect to L l, then F is proper along any one-parameter subgroup λ of G.
Next we recall the Chow coordinate and Hyperdiscriminant of M: Note that G(N n, N + 1) consists of all subspaces in CP N of dimension (N n 1). We define Z M = { P G(N n, N + 1) P M }.
This Z M is an irreducible divisor of G(N n, N + 1) and determines a homogeneous polynomial R M C[M (n+1) (N+1) ], deg(r M ) = (n + 1)d. Here, M k l denotes the space of all k l matrices. We call R M the Chow coordinate of M.
Next consider the Segre embedding: M = M CP n 1 CP N CP n 1 P(M n (N+1) ). We define Y M = { H P(M n (N+1) ) p, s.t. T p M H }.
This Y M is a divisor in P(M n (N+1) ) and determines a homogeneous polynomial M C[M n (N+1) ], deg( M ) = (n(n + 1) µ)d. Here, µ is determined by c 1 (M). We call M the hyperdiscriminant of M.
Put r = (n + 1)d d, where d = (n(n + 1) µ)d, and V = C r [M (n+1) (N+1) ], W = C r [M n (N+1) ]. Here C r [C k ] denotes the space of homogeneous polynomials of degree r on C k.
For L l, set R(M) = R d M and (M) = (n+1)d M, then we associate M with the pair (R(M), (M)) V W, S. Paul proved: F(σ) a n p R(M), (M) (σ) C, σ G, Here, a n > 0 and C are uniform constants.
For each λ, p R(M), (M) (λ(t)) is equal to (w λ (R(M)) w λ ( (M))) log t 2 + O(1). Since F is bounded from below, we see w λ (R(M)) w λ ( (M)) 0. So (R(M), (M)) is K-semistable as a pair.
Furthermore, we have lim F(λ(t)) = w λ(r(m)) w λ ( (M)) > 0. t 0 It was proved before. J(σ) p R(M),I r(σ) C, σ G. where I gl is the identity and I r U = gl r.
For any one-parameter subgroup λ, we also have p R(M),I r(λ(t)) = (r w λ (I) w λ (R(M))) log t 2 + O(1). It follows lim J(λ(t)) = w λ(r(m)) r w λ (I) > 0. t 0
Summarizing the above, we see that (R(M), (M)) is K- stable as a pair. Then it follows from our main theorem that F is proper modulo J with respect to L l.
Now we discuss the proof of our main result. For your convenience, we restate it here: If (v, w) is K-stable, then there is an integer m > 0 such that for any one-parameter subgroup λ, m (w λ (v) w λ (w)) w λ (v) q w λ (I).
First, since (v, w) is K-semistable, we have G[v, w] G[v, 0] =. This implies p v,w (σ) = log σ(w) 2 log σ(v) 2 c.
Next, we interpret the K-stability in terms of effectiveness of certain line bundle. Let π : P(V, W) P(V W) be the blow-up of P(V W) along subvarieties P(V {0}) and P({0} W). The G-action can be lifted to P(V, W), so does the orbit G[v, w]. Let X be the closure of the orbit π 1 (G[v, w]) in P(V, W).
Two natural projections: and π V : P(V, W) P(V {0}) π W : P(V, W) P({0} W)). Define a line bundle over X: L = π V H 1 V π W H W.
There is a natural section S v,w of L over G([v, w]): S v,w (σ) = π V σ(v) π W σ(w) 1, σ G. The Hermitian norms on V and W induce a Hermitian metric on L, we observe p v,w (σ) = log S v,w 2 L.
Since p v,w is bounded from below, we have sup σ G S v,w L (σ) <. Hence, S v,w can be extended to be a holomorphic section over X.
The K-stability can be expressed in terms of S v,w in a similar way. Set U = gl q and u = I q. Then we have p v,u (σ) = q log σ 2 log σ(v) 2 c. Here σ is actually the Hilbert-Schmidt norm of σ gl.
As above, we have a blow-up variety P(V, U) and a line bundle L = π V H 1 V π U H U. We also have a holomorphic section S v,u on G([v, u]) and a Hermitian metric L on L satisfying: p v,u (σ) = log S v,u 2 L.
It suffices to prove that if (v, w) is K-stable, then there is an integer m > 0 such that for any one-parameter subgroup λ, S v,w m L C λ S v,u L. Since each one-parameter subgroup λ is contained in a certain maximal torus T G, we only need to prove the above inequality on each maximal torus.
Let T be the compactification of T([v, w], [v, u]) P(V, W) P(V, U). Both L and L can be pulled back to T and have two induced sections, still denoted by S v,w and S v,u. We need: There is an integer m > 0, independent of T, such that S v,w m L C T S v,u L on T.
We prove this in two steps: 1. For any irreducible divisor D T\T([v, w], [v, u]), S v,w vanishes along D whenever S v,u = 0 on D. This is done by using a variant of Richardson s lemma and the K-stability. 2. There is an upper bound on vanishing order of S v,u along any irreducible D. This is done by using the Lelong equation and the fact that such T s form a bounded family.