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3 Constant Scalar Curvature Kähler Metrics and Stabilities Li, Chi Supervisor: Prof. Tian, Gang School of Mathematical Sciences, Peking University June, 2007 Submitted in total fulfilment of the requirements for the degree of Master in Pure Mathematics

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7 Kähler Kähler-Einstein, Kähler- Einstein.. Futaki : Futaki. Futaki 0., Kähler-Einstein. Tian Kähler-Einstein K-. K- Futaki., Futaki, K. Chow. Donaldson Bergman Chow.. K- K-, Futaki. Donaldson Chow K-., Chow balanced, Chow Donaldson. Donaldson, Bergman. Bergman Hilbert, Donaldson Futaki Chow, Chow K-. Futaki Futaki,..,, Futaki, Bergman

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9 Abstract One of the major problems in Kähler geometry is the existence of Kähler-Einstein metrics. The existences of Kähler-Einstein metrics with nonpositive scalar curvature have been confirmed. But Kähler-Einstein metrics with positive scalar curvature do not necessarily exist. Futaki defined an important holomorphic invariant: Futaki invariant. The necessary condition for existence is the vanishing of Futaki invariant. Compared with the case of vector bundles, it is conjectured that the existence of Kähler-Einstein metric is equivalent to some kind of stability of the manifold in the Geometric Invariant Theory. Tian first proved that the existence of Kähler-Einstein metric with positive scalar curvature imply K- stability which he defined. In the definition of K-stability, the Futaki invariant plays a fundamental role. More generally, one considers the existence of Kähler metrics with constant scalar curvature. Both the Futaki invariant and K-stability extend to this case. There is also a notion of Chow stability. Donaldson proved that under some condition, the existence of constant scalar curvature metric implies asymptotic stability of the underlying manifold. Both stabilities have convex functionals and weights under group actions. In the case of K-stability, the functional is K- functional and the weight is just the Futaki invariant. Using the calculation of Donaldson, it can be showed that the asymptotic Chow stable implies semi K-stable. In this article, we explain from the definition of Chow stability to the criterion by balanced metrics, and we can get the Chow weight and the functionals which Donaldson used in his papers. We explain the main idea of the proof of Donaldson s theorem and how the expansion of Bergman kernel is used. We can use the Bergman kernel to give the asymptotic expansion of the Hilbert weight, get the algebraic definition of Futaki invariant by Donaldson and the fact that the Futaki invariant is the leading coefficient in the expansion of the Chow weight associated to a 1psg action, thus we explain the result that the asymptotic Chow stable implies semi K-stable. We also use the algebraic definition to give the computation of the Futaki invariant of complete intersections and test that the result is the same as that obtained using localization formula. This article is a summary of what I learned in this subject. Keywords: Constant scalar curvature metrics, Stability, Futaki invariant, Bergman kernel

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11 Abstract i iii v, 1 Chow balanced 5 Chow Bergman Donaldson Futaki, Hilbert Chow Futaki Hilbert Futaki K K A 35 A A.2 Bott-Chern Dolbeault

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13 , (, ω) n Kähler,, Kähler 1 ω = g 2π i jdz i d z j i,j g = i,j g i jdz i d z j Ricci c 1 (), 1 1 Ric(ω) = R 2π i jdz i d z j 2 = 2π z i z log det(g k l)dz i d z j j i,j i,j S = i,j g i j R i j = nric(ω) ωn 1 ω n Kähler Kähler-Einstein. Einstein, ω Kähler- Ric(ω) = λω (1.1) λ = 1, 0, 1. λ, c 1 (). c 1 () < 0 Aubin Yau. c 1 () = 0 Yau Calabi.. c 1 () > 0, λ = 1., Kähler-Einstein. Matsushima[16] Kähler-Einstein Aut() reductive. Futaki[10], Kähler-Einstein, 0., Hermitian-Einstein, Hitchin-Kobayashi. Yau Kähler- Einstein. Tian[21] K,. K- Futaki. ω c 1 (), -, h ω C () Ric(ω) ω = 1 h 2π ω, ω S n = ω h ω. ω Kähler-Einstein. Kähler [ω] H 1,1 (, R). φ C (), 1 ω φ = ω + 2π φ P (, ω) = {φ C () ω φ > 0}

14 K (, [ω]) = {ω φ φ P (, ω)} φ P (, ω) S(ω φ ). S S Kähler. V = ωn. S = 1 S(ω)ω n = nc 1()[ω] n 1, [] V [ω] n, [] [ω] H 1,1 (, R) H 2 (, Z), L, c 1 (L) = [ω]. (, L) polarized. h L Hermitian, Ric(h) = 1 2π log e L 2 h e L. φ C (), h φ = he φ, Ric(h φ ) = Ric(h) + 1 2π φ., Hermitian h, ω = Ric(h). L, k, Kodaira. I kl : P(H 0 (, kl) ) (1.2) z {s H 0 (, kl) s(z) = 0} I kl O(1) = kl. H0 (, kl) {s α }, H 0 (, kl) = C N k+1 I {sα} : CP N k (1.3) z [s 0 (z) : : s Nk (z)] CP N k Fubini-Study. {z α }, ω F S = 1 2π log(1 + N k α=1 z α 2 ) (1.4) Kähler 1 k (I {s α}) ω F S. Hermitian h, ω = Ric(h). 1 (I {sα}) 1 ω F S = ω + 2π k N k log s α 2 h k P k (, ω) = {φ = 1 N k k log s α 2 h {s α} H 0 (, kl) } (1.5) k α=0 α=0 K k (, ω) = { 1 k (I {s α}) ω F S {s α } H 0 (, kl) } Tian[20] Kähler ω k K k, {s α } h H 0 (, kl). ω k Bergman., P (, ω) 2

15 , P k (, ω). Kähler : Bergman (plurisubharmonic function)., ω. Donaldson[6] ω k K k, balanced. Zhang[25] balanced Chow. Luo[14], Paul[19]. Donaldson, Chow. Kähler-Einstein balanced..,, properness. Chow, φ C (), C () 0 φ {φ t } Fω(φ) 0 = 1 1 d V dt φ tωφ n t (1.6) 0 ω c 1 (L), F 0 ω Bott-Chern, A.7 {φ t }. K-, K-, Tian[21] Kähler-Einstein properness., Hilbert,.. Kähler,., proper, 0. Futaki,., Chow,, Bergman Donaldson., Futaki,, Futaki,.. Bott-Chern,. 3

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17 Chow balanced Kodaira (1.2), CP N. dim = n, deg = d. d = ωn F S, d V. Chow. Grassmannian Gr(N n 1, CP N ) CP N N n 1, m + 1 = (N n)(n + 1). Q, Plücker I P l : Gr(N n 1, CP N ) P( N n C N+1 ), IP lo(1) = det(q). Z() = {V Gr(N n 1, CP N ) V } Γ() = {(z, V ) Z() z V } Γ(CP N ) = {(z, V ) Gr(N n 1, CP N ) z V } π 1 π Γ() 2 Z() CP N π 1 Γ(CP N ) π 2 Gr(N n 1, CP N ) (2.1) Z() N n 1, π 2. π 1 Gr(N n 2, CP N 1 ), dimz() = dimγ() = dim + dimgr(n n 2, CP N 1 ) = (N n)(n + 1) 1 = m Gr = Gr(N n 1, CP N ). CP N U = CP N n 2 W = CP N n. F (U, W ) = {V Gr(N n 1, CP N ) U V W } = CP 1 U, W, U =, (W ) = d. F (U, V ) d (N n 1), (F (U, W ) Z()) = d. Z() Gr d. f H 0 (Gr, O(d)), Z() = f. O(d) det(q) = O(1) d. f Chow. [f] P[H 0 (Gr, O(d))] Chow. SL(N + 1, C) (CP N, O(1)), (Gr(N n 1, CP N ), O(1)), H 0 (Gr, O(d)) P[H 0 (Gr, O(d))] Chow, f H 0 (Gr, O(d)) SL(N + 1, C), f. Chow, f 0.,,, Hilbert.

18 2.2. f H 0 (Gr, O(d)), Chow f Ch log f 2 Ch = 1 log f 2 h ω m+1 D d Gr F S Gr D = Gr ωm+1 Gr. f 2 h d F S, ω Gr Plücker O(d) Gr Fubini-Study, 1. f Gr, log f 2 h d F S, f. cf 2 Ch = c f 2 Ch. : σ SL(N + 1, C), f σ σ f, f σ σ() Chow. SL(N + 1, C) F (σ) = log f σ 2 Ch (2.2) SU(N + 1), F (σ) S = SL(N + 1, C)/SU(N + 1). F (σ) S. S. sl(n + 1, C) = su(n + 1) + 1su(N + 1), SL(N + 1, C) = SU(N + 1) C σ(e t ) : C SL(N + 1, C). A 1su(N + 1) σ(e t ) = exp(ta) (2.3) A = A, tr(a) = 0 (2.4) Z = (Z 0,, Z N ) C N+1, Z 2 = N α=0 Z α 2. Z [Z] CP N. σ ω F S = ω F S + 1 2π d dt σ ω F S = φ σ φ σ ([Z]) = log 1 2π φ σ, φσ ([Z]) = d σ Z 2 log dt Z (Phong-Sturm[17], S. Paul[19]). d dt log f σ 2 Ch = (n + 1) φ σ (σ ω F S ) n = 2(n + 1) σ Z 2 Z 2 (2.5) = 2 Z σ AσZ σ Z 2 (2.6) σ() Z AZ Z 2 ωn F S (2.7).. Plücker, ω Gr P ( N n C N+1 ) Fubini-Study Gr. Φ σ σ ω Gr = ω Gr + 1 Φ 2π σ. d dt log f σ 2 Ch = 1 1 log f 2 h + 1) D d F S(m t=0 Gr 2π log Φ σ ωgr m ( 0) = (m + 1) 1 Φ σ ωgr m (2.8) D Z() 6

19 CHOW BALANCED Poincaré-Lelong : 1 2π log f 2 h = d F S Z() (2.1) 1 D π 1 π2ω m+1 Gr = ω n+1 F S π 1 π 1 Gr(N n 2, CP N 1 ). t (m + 1) 1 D π 1 π 2( Φ σ ω m Gr) = (n + 1) φ σ ω n η = (m + 1) 1 D π 1 π 2( Φ σ ω m Gr ) (n + 1) φ σ ω n (n,n), η σ(e t ) CP n 0, η, η = 0. (m + 1) 1 Φ σ ωgr m = (m + 1) 1 π D Z() D 2( Φ σ ωgr) m (π 2 ) Γ() = (m + 1) 1 π 1 π D 2( Φ σ ωgr) m = (n + 1) φ σ ωf n S (2.9) 2. (2.8) Z() F (σ) S,. (2.9) Z(),. (N + 1) Hermitian M() M() αβ = t = σ(e t ), (2.7) Z α Zβ Z 2 ωn F S (2.10) d dt log f σ 2 Ch = d (n + 1) d dt F 0 ω F S (φ σ ) = 2(n + 1)tr(M( t )A) (2.11), d V. F (σ) = log f σ 2 Ch = d (n + 1)F 0 ω F S (φ σ ) (2.12) 2.2. tr(m( t )A) t R. O(1), tr(m( t )A) t.. (2.3), d dt tr(m( t)a) = d Z σ AσZ (σ ω dt σ Z 2 F S ) n = 2 ( Z A 2 Z (Z AZ) 2 Z 2 Z 4 σ() 7 1 )ωf n S n 2π Z AZ Z 2 Z AZ Z 2 ω n 1 F (2.13) S

20 . σ. SU(N + 1), A = diag(λ 0,, λ N ), λ α R, N α=0 λ α = 0. (2.3) CP N v, {w α = Z α Z 0 α = 1,, N}, v = N (λ α λ 0 )w α w α α =1 Im(v) Killing, Re(v) = J(Im(v)). N θ A = Z AZ α=0 = λ α Z α 2 Z 2 N α=0 Z (2.14) α 2 ( ), CP N Fubini-Study Kähler, 2Im(v) 1 θ 2π A Hamilton, 2 i Im(v) ω F S = 1 2π dθ A 1 2π θ A = i v ω F S (2.15) Z A 2 Z Z 2 (Z AZ) 2 Z 4 = g F S (v, v) = v 2 = θ A 2 ω F S (2.16) CP N, CP N. v = v T + v T CP N = T T, 1 2π n Z AZ Z Z AZ 2 ω n 1 Z 2 F S = θ A 2 ωf n S = v T 2 ωf n S (2.17) (2.16),(2.17) (2.13) ( 4 ω n F S ) θ A 2 θ A 2 = v 2 v T 2 = v 2 0 (2.18) (2.18) (2.13) 0 v. O(1), tr(m( t )A) t. 3. Z α 2 F S = Z α 2 Z = Z α 2 2 β Z β 2 Z α CP N O(1) Z α 2 F S Fubini-Study. N N N λ 2 α Z α 2 F S ( λ α Z α 2 F S) 2 ( λ α Z α 2 F S) 2 ω F S = α=0 α=0 α=0 N λ α Z α h i i Z α hz α 2 F S α=0 O(1) Fubini-Study. h i i = h Study. (3.7) (3.8). 8 (2.19) g i j, g = g z j z i F S Fubini-

21 CHOW BALANCED (2.11) F (σ) = log f σ 2 Ch SL(N + 1)/SU(N + 1). O(1), F (σ). SL(N + 1)/SU(N + 1), F (σ) 2.4. : (1) Chow, (2) F (σ) SL(N + 1, C)/SU(N + 1) proper, σ ( ), F (σ) +. (3) F (σ) S = SL(N + 1, C)/SU(N + 1), σ S A 1su(N + 1), σ() Z AZ Z 2 ωn F S = tr(m(σ())a) = M (N + 1) (N + 1) Hermitian, M, A 1su(N + 1), tr(ma) = αβ M αβ A βα = balanced, M(). tr(m) = ωf n S = d balanced I N+1 N M() = d N + 1 I N Chow, σ SL(N+1, C)/SU(N+1), σ() balanced. Hilbert. σ(e t ), σ(e t ) Z(). Chow f Z(), σ(e t ) 1 Cf. (2.7), t 0 w = (n + 1) f σ(et ) = e ta f = e tw f θ A ω n+1 F S 9 = (n + 1)tr(M()A) (2.20)

22 Chow, t 0, tr(m( t0 )A) = , t > t 0, tr(m( t )A) > tr(m( t0 )A) = 0 t,. f Chow. P(H 0 (Gr, O(d))) [f ] = lim t σ(e t )[f], f Chow. C Cf, w ch (σ), Chow. Hilbert. w ch (σ) = (n + 1)tr(M( )A) > (Hilbert ). ( ), : SL(N + 1, C) σ : C SL(N + 1, C), w ch (σ) > 0( 0).,. GL(N +1, C)/U(N +1), F (σ) GL(N + 1, C)/U(N + 1). σ GL(N + 1, C), σ = (det(σ)) 1 N+1 σ SL(N + 1, C), (2.5), (2.12) F 0 ω (φ + c) = c + F 0 ω(φ) F (σ) = F ( σ) = d (n + 1)Fω 0 F S (φ σ ) = (n + 1)( d Fω 0 F S (φ σ ) 2d log det(σ)) (2.21) N + 1, A gl(n + 1, C), A = A tr(a) N+1 I N+1 sl(n + 1, C), σ(e t ) = exp(ta). σ(e t ) Chow (n + 1)tr(M( )A) = (n + 1)(tr(M( )A) d tr(a)) (2.22) N

23 Chow 3.1 Bergman polarized (, L, ω). h L Hermitian, ω = Ric(h). h H 0 (, kl), s i H 0 (, kl), i = 1, 2. s 1, s 2 Hilb(hk ) = 1 (s 1, s 2 ) h k(kω) n n! Hilb(h k ) {s α α = 0,, N k }, N k + 1 H 0 (, kl) Bergman B(h k ) N k B(h k )(z) = s α (z) 2 h k 4. P k : Γ(, kl) H 0 (, kl), Γ(, kl). α=0 N k B(h k )(z, w) = s α (z) s α(w) α=0 P k, B(h k )(z) = B(h k )(z, z). B(h k )(z). h B(h k ), B(kω), ω = Ric(h)., B k = B k (h) = B(h k ). B k k. 1 N k B k (z)(kω) n = s α 2 Hilb(h n! k ) = N k + 1 α=0 N k +1. L, k, K +kl, Kodaira, L : H p (, kl) = H p (, Ω n ( K + KL)) = 0, 1 p n Riemann-Roch k H 0 (, kl) N k + 1 = dimh 0 (, kl) = e kc1(l) T d() (3.1) T d() Todd, (c i = c i ()) T d() = 1 + c c2 1 + c 2 + c 1 c 2 + (3.2) 12 24

24 3.1 BERGMAN (3.1) N k + 1 = C 0 k n + C 1 k n 1 + C 0 = c 1(L) n, [] = 1 ω n = V n! n! n! C 1 = 1 2n! nc 1(L) n 1 c 1 (), [] = 1 nric(ω) ω n = V 2n! 2n! V, S S. B k 1 V Sω n = 1 2n! V S B k = a 0 + a 1 k 1 + O(k 2 ) a i. : 3.1. Kähler ω, k C : B k (ω) = a 0 (ω) + a 1 (ω)k 1 + (3.3) r, N 0 N B k (ω) a i (ω)k i Cr () C r,n,ω k N 1 (3.4) i=0 a i (ω) ω. a 0 (ω) = 1, a 1 (ω) = 1 2 S(ω) (3.4), r, N, s ω ( )C s, C r,n,ω ω. 5. Tian [20] Hörmander Peak Section, a 0, C 2. Ruan[18] Bochner C, Lu[12] Tian Peak Section a 1,. Zelditch[24] Szegö, Dai-Liu-Ma[4]. {s α } Kodaira (1.3), k Bergman ω k = 1 k I {s ω α} F S ω F S CP N Fubini-Study., 3.2 (Tian). k, ω k ω. 12

25 CHOW. kl e L, Fubini-Study 1 ω k = 2πk Nk α=0 log s α 2 h 1 k = ω e L 2 2πk log B k h k r, 1 log B k k = 1 log(1 + k O(k 1 )) Cr 0, k P k (, ω) P (, ω).. φ P (, ω), h φ = he φ, Ric(h φ ) = ω φ. H 0 (, kl) Hilb(h φ ), P k (, ω) φ k = 1 N k N k k log s α 2 h = log s k α 2 h + φ = 1 k φ k log B k(h φ ) + φ (3.5) α=0 B k r > 0, φ k C r (ω φ ) φ, φ, ω ω φ C r, φ k C r (ω) φ. α=0 Bergman balanced. 3.2 Donaldson (, L) balanced, H 0 (, kl) {s α }, I {sα}() banlanced. h F S O(1) Fubini-Study, I {s h α} F S, I {s ω n!(n α} F S balanced,. { k +1) V k s α } Hilb(h F S ), N N s α 2 Z α 2 h F S = Z = 1 2 ω F S Bergman α=0 B(ω F S )(z) = N k α=0 α=0 n!(n k + 1) V k s α 2 h F S = n!(n k + 1) V k, h k kl Hermitian B(h k ). {s α } Hilb(h k ), 1 (I {sα}) ω F S = ω hk + 2π B(h k ) = ω hk I {sα}() balanced (, kl) balanced, kl Hermitian h k, Bergman B(h k ). Bergman (3.4) : 13

26 3.2 DONALDSON 3.5. k, (, kl) balanced h k, ω k = 1 k Ric(h k) C ω, ω. Aut(, L), 2.3 F (σ) = d F 0 ω F S (φ σ ) SL(N + 1, C)/SU(N + 1). balanced balanced, 3.6. Aut(, L), (, L) balanced, H 0 (, L) I L () balanced U(N + 1) R. L balanced h, Kähler ω = Ric(h). Donaldson 3.7 (Donaldson[6]). Aut(, L), ω c 1 (L). k, (, kl) balanced, balanced h k, balanced ω k = 1 k Ric(h k) C ω. Donaldson. H 0 (, kl), {s α } Kodaira I {sα}() balanced, M(I {sα}()) αβ = I {sα} () Z α Zβ s α s β 1 Z 2 ωn F S = γ s γ ( 2 2π log γ s γ 2 ) n, M = 0.,. Γ(kL) kl. C () Γ(kL). f C (), R(f) : H 0 (, kl) Γ(kL) s f i i s + fs Γ(kL) N k+1 = (N k +1) {}}{ Γ(kL) Γ(kL) C () Γ(kL) Nk+1. SL(N k + 1, C) Γ(kL) Nk+1. SL(N k + 1, C),. Γ(kL) Nk+1 B = {{s α } {s α } H 0 (, kl) } {s α } B, A 1su(N k + 1), s α s β θ A = θ A,{sα} = A αβ γ s γ 2 14

27 CHOW 1su(N k + 1) τ({s α }) : 1su(N k + 1) Γ(kL) N k+1 A {A αβ s β R(θ A,{sα})s α } = {A αβ s β θ i A i s α θ A s α } 1su(Nk + 1) A, B su = tr(ab) Γ(L) N k+1 {s α}, {s α} {sα} = Re α s αs α 1 β s β ( 2 2π log β s β 2 ) n τ({s α }) τ({s α }) : Γ(kL) N k+1 1su(N k + 1). {s α }, σ SL(N k + 1, C) σ {s α } = {σ β αs β } balanced. σ(t) SL(N k + 1, C)/SU(N k + 1). {s α (t)} = σ(t) {s α }, t = I {sα(t)}(), M(t) = M( t ). d M(t) = M(t) (3.6) dt (3.6) t > 0, M(t) = e t M(0) t,, M( ) = 0. {s α ( )} = σ( ) {s α } balanced. A(t). A(t) = σ(t) 1 d dt σ(t) t, SU(N k + 1), A(t) A = diag(λ 1 (t),, λ Nk (t)). B 1su(N k + 1), d dt tr(m(t)b) = d dt tr(m(t)b) = B d s α s β 1 βα dt γ s γ ( 2 2π log s γ 2 ) n γ = B βα ( λ αs α s β γ s γ s αs β (λ γ s γ, s γ ) 1 2 γ s γ 2 γ s γ )( 2 2π log s γ 2 ) n γ s α s β 1 +B βα γ γ s γ (n 2 2π λ γ s γ 2 1 γ s γ ) ( 2 2π log s γ 2 ) n 1 γ = ( λ αs α B αβ s β 1 γ s θ γ 2 A θ B g i jθ Aθ j i B )( 2π log s γ 2 ) n (3.7) γ (λ α s α θ i = A i s α θ A s α )(B αβ s β θ i B i s α θ B s α ) 1 γ s ( γ 2 2π log s γ 2 ) n γ = τ(t)(a), τ(t)(b) {sα(t)} ( τ(t) = τ({s α (t)})) (3.8) = τ(t) τ(t)(a), B su = tr[((τ(t) τ(t))(a))b] 15

28 3.2 DONALDSON d dt M(t) = τ(t) τ(t)(a). (3.6), (3.6) SL(N k + 1) τ τ(a) = M A(t) = (τ(t) τ(t)) 1 M(t) σ(t) 1 d dt σ(t) = (τ(t) τ(t)) 1 M(t) (3.9) (3.9) [0, ], M(0) (τ(t) τ(t)) 1. Donaldson, Bergman B k Kähler ( nearly balanced ), H 0 (, kl) {s α }, M(0). t > 0, (τ(t) τ(t)) 1, [0, ), (3.9) SL(N k + 1, C), t,. nearly balanced, S(ω φ ) φ, : φ t P (, ω), d dt S(ω φ t ) = d dt (gi j (log det(g)) i j) = g i l φk l g k j R i j ( φ k k) i i = φ i j R i j φ (3.10) = φ i j R i j ( φ ki ki + ( φ j R j ki k ),i ) = φ ki ki + φ j R k i i k,j = φ ij ij + φ i S i (3.11) = φ ij ij + φ i S i (3.12) S Kähler ω, φ S(ω φ ) DS ω δφ = (δφ) ij ij (δφ) ij ij = 0 (δφ) i L. Aut(, L), z i L, DS ω. Donaldson p ω = ω + k i φ i B k (ω) DS ω, p, φ i, k i, i = 2,, p + 1,, balanced. i=1 16

29 Futaki, Hilbert Chow 4.1 Futaki Futaki. S ω, S S. h C () S S = ω h. v, f (ω, v) = v(h)ω n (4.1) i v ω = 0, θ v = θ v (ω) C (), (0, 1) α, 2π 1 i v ω = α + θ v (4.2) α α ī,j = 0, α ī,ī = 0 (4.3) α i,i = 0, Futaki (4.1) f (ω, v) = (S S)θ v ω n (4.4) 4.1. f (ω, v) ω [ω]. f([ω], v).. (Calabi) φ t P (, ω), ω t = ω φt. t h(ω t ) = S(ω t ) S t, (3.10) ḣ φ i j h i j = φ i j R i j φ (4.5) f = φ i j (R i j h i j), Bianchi (R i j h i j),i = R i k k j,i h i i j = R i k k i, j ( h) j = S j (S S) j = 0 (4.6) (R i j h i j), j = 0 (4.7) fωn = φ j ω n (R i j h i j),i = 0, 1 f C () 1 f = f. (4.5) ḣ = φ 1 f + c t

30 4.1 FUTAKI c t. (4.1) t, α i,i = 0, d dt f(ω t, v) = = [(θ i v + α i )( φ 1 f) i + v(h) φ]ω n = [ θ v φ + θ v f + v(h) φ]ω n = [( θ v + v(h)) φ j j + θ φi j v (R i j h i j)]ω n = [( θ v + v(h)) j + (θ v (R i j h i j)) i ] φ j ω n v, v i, j = 0, α i,i = 0 ( θ v ) j = θ v i i j = (v i α i ),i j = v i,i j = v i, ji + v k R k i ji = v i R i j (4.8) (v(h)) j = v i h i j, (θ v (R i j h i j)),i = θ v,i (R i j h i j) ( (4.6)) (4.3), (4.7) d dt f(ω t, v) = (v i θ i v )(R i j h i j) φ j ωt n = (α i (R i j h i j)), j n φωt = 0 6. (A.16) α=0, 4.1, θ v θ v i j = 0, α 0, : (4.4) t, (3.11) d (S S)θ v ωt n = dt = = 0 θ v (ω φ ) = θ v (ω) + φ [( φ ij ij + φ i S i )θ v + (S S)(v( φ) + θ v t φ)]ω n t [ φ ij θ v ij + ((S S)θ v φi ) i ]ω n t, α = 0. v L, (, L). v Lie(Aut(, L)).. (4.4) f (ω, v) = (A.16), (4.8) (ω + θ v ) n (Ric(ω) θ v ) S n + 1 (ω + θ v) n+1 (4.9) ( i v )(ω + 1 2π θ v) = 0, ( i v )(Ric(ω) 1 2π θ v) = 0 18

31 FUTAKI, HILBERT CHOW A.10 f (ω, v) ω [ω] = c 1 (L). f (c 1 (L), v) = f (ω, v). (4.9) Tian, Tian CM CM, Futaki CM. Ding-Tian[5] Futaki (normal). Donaldson[7] Futaki. Hilbert, Futaki, Donaldson Futaki. 4.2 Hilbert Futaki v (, L) C, σ(e t ). S 1 C. C H 0 (, kl) : e t C, s H 0 (, kl), (e t s)(z) = e t s(e t z) (4.10) N k H 0 (, kl) = Cs α (4.11) N k + 1 = dimh 0 (, kl), e t s α = e tλα s α. C H 0 (, kl) N k w k = α=0 {s α } CP N k = P(H 0 (, kl) ), kl CP N k. CP N k {Z α }, {Z α }, s α Z α. C GL(N + 1, C) σ(e t ) = e ta k CN k+1 = H 0 (, kl). C Nk+1, A k = diag( λ 0,, λ Nk ), σ(e t ) = diag(e tλ0,, e tλ N k ) (4.12) α=0 λ α w k = tr(a k ) (4.13) w k. S 1, L S 1 Hermitian h, ω h = Ric(h) > 0. S 1 ω h S 1 h ω h, S 1 H 0 (, kl) Hilb(h k ). λ α 0, (4.11) {s α }. L L v v L, L h Hermitian. L kl v = kl v 19 + kθ v

32 4.2 HILBERT FUTAKI θ v = µ L (v) ( ). L kl v 1 i v ω h = 2π θ v s α = λ α s α, λ α s α = kl s α kθ v s α. v λ α s α 2 h k = (λ αs α, s α ) h k = ( kl v s α + kθ v s α, s α ) h k = v( s α 2 h k) kθ v s α 2 h k kl,, s α, kl v s α = 0. N k N k w k = λ α = λ α s α 2 Hilb(h k ) = 1 λ α s α 2 h n! k(kω h) n α=0 α=0 α = 1 s α n! (v( 2 h k) + kθ v s α 2 h k)(kω h) n α α = 1 θ v (kb k (h) ωh B k (h))(kω h ) n (4.14) n! Bergman (3.3) w k = k n+1 i 1 θ v (a i a i 1 )ωh n (4.15) n! i=0 = ( kn+1 θ v ωh n + kn Sθ v ωh n + ) (4.16) n! 2n! (4.14). σ(e t ) h h σ, ω σ h = σ ω h. s, (4.10), s 2 σ h(z) = σ s(z) 2 h(σ z) = σ s(σ 1 σ z) 2 h(σ z) = (σ σ s 2 h)(z) s α, s β Hilb(σ h k ) = 1 n! = 1 n! = 1 n! (s α, s β ) σ h kσ (kω h ) n σ ((σ s α, σ s β ) h k)σ (kω h ) n (σ s α, σ s β ) h k(kω h ) n ( σ() = ) = σ s α, σ s β Hilb(h k ) Hilb(h) ( s α, s β Hilb(h) ). σ {s α } σ 1. Hilb(σ h k ) = σ 1 Hilb(h k )(σ 1 ) (4.17) (σ 1 ). log det Hilb(σ h k ) = 2 log det σ + log det Hilb(h k ) (4.18) 20

33 FUTAKI, HILBERT CHOW (4.12) (4.13), t w k = d dt log det(σ) = 1 d 2 dt log det Hilb(σ h) (σ h) 1 d dt σ h = 2θ v. (4.14). Hermitian h t = he φt, ω ht = Ric(h t ) > 0. d log det Hilb(h k t ) = tr dt t= ( Hilb(h k 0) 1 d ) dt Hilb(h k t ) t=0 = d N k 1 s α 2 h (kω dt n! k h0 ) n ( {s α } Hilb(h k 0) ) 0 α=0 = 1 N k s α 2 h ( k n! φ + k ωh0 φ)(kωh0 ) n 0 α=0 = 1 φ(kb(h k n! 0) ωh0 B(h k 0)(kω h0 ) n (4.19) Donaldson. O(1) CP 1, P O(1). Kähler = P S 1 π CP 1 L = P S 1 L. CP 1 P S 1 (H 0 (, kl)) = R 0 π O(kL) = IndD k D k = kl kl + ( ) Dirac. w k = c 1 (P S 1 (H 0 (, kl))) = c 1 (IndD k ) Family Riemann-Roch w k = c 1 (IndD k ) = ( e k c1(l) T d(t )) [2] (4.20) T = T 1,0,. ξ S 1. v = Jξ+ 1ξ. Θ Fubini-Study P, CP 1 2, t = 1 1 2π Θ = 2π log det(1 + z 2 ) = 1 π dx dy (1 + x 2 + y 2 ) 2 [t], [CP 1 ] = CP 1 t = 1. c 1 (L) de Rham 1 2π (RL + µ L (ξ)θ) = ω µ L (v)t = ω θ v t (4.21) 21

34 4.2 HILBERT FUTAKI c(t ) det( π RT u(1) (ξ)), c 1 (T ) [ w k = R T u(1) (ξ) = RT ( ξ) T 1,0 Θ = R T 1( v)θ (4.22) Ric(ω) + 1 2π div(v)θ = Ric(ω) + θ vt (4.23) e k (ω θvt) T d ( 1 2π (RT )] 1( v)θ) [2] (4.24) Ω δ γ = α,β R γ δ α βdz α d z β (4.25) (3.2) (4.22) ( 1 T d 2π (RT ) 1( v)θ) (4.24) w k k linear = ( 1 2π tr(ω) + θ vt) (( 1 2π )2 (3tr(Ω) 2 Ω β αω α β)+ 1 2π (6 θ vtr(ω) 2Ω β αv α,β)t)+ w k = i=0 D i k n+1 i (4.26) D 0 = 1 (ω θ v ) n+1 1 = + θ v ) (n + 1)! (n + 1)! (ω n+1 = 1 θ v ω n (4.27) n! D 1 = 1 (ω θ v ) n (Ric(ω) + θ v ) = 1 (ω + θ v ) n (Ric(ω) θ v ) = 1 Sθ v ω n 2n! 2n! 2n! (4.28) D 2 = ( 2π )2 (ω θ v ) n 1 (( 24(n 1)! 2π )2 (3(T r(ω)) 2 Ω β αω α β) + 2π (6 θ vtr(ω) 2Ω β αv α,β)) = 1 θ v ( 1 n! 24 (R α β αγ δ γ δr β 4R β α R α β + 3S 2 ) S 6 )ωn = 1 θ v ( 1 n! 24 ( R 2 4 Ric 2 + 3S 2 ) S 6 )ωn (4.4) (4.27), (4.28) f (c 1 (L), v) = n!(s D 0 2D 1 ) (4.29), Donaldson Futaki. Futaki. 22

35 FUTAKI, HILBERT CHOW 4.2. c. (1) σ(e t ) σ(e ct ), Futaki c. (2) σ(e t ) σ(e t ) e ct, Futaki. 7., (4.26) D i D i = 1 n! θ v b i ω n b i. (4.15), b i = a i a i 1, Bergman, b i. a 2 = b 2 + a 1 = 1 24 ( R 2 4 Ric 2 + 3S 2 ) + S 3 Lu[12]. θ v, Riemann-Roch e kω h T d(rt (h)) L h Bott-Chern ( A.7, ) (h 1 ḣ)(kb k (h) B k (h))(kω h ) n 4.3 (complete intersection)hilbert, Futaki Chow. Futaki Lu[13] 4.1. F d, = {Z = [Z 0, Z 1,..., Z N ] CP N F (Z) = 0} S() = C[Z 0,, Z N ]/(F ) = S k () (F ) F, (F ) = k=0 I k, I k F k. S k () = H 0 (, kh). 0 I k () Sym k ((C ) N+1 ) H 0 (, kh) 0 (4.30) H CP N, c 1 (H) Kähler. K 1 k=0 + [] = K 1 CP N 23

36 4.3 [] = dh, CP N = (N + 1)H, K 1 = (N + 1 d)h. S = (N 1)(N + 1 d) σ(e t ) : C SL(N + 1, C), v σ(e t ). σ(e t ), v v F = µ F σ(e t ) SL(N+1, C), C (C ) N+1 λ = 0. H 0 (, kh) w k Sym k ((C ) N+1 ) I k (). w k = λ k( ) ( k+n N N + 1 λ (k d)( ) k d+n ( ) ) N k d + N + µ N + 1 N = kn N! µ kn 1 µ(n + 1 2d) 2(N 1)! (4.29) (d 1)(N + 1) f (c 1 (H), v) = µ (4.31) N Futaki. F 1, F 2,, F r d 1, d 2,, d r. σ(e t ), = r {F i = 0} i=0 S() = C[Z 0,, Z N ]/(F 1,, F r ) = vf i = µ i F i, i = 1,, r S k () (4.30), w k = λ k( ) k+n r N N + 1 ( 1) α 1 [λ (k (d i 1 + d i2 + + d iα )) ( k (d i1 N + 1 α=1 1 i 1<i 2< <i α r ( ) k (di1 + d i2 + + d iα ) + N +(µ i1 + µ i2 + + µ iα ) ] N r ( ) λ=0 k = ( 1) α 1 (di1 + d i2 + + d iα ) + N (µ i1 + µ i2 + + µ iα ) N α=1 = 1 N! N p=0 k N p 1 i 1<i 2< <i α r p q=0 ( 1) q ( N p + q q k=0 ) +di2 + +diα )+N N ) c p q (N)a q (4.32) c s (N) = j 1 j 2 j s 1 j 1<i 2< <j s N 24

37 FUTAKI, HILBERT CHOW N 2s, a q = c 2 = c 0 = 1, c 1 = 1 i<j N r ( 1) α 1 α=1 N j = j=1 N(N + 1) 2 i j = 1 N(N + 1)(N 1)(3N + 2) 24 1 i 1<i 2< <i α r q < r 1, a q = 0. q = r 1, r : (µ i1 + + µ iα )(d i1 + + d iα ) q a r 1 = ( 1) r 1 (r 1)!(µ 1 d 2 d r + µ 2 d 1 d 3 d r + + µ r d 1 d r 1 ) = r r ( 1) r 1 µ i (r 1)! d i d i i=1 i=1 a r = 1 2 ( 1)r 1 r!((µ µ r )d 1 d r + (µ 1 d 2 d r + + µ r d 1 d r 1 )(d d r )) = 1 µ i 2 ( 1)r 1 r! d i ( µ i + d i ) d i i i i i (4.32) w k = ( kn r+1 N ( 1) r 1 N! r 1 = k N r+1 r r d i (N r + 1)! i=1 i=1 ) ( ) a r 1 + kn r N 1 N(N + 1) [( 1) r 1 N! r 1 2 µ i k N r + 1 d i 2 (N r)! i d i [(N + 1 i d i ) i a r 1 + ( 1) r ( N r µ i d i i µ i ] ) a r ] f (c 1 (H), v) = (N r)(n + 1 i d i) N r + 1 = i d i ( i d i µ i N + 1 i d i N r + 1 i i i µ i d i i µ i d i ) d i [(N + 1 i d i ) i µ i d i i, r = 1, (4.31). σ SL(N + 1, C), (2.20) (4.27), Chow : w ch (σ) = i d i i µ i d i µ i ] Ding-Tian[5] F = Z 0 Z Z 2 Z 3 (Z 2 Z 3 ), σ(e t ) = diag(1, e 3t, e 2t, e 2t ), v σ. K 1 = H.. 4.2(2), f (c 1 (), v) = f (c 1 (H), v). 25

38 4.3 Futaki, σ(e t ) = σ(e 4t ) e 7t = diag(e 7t, e 5t, e t, e t ) SL(4, C) Futaki, ṽ σ. N = 3, d = 3, µ = 3, (4.31) Futaki f (c 1 (H), ṽ) = 8. f (c 1 (), v) = f (c 1 (H), v) = 2.. p 0 = [1, 0, 0, 0]. C 2 /Γ, Γ D 4 SU(2). Γ = 8. π : C 2 (z 1, z 2 ) [1, (z4 1 z2) 4 (z 1 z 2 ), (z2 1 + z2) 2 2, (z 1 z 2 ) 2 ] 4 4 σ(e t ) C 2, (z 1, z 2 ) (e t 2 z1, e t 2 z2 ). π v = 1(z 2 1 z1 + z 2 z2 ). v 5 [1, 0, 0, 0], [0, 1, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 1, 1]. Ding-Tian[5] (v 0 ) f (c 1 (), v) = 1 n + 1 z v(z)=0 1 (div z (v)) n+1 Γ z det( v Tz) (4.33) f (c 1 (), v) = 1 3 ( ( 2)3 + 3 ( 1) ) = F = Z 0 Z Z 1 Z Z 3 3, σ(e t ) = diag(1, e 6t, e 3t, e 4t ). K 1 = H. Futaki, σ(e t ) = σ(e 4t ) e 13t = diag(e 13t, e 11t, e t, e 3t ). N = 3, d = 3, µ = 9. (4.31) f (c 1 (H), ṽ) = 24. f (c 1 (), v) = f (c 1 (H), v) = 6. [1, 0, 0, 0]. C 2 /Γ, Γ SU(2). Γ = 24. ( F.Klein Lectures on the Icosahedron ) π : C 2 (z 1, z 2 ) [1, (z z 2 1z z 4 2) 3, 2( 3) 3 4 z1 z 2 (z 4 1 z 4 2), (z z 4 1z z 8 2)] σ(e t ) C 2 (z 1, z 2 ) (e t 2 z1, e t 2 z2 ). π v = 1 2 (z 1 z1 +z 2 z2 ). v 3 [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]. (4.33) f (c 1 (), v) = 1 3 ( ( 5)3 + ( 2) ) =

39 5.1 K K φ P (, ω), K Mabuchi[15], ν ω (φ) = 1 φ t P (, ω) 0 φ. 8. K R K 1 (h)r L,n (h) dt (S(ω φt ) S) d dt φ t ωφ n t S n+1 RL,n+1 (h) Bott-Chern ( φω n φ n φric(ω φ ) ω n φ + S φω n φ) A.7 φ t. L K 1 K 1 L. Tian restricted Bott-Chern. v, v σ(e t ). φ = θ v, d dt ν ω(φ t ) = (S S)θ v ω n = f ([ω], v), σ(e t ) ω = ω + 1 2π φ t, K P (, ω). Chen[2] Kähler K (, [ω]) (metric space). P (, ω) : γ = {φ t }, L(γ) = 1 0 dt φ 2 ωφ n φ φ 2 g φ = φ φ i φi = 0 (5.1) 5.1. ν ω (φ).,.. φ t P (, ω), φ t (5.1), d 2 dt F ω(φ) 0 = ( φ + φ φ)ω 2 2 φ = ( φ φ i φi )ωφ n = 0

40 5.1 K K (3.11) d 2 dt ν ω(φ) = ( φ ij 2 ij + φ i S i + S φ + S φ φ)ω φ n S d2 dt F ω(φ 0 2 t ) = ( φ ij φij S( φ φ i φi ))ωφ n = φ ij φij ωφ n 0 0 φ i j = 0, i φi,. z i Kähler-Einstein. ω c 1 (), { Ric(ω) ω = 1 h 2π ω ω n = (5.2) ehω ωn Kähler-Einstein Monge-Ampere : ω n φ = e hω φ ω n (5.3) (5.3) Euler-Langrange, Ding F ω (φ) = Fω(φ) 0 log( 1 e hω φ ω n ) (5.4) V F 0 ω.. φ t = tφ. F 0 ω(φ) = 1 V 1 0 dt φ(ω + t φ) n n 1 1 n + 1 ( ) n + 1 = i + 1 n j=i ( ) j i ( ) n + 1 φ i + 2 φ ( φ) i ω n 1 i j 1 n ) = 1 φω n 1 V V i=0 = 1 φω n n 1 ( k φ V V n + 1 i φ ( φ) i ω n 1 i i=0 j=i+1 k=i = 1 φω n n 1 n k ( ) φ V V n + 1 φ k ( ( k=0 j=k+1 i φ) i ω k i ) ω n 1 k i=0 = 1 φω n + 1 n 1 n k φ V V n + 1 φ ωφ k ω n 1 k k=0 = 1 φω n + J ω (φ) (5.5) V 28

41 , J ω (φ) = 1 n 1 n 1 n k φ V n + 1 φ ωφ k ω n 1 k 0 k=0 k=0 F 0 ω(φ) 1 V φω n (5.6) Kähler-Ricci, Tian Kähler-Einstein F ω (φ) P (, ω) properness, [21], [22]. ν ω (φ) F ω (φ) cocycle,. 1 ν ω (φ) = n dt 0 = log ωn φ ω n ωn φ + = F ω (φ) + h ω ω n φ(ric(ω φ ) Ric(ω) + Ric(ω) ω + ω ω φ ) ω n 1 φ h ω (ω n ωφ) n + h ωφ ω n φ φω n φ + F 0 ω(φ) F ω (φ) properness ν ω (φ) properness. Tian[21] properness Kähler- Einstein K-. Donaldson[7] K (, L) test configuration : 1. C 2. C L 3. C π : C, C C. t 0, t = π 1 (t), ( t, L t ) (, L). Tian, L = K 1,, 0. Kähler-Einstein. C 0, v. 0, Ding-Tian[5] Futaki f 0 (c 1 (), v). 5.2 (Tian). (, L) K- (K- ),, test configuration, f 0 (c 1 (L), v) ( ). k, kl C test configuration CP N k, CP N k t, C GL(N k + 1, C) σ(e t ) = e ta k. t = σ(e t )(), 0 = lim t 0 t. (4.15) (4.26) tr(a k ) = D 0 k n+1 + D 1 k n + 29

42 5.2 A k 0 v. tr(a k ) 0. A k = A k tr(a k) N k + 1 I N k +1 σ k (e t ) = exp(ta k ) SL(N k +1, C). M k ( 0 ) 0 CP N k M( 0 )(2.10) (4.16) (4.27) tr(m k ( 0 )A k ) = tr(m k ( 0 )) = k n V = k n n!c 0 µ kl (v)ω n k = k n+1 µ L (v)ω n = k n+1 n!d 0 (2.22) σ k Chow ( [9]) ( w ch (σ k ) = (n + 1)tr(M k ( 0 )A k ) = (n + 1) tr(m k ( 0 )A k ) tr(m ) k( 0 ))tr(a k ) N k + 1 ( ) = (n + 1) k n+1 n!d 0 kn n!d 0 (D 0 k n+1 + D 1 k n + ) C 0 k n + C 1 k n 1 + = (n + 1)!(k n C 1D 0 C 0 D 1 C 0 + ) = (n + 1)k n f 0 (c 1 (L), v) + (5.7) k, Chow, w ch (σ k ) > 0, k >> 1. (5.7) f 0 (c 1 (L), v) Chow K-., Donaldson Aut(, L), Chow. (, L) K-. Chow F (σ) = V F 0 ω F S (φ σ ) 5.2 2V log det(σ), σ GL(N + 1, C)/U(N + 1) N + 1 properness.,. CP N, O(1). O(1) Fubini-Study F S. H 0 (, O(1)) Z α. GL(N + 1, C)/U(N + 1) C N+1. H, {s α } H, s V α = s n!(n+1) α. {Z α } {s α} σ s α = σαz β β, U(N + 1), σ GL(N + 1, C)/U(N + 1), 30

43 det(σ) R. Hilb(H) {(Z α, Z β ) H }, det Hilb(H) = det(σ) 2. {s α } O(1) F S(H), s α 2 F S(H) = s α 2 F S N β=0 s β 2 F S = s α 2 F Se φ H (5.8) φ H = log N s β 2 F S = log β β=0 γ σ γ β Z γ 2 F S = log σ Z 2 Z 2 = φ σ F 0 ω, ω F S ω = Ric(h). F 0 ω F S (h), F (σ) F (H) = V F 0 ω(f S(H)) + Fω(F 0 S(H)) = Fω( 0 log F S(H) ) h V log det H (5.9) N + 1 H F (H), H s α balanced. h = F S(H), h O(1) balanced, Hilb(F S(H )) = H, F S(Hilb(h )) = h (5.10) (5.9) P (, ω). Chow (2.22), (2.20), (4.13) (4.14) Chow θ v (h)ωh n V θ v (B(h) ωh B(h))ωh n N + 1 θ v Kähler, φ t, (4.19) : φ P (, ω), F (φ) = V F 0 ω(φ) + V N + 1 log det Hilb(h φ) Hilb(h φ ) = ( s α, s β ) (N + 1), {s α } H 0 (, L). P (, ω) GL(N + 1, C)/U(N + 1). 5.3 (Donaldson[8]). (1) F (φ) balanced, Bergman B(h φ ). (2) F (φ) F (Hilb(h φ )), F (H) F (F S(H)). (3) balanced F (φ).. (1) (4.19) d dt F (φ) = φωφ n V 1 φ(b(h φ ) φ B(h φ ))ωφ n N + 1 n! = V 1 φ(b(h φ ) ωφ B(h φ ) c)ωφ n N + 1 n! 31

44 5.2, Bergman B(h φ ) c = 1 (B(h φ ) ωφ B(h φ ))ωφ n = V (N + 1)n! V φ F (φ), ωφ (B(h φ ) c) = B(h φ ) c., B(h φ ) = c. h φ balanced. (2) {s α } Hilb(h φ ), s V α = s n!(n+1) α. (5.8) s α 2 F S(Hilb(h φ )) = s α 2 h φ N β=0 s β 2 h φ 1 V ( F (Hilb(h φ )) F (φ)) = Fω 0 φ (F S(Hilb(h φ ))) ( Fω ) 0 1 log( s V α 2 h φ )ωφ n ( (5.6)) α log( 1 s V α 2 h φ ωφ) n ( log ) α 1 = log( s α 2 h (N + 1)n! φ ωφ) n = 0, {s α } H, s V α = s n!(n+1) α. F (F S(H)) F V (H) = (log det Hilb(F S(H)) log det(h)) N V log( s α 2 F S(H) (N + 1)n! ωn F S(H) ) α = V log( 1 s V α 2 F S(H) ωn F S(H) ) = 0, N + 1 Hermitian A, (det A) 1 N+1 tr(a) N + 1 α α 1 tr(a) log det A log N + 1 N + 1 (3) balanced H F, (2) (5.10) F (φ) F (Hilb(h φ )) F (H ) = F (F S(H )) = F (h ) 32

45 L kl, k Chow F k (φ) = V k F 0 kω(kφ) + V k N k + 1 log det Hilb(hk φ) φ t P (, φ), (4.19) Bergman ( 3.1), d dt F k(φ) = k φ(kω φ ) n kn V 1 φ(kb(h k N k + 1 n! φ) ωφ B(h k φ))(kω φ ) n = k n+1 φωφ n V k n V k n + 1SV φ(k n kn S(ω φ)k n + )ωφ n = k n φ(s S)ωφ n + O(k n 1 ) (5.7). Bergman, r, s P (, ω) C s+2 φ, F k (φ) = k n ν ω (φ) + O(k n 1 ) (5.11) 5.3 balanced h k F k (φ). 3.7 (5.11), 5.4 (Donaldson[8]). c 1 (L) ω, Aut(, L), ω K., Kähler, Chen-Tian[3] K-. 33

46 5.2 34

47 A A.1 G, EG BG G. EG. G m. G = EG G = (EG )/G, G BG. A.1. H G() = H ( G ), H G ({pt}) = H (BG), {pt}. de Rham. g G, g. ξ g, ξ ξ (, ξ ξ): ξ (x) = d exp(tξ) x dt t=0 S(g ) = i Sym( i g ) g, A() = m i=0 Γ( i T ). S(g ) A(). α S i (g ) A j (), deg α = 2i + j., G S(g ) A(). α S(g ) A() g A(). g G, ξ g, (g α)(ξ) = (g 1 ) (α(ad g 1ξ)). A.2. S(g ) A() G. α S(g ) A() g G, ξ g, α(ad g 1(ξ)) = g α(ξ). A G (). A.3. (d g α)(ξ) = d(α(ξ)) i ξ α(ξ) =: d ξ (α(ξ)),. deg d g α = deg α + 1. A.1. d g A G (), A G () d 2 g = 0.. α G- (Ad g 1ξ) = (g 1 ) ξ d g α G-. (d 2 gα)(ξ) = (di ξ + i ξ d)α(ξ) = L ξ α(ξ) = 0 (A.1)

48 A.1 (A G, d g ), de Rham, H (A G, d g ) de Rham., Chern-Weil. A.4. G M, M η (basic), (horizontal) G, ξ g, i ξm η = 0, g G, g η = η. A(M) bas. 9. M/G,, M/G M. M, A(M) D = h d, h. (A(M) bas, D) = (A(M/G), d). P B G, θ P, Θ. g G f, f(θ) P, B. Chern-Weil CW : S(g ) G A(P ) bas an invariant polynomial f f(θ), Chern-Weil CW : (S(g ) A()) G (A(P )) bas α h(α(θ)) h. g {ξ i }, θ = θ i ξ i. ξ i P, h : A(P ) A(P ) hor ω i (1 θ i i ξi )ω (A.2) A.2. CW d g = D CW. CW, CW : H (A G (), d g ) H (P G ) (A.3) A.3. (P, B) = (EG, BG), (A.3). A.1. G = S 1, BG = CP, = {pt}. CW : C[t] = H (BG). de Rham.,. A.4. CW (S(g ) A()) G = (A(P )) bas A(P G ) R R π =R S(g ) G CW (A(P )) bas = A(B) (A.4) 36

49 A. α (S(g ) A()) G, (A.2) Θ, h(α(θ)) = α(θ) + β f(θ, Θ), β A() dim. CW (α) = h(α(θ)) = α(θ) = ( α)(θ) = CW ( α) Chern-Weil. A.5. E G, G E, G, x, g G, g x : E x E g x. E E G, g G, g E = E. A.3, (A.1), A.6. S(g ) A(, E) 1, s S(g ) A(, E), ( E g s)(ξ) = E (s(ξ)) i ξ (s(ξ)) =: E g (ξ)(s(ξ)) S(g ) A(, E) 2 R E g (ξ) = ( E g (ξ)) 2 + L E ξ L E ξ, s S(g ) A(, E), (L E ξ s)(x) = d dt t=0 exp( tξ) s(exp(tξ) x) s(x) t A.5. (1) R E g G, R E g (S(g ) A(, End(E))) G. (2) (Bianchi ) [ E g, R E g ] = 0.. (1) R E g (ξ) = ( E g (ξ)) 2 + L E ξ = E,2 [ E, i ξ ] + L E ξ = R E + L E ξ E ξ = R E + µ E (ξ) µ E (ξ) = L E ξ E ξ. R E (A 2 () End(E)) G, µ E (S(g ) End(E)) G, (1). (2) g E = E, g(t) = exp(tξ), t, ξ g, [ E, L E ξ ] = 0. Cartan [i ξ, L T ξ ] = [i ξ, di ξ + i ξ d] = 0. [ E g, R E g ](ξ) = [ E g (ξ), ( E g (ξ)) 2 + L E ξ ] = [ E i ξ, L E ξ ] = 0 37

50 A E G, ξ E ξ. E, ξ ξ. ξ V = ξ ξ, s E x, ξ V (s) = µ E (ξ)(s), Bianchi, ξ g [ E, µ E (ξ)] = i ξ R E (A.5) A.2. (1) (, ω), L, L 1 2π L,2 = ω, G (, L), g G, g L = L, g ω = ω. (A.5) dµ L (ξ) = 2π 1 i ξ ω 1 2π µl : g C (). (A.6) (2) E = T. g G, = T Levi-Civita, L = L T., µ T (ξ) = L ξ ξ = ξ Γ(End(T )) (A.5) ξ = i ξ R Γ(T End(T )) Kähler, J. J, J = 0 RJ = JR Jξ = i ξ RJ (A.7) (, g, J) Ric (1,1),, u, v T, (A.7) Ric(u, v) = 1 1 tr(w R(u, v)jw) 2π 2 1 2π d(div Jξ ) = 2i ξ Ric (A.8) ξ = ξ 1,0 + ξ0,1 T C = T 1,0 T 0,1, div(ξ 1,0 ) + div(ξ0,1 ) = div(ξ ) = 0, (A.8) 1 2π div(ξ 1,0 ) = i ξ 1,0 Ric (A.9) E, EG G E EG G, c(eg G E) H even (EG G ). : c(r E g ) = det(1 + Chern-Weil π RE g )

51 A A.6. c(r E g ) (S(g ) A()) G, H G () G E. Chern-Weil CW (A.3), H even (EG G, R), c(r E g ) c(eg G E) A.2 Bott-Chern Dolbeault, E, rke = r. h E Hermitian. h E E. E h,. E {s i }, h = (h ij ) = ((s i, s j ) h ), E (1,0) : θ = h 1 h E (1,1) Θ = dθ + θ θ = h 1 h h 1 h + h 1 h E = E + E T = T (1,0) + T (0,1), E,2 = 0, E,2 = 0, R E = E E + E E = [ E, E ] (A.10) f k+1 (A) = tr(a k ), A gl(r, C). R E h, f k+1 (h) = tr((r E ) k+1 ). Chern-Weil f k (h) (k, k), H k+1,k+1 () h. h t Hermitian, d dt tr(re,k+1 ) = (k + 1)tr(R E,k [ E, d dt E ]) = k tr(r E,k [ E, h 1 ḣ]) = k tr(r E,k h 1 ḣ) tr(r E,k+1 (h 1 )) tr(r E,k+1 (h 0 )) = (k + 1) 1 0 tr(r E,k h 1 ḣ)dt A.7. (k+1) 1 0 tr(re,k h 1 ḣ)dt h 0 h 1 f k+1 Bott-Chern, f k+1 (h 0, h 1 ). A.7. f n+1 (h 0, h 1 ), h 0 h 1.. h t h 0 h 1 Hermitian, h = h s,t = (1 s)h t + s h t. ḣ 39

52 A.2 BOTT-CHERN DOLBEAULT t. R E, E, E R,,, d ds tr(rn h 1 ḣ) = ntr(r n 1 [, 1 dh [, h ds ]]h 1 ḣ) + tr(r n d ds (h 1 ḣ)) = α + ntr(r n 1 1 dh [, h ds ][, h 1 ḣ]) + tr(r n ( d dh dh (h 1 ) [h 1 dt ds ds, h 1 ḣ])) = α + β ntr(r n 1 1 dh h ds [, [, h 1 ḣ]]) + tr(r n d dh (h 1 dt ds )) tr(rn 1 dh [h ds, h 1 ḣ]) = α + β tr([r n 1 dh h ds, h 1 ḣ]) + ntr(r n 1 1 dh h ds [, [, h 1 ḣ]]) + tr(r n d dh (h 1 dt ds )) = α + β + d dt tr(rn h 1 dh ds ) α = ntr(r n 1 1 dh [, h ds h 1 ḣ]), β = ntr(r n 1 1 dh h [, h 1 ḣ]). ds dh ds t=0 = dh ds t=1 = 0, d 1 tr(r n h 1 ḣ)dt = ds d dt tr(rn 1 dh h ds )dt = 0 Dolbeault. G = Aut(), g = Lie(Aut()). A p, () = Γ( n p,q q=0 T ), A p, G () = S(g ) A p, (). g : α A p, G (), v g, ( g α)(v) = ( i v )α(v) ( i v ) 2 = ( i v + i v ) = 0 (A.11) (A p, G (), g ), Dolbeault, H p, G (). 11. de Rham,, G (A.11). v, v E, v Lie(Aut(, E)),, G g Aut(, E) Lie(Aut(, E)). A.6 E g (v) = E i v, R E g (v) = E,2 g + L E v A.8. R E g (v) = E,2 + L E v E v = R E + µ E (v) (A.12) [ E i v, R E g (v)] = 0 (A.13) 40

53 A. [ E i v, Rg E (v)] = [ E i v, ( E i v ) 2 ] + [ E, L E v ] [i v, L E v ] : [ E i v, ( E i v ) 2 ] = [[ E i v, E i v ], E i v ] [ E i v, E i v ] = [ E i v, E ] = R E E v = ( E i v ) 2 (A.10), v (1,0). 0., v, E. E E. E [ E, L E v ] = 0., Cartan L T = di + id, [i v, L E v ] = (A.12), [ E, R E ] = 0, (A.13) [ E, µ E (v)] = i v R E (A.14), f k+1 (h, v) = tr((r E g (v)) k+1 ) = (k + 1)tr(R E,k µ E (v)) A.9. ( i v )f k+1 (h, v) = 0, f k+1 (h, v) H k,k G () h.. (A.13), [ E i v, (R E g (v)) k ] = 0, ( i v )tr((r E g (v)) k ) = 0. h t Hermitian, E t. (A.12) t, (A.10), R E t = E,2 t = [ E, E t ], E E, d dt RE g (v) = [ E i v, E t ] (A.15) d dt f k+1(h t, v) = (k + 1) tr(ṙe g (v)(rg E (v)) k ) = (k + 1)tr([ E i v, E t ](Rg E (v)) k ) = (k + 1) tr([ E i v, E t (Rg E (v)) k ]) ( (A.13)) = (k + 1) ( i v )tr( E t (R E g (v)) k ) = ( i v )β t (v) A.10. f n+1(h, v) h. F (v). A.11. β A() = Γ( i i T ), ( i v )β = 0. 41

54 A.2 BOTT-CHERN DOLBEAULT. β = n p,q=1 = β p,q, ( i v )β = β n,n 1 i v β n+1,n = dβ n,n 1 = 0 w g = Lie(Aut(, E)), g t = exp(tw), h t = gt h, v t = Ad g 1v, f t n+1 (h t, v) = gt (f(h, v t )).?? F (v) = f n+1 (h t, v) = gt f n+1 (h, v t ) = f n+1 (h, v t ) = F (Ad g 1v) t t A.12. v, w Lie(Aut(, E)), F ([w, v]) = 0. A.3. L Kähler (, ω), h Hermitian 1 Ric(h) = ω, (A.14) 2π 1 2π µ L (v) = i v ω (A.16) µ L (v) θ v. h t = he φt Hermitian, Ric(h t ) = Ric(h) + φ t. (A.15) d 1 1 dt (ω + 2π µl (v)) = 2π ( i v ) φ T Kähler L T v, (A.14) (A.15) v = i v R T 1 2π µ L (v) = i v Ric(ω) (A.17) d 1 dt (Ric(ω) µl (v)) = 2π ( i v ) φ C (, L), S 1 C. S 1 L h, ξ, ξ Killing, v = Jξ+ 1ξ 2 C. ξ 1,0 = 1v. µ L (ξ) = 1µ L (v) (A.18) ( ξ) T 1,0 = 1 v (A.19) (A.6) (A.16), (A.9) (A.17). 42

55 A A.4. CP N, L = H. λ α R, 0 α N, λ 0 e tλ0 A =..., σ(t) = exp(ta) =... λ N A σ(t) C N+1, CP N. A σ(t), CP N v, Z = [Z 0, Z 1,, Z N ], {z α = Z α Z 0 ; α = 1,, N} N v = (λ α λ 0 )z α z α α =1 H CP N, C N+1 CP N, [Z] 1 C Z. σ(t) H, H. Z α H, e tλ N 2 F S H Fubini-Study, 2 F S Hermitian L, σ(e t ) Z α = e tλα Z α L v Z α = λ α Z α Z α 2 F S = Z α 2 N Z β 2 β=0 L v Z α = v(log Z α 2 F S)Z α = N λ β Z β 2 β=0 Z N α + λ α Z α Z β 2 β=0 µ L (v)z α = L L v Z α L v Z α = β λ β Z β 2 β Z β 2 Z α Z α L, µ L (v) = β λ β Z β 2 β Z β 2 (A.20) (A.16) ( ) 1 β 2π λ β Z β 2 β Z = i β 2 v ω F S 43

56 A.2 BOTT-CHERN DOLBEAULT A.2 U(N + 1), CP N u(n + 1) ι : CP N u(n + 1) [Z 0,, Z N ] ( ) Zα Z β 1 Z 2 44

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