Remarks on logarithmic K-stability

Remarks on logarithmic K-stability arXiv:1104.0428v1 [math.DG] 3 Apr 2011 Chi Li ABSTRACT: We make some observation on the logarithmic version of K-...

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Remarks on logarithmic K-stability

arXiv:1104.0428v1 [math.DG] 3 Apr 2011

Chi Li ABSTRACT: We make some observation on the logarithmic version of K-stability.

1

Introduction

−1 Let (X, J) be a Fano manifold, that is, KX is ample. The basic problem in K¨ahler geometry is to determine whether (X, J) has a K¨ ahler-Einstein metric (cf. [12]) On way to attack this problem is to use continuity method. Fix a reference K¨ahler metric ω ∈ c1 (X). Its Ricci curvature Ric(ω) also lies in c1 (X). So there exists hω ∈ C ∞ (X) such that Z Z ¯ ω, ωn e hω ω n = Ric(ω) − ω = ∂ ∂h X

X

Consider the following family of Monge-Amp`ere equations. ¯ t )n = ehω −tφ ω n (ω + ∂ ∂φ

(∗)t

This is equivalent to the equation for K¨ahler forms: Ric(ωφ ) = tωφ + (1 − t)ω

(1)

By Yau’s theorem [16], we can always solve (∗)t for t = 0. If we could solve (∗)t for t = 1, we would get K¨ ahler-Einstein metric. However, it was first showed by Tian [13] that we may not be able to solve (∗)t on certain Fano manifold for t sufficiently close to 1. Equivalently, for such a Fano manifold, there is some t0 < 1, such that there is no K¨ahler metric ω in c1 (X) which can have Ric(ω) ≥ t0 ω. The existence problem of K¨ ahler-Einstein metric is a special case of the existence problem of constant scalar curvature K¨ ahler (cscK) metric. For the latter, we fix an ample line bundle L on (X, J). We have the following folklore conjecture. For the definition of K-stability, see [14], [3] or Definition 4. Conjecture 1 (Tian-Yau-Donaldson). ([14],[3]) There is a smooth constant scalar curvature K¨ ahler metric in c1 (L) on (X, J) if and only if (X, J, L) is K-stable. Return to the continuity method (∗)t and let R(X) = sup{t : (∗)t is solvable }. Sz´ekelyhidi proved that Proposition 1 ([10]). R(X) = sup{t : ∃ a K¨ ahler metric ω ∈ c1 (X) such that Ric(ω) > tω} In particular, R(X) is independent of reference metric ω. There is another continuity method we can try. Let Y ∈ | − KX | be a general element, then Y is a smooth Calabi-Yau hypersurface. The K¨ahler-Einstein metric with cone singularity along Y of cone angle 2πβ is a solution to the following distributional equation Ric(ω) = βω + (1 − β){Y } 1

(2)

Conjecture 2 (Donaldson). There is a cone-singularity solution ωβ to (2) for any parameter β ∈ (0, R(X)). If R(X) < 1, there is no solution for parameter β ∈ (R(X), 1). The purpose of this note is to discuss the logarithmic version of K-stability and prove the following result. Theorem 1. Let X△ be a toric Fano variety with a (C∗ )n action. Let Y be a general hyperplane section of X△ . When β < R(X△ ), (X△ , βY ) is log-K-stable along any 1 parameter subgroup in (C∗ )n . When β = R(X△ ), (X△ , βY ) is semi-log-K-stable along any 1 parameter subgroup in (C∗ )n and there is a 1 parameter subgroup in (C∗ )n which has vanishing log-Futaki invariant. When β > R(X△ ), (X△ , βY ) is not log-K-stable. This explains and generalizes slightly the calculation in [4] and gives some evidence for the Conjecture 2 (Combined with Conjecture 3). We prove the above result by calculating R(X△ ) and log-Futaki invariant explicitly. R(X△ ) was calculated in [6] based on Wang-Zhu’s work [15]. The main formula for log-Futaki invariant is (19). A toric Fano manifold X△ is determined by a reflexive lattice polytope △ (For details on toric manifolds, see [8]). For example, let Blp P2 denote the manifold obtained by blowing up one point on P2 . Then Blp P2 is a toric Fano manifold and is determined by the following polytope. Any such polytope △ contains the origin O ∈ Rn . We denote the barycenter of △ by Pc . If −−→ Pc 6= O, the ray Pc + R≥0 · Pc O intersects the boundary ∂△ at point Q. Theorem 2. [6] If Pc 6= O,

OQ R(X△ ) = Pc Q

Here OQ , Pc Q are lengths of line segments OQ and Pc Q. In other words, Q=−

R(X△ ) Pc ∈ ∂△ 1 − R(X△ )

If Pc = O, then there is K¨ ahler-Einstein metric on X△ and R(X△ ) = 1.

@ @ @ @ @ @ P O rr c @ @ @ @r @ [email protected] @ @ @

Acknowledgement: I would like to thank Professor Gang Tian for helpful discussion and constant encouragement. In particular, he suggested the log-K-stability and told me the result in [7] which provides the starting example of the logarithmic version of Tian-Yau-Donaldson conjecture.

2

2

Log-Futaki invariant

In this section, we recall Donaldson’s definition of log-Futaki invariant (6). Let (X, L) be a polarized projective variety and D be a normal crossing divisor: D=

r X

αi Di

i=1

with αi ∈ (0, 1). From now on, we fix a Hermitian metric k · ki = hi and defining section si of the line bundle [Di ]. Assume ω ∈ c1 (L) is a smooth K¨ ahler form. We define √ √   −1 ¯ −1 ¯ P(ω) = ωφ := ω + ∂ ∂φ; φ ∈ L∞ (X) ∩ C ∞ (X\D) such that ω + ∂ ∂φ ≥ 0 2π 2π by

Around any point p ∈ X, we can find local coordinate {zi ; i = 1, · · · , n}, such that D is defined r

p αi {zi = 0} D = ∪i=1

where rp = ♯{i; p ∈ Di }. ahler metric on (X, D), if around p, ω is Definition 1. We say that ω ˆ ∈ P(ω) is a conic K¨ quasi-isometric to the metric rp X dzi ∧ d¯ zi i=1

|zi |2αi

n X

+

j=rp +1

dzj ∧ d¯ zj

We will simply say that ω ˆ is a conic metric if it’s clear what D is. Geometrically, this means the Riemannian metric determined by ω has conic singularity along each Di of conic angle 2π(1 − αi ). Remark 1. Construction of K¨ ahler-Einstein metrics with conic singularites was proposed long time ago by Tian, see [11] in which he used such metrics to prove inequalities of Chern numbers in algebraic geometry. One consequence of this definition is that globally the volume form has the form Ω 2αi i=1 ksi ki

ω ˆ n = Qr

where Ω is a smooth volume form. For any volume form Ω, let Ric(Ω) denote the curvature of the −1 Hermitian metric on KX determined by Ω. Then, by abuse of notation, Ric(ˆ ω)

= Ric(ˆ ω n ) = Ric(Ω) +

√ r r r X X −1 X αi {Di } αi c1 ([Di ], hi ) + αi ∂ ∂¯ log ksi k2i = Ric(Ω) − 2π i=1 i=1 i=1

= Ric(Ω) − c1 ([D], h) + {D}

(3)

αi r i where h = ⊗ri=1 hα i and s = ⊗i=1 si are Hermitian metric and defining section of the R-line bundle r αi [D] = ⊗i=1 [Di ] . Here we used the Poinc´are-Lelong identity: √ −1 ¯ ∂ ∂ log ksi k2i = −c1 ([Di ], hi ) + {Di } 2π

3

where {Di } is the current of integration along the divisor Di . The scalar curvature of ω ˆ on its smooth locus X\D is n(Ric(Ω) − c1 ([D], h)) ∧ ω ˆ n−1 nRic(ˆ ω) ∧ ω ˆ n−1 ¯ˆ = S(ˆ ω ) = gˆij R i¯ j = ω ˆn ω ˆn So if S(ˆ ω) is constant, then the constant only depends on cohomological classes by the identity: nµ1 :=

−n(KX + D) · Ln−1 V ol(D) n(c1 (X) − c1 ([D])) ∧ [c1 (L)]n−1 = = nµ − n n c1 (L) L V ol(X)

Here nµ =

(4)

n c1 (X) · c1 (L)n−1 −nKX · Ln−1 = c1 (L)n Ln

is the average scalar curvature for smooth K¨ahler form in c1 (L). And Z Z c1 (L)n−1 D · Ln−1 Ln c1 (L)n V ol(D) = = , V ol(X) = = (n − 1)! n! n! D (n − 1)! X

Now assume C∗ acts on (X, L) and v is the generating holomorphic vector field. Recall that the ordinary Futaki-Calabi invariant ([5], [2]) is defined by Z ωn θv (S(ω) − nµ) F (c1 (L))(v) = − n! X

where θv satisfies

¯ v ιv ω = ∂θ

Now assume ω ˆ ∞ ∈ P(ω) is a conic metric and satisfies S(ˆ ω∞ ) = nµ1

(5)

Assume D is preserved by the C∗ action. Let’s calculate the ordinary Futaki invariant using the ˆ ω∞ , v). Then near p ∈ D, v ∼ Prp ci zi ∂z + v˜ with v˜ = o(z1 · · · zr ) conic metric ω ˆ ∞ . Let θˆv = θ(ˆ p i i=1 Prp |zi |2(1−αi ) . holomorphic. θˆv ∼ i=1 We then make use of the distributional identity (3) to get Z ω ˆ n−1 F (c1 (L))(v) = − θˆv (nRic(ˆ ω∞ ) − nµˆ ω∞ ) ∧ ∞ n! X Z ω ˆ n−1 θˆv [(nRic(Ω) − nc1 ([D], h) − nµ1 ω ˆ ∞ ) + n{D} − (nµ − nµ1 )ˆ ω∞ ] ∧ ∞ = − n! X Z Z Z n n−1 n ω ˆ ω ˆ ω ˆ = − θˆv (S(ˆ ω∞ ) − nµ1 ) ∞ − {D}θˆv ∞ + (nµ − nµ1 ) θˆv ∞ n! (n − 1)! n! X X X  Z Z n n−1 V ol(D) ω ˆ ω ˆ − θˆv ∞ = − θˆv ∞ (n − 1)! V ol(X) n! X D So we get 0 = F (c1 (L))(v) +

V ol(D) ω ˆ n−1 − θˆv ∞ (n − 1)! V ol(X) D

Z

Z

n

ω ˆ θˆv ∞ n! X



Since the two integrals in the above formula is integration of (singular) equivariant forms, they are independent of the chosen K¨ ahler metric in P(ω) with at worst conic singularities. In particular, we can choose the smooth K¨ ahler metric ω, then we just discover the log-Futaki invariant defined by Donaldson: 4

Definition 2. [4] F (c1 (L), D)(v) = F (c1 (L))(v) +

V ol(D) ω n−1 − θv (n − 1)! V ol(X) D

Z

ωn θv n! X

Z



(6)

Remark 2. This differs from the formula in [4] by a sign. And we think of D as a cycle with real coefficients, so if we replace D by (1 − β)△, we have the same formua as that in [4].

3

log-K-energy and Berman’s formulation

We can integrate the log-Futaki-invariant to get log-K-energy Z 1 Z Z 1 Z Z Z ωn V ol(D) 1 ω n−1 ωn νω,D (φ) = − dt (S(ωt ) − S)φ˙ t + dt − φ˙ t dt φ˙ t n! V ol(X) 0 n! 0 X 0 D (n − 1)! X  Z 1 Z √ n−1 −1 ¯ V ol(D) 0 ω = νω (φ) + ∂ ∂ log ksk2 + c1 ([D], h) φ˙ t + F (φ) 2π (n − 1)! V ol(X) ω 0 X Z ωφn V ol(D) 0 log ksD k2 Fω (φ) + JωχD (φ) + (7) = νω (φ) + V ol(X) n! X where χD = c1 ([D], h) is the Chern curvature form. The functionals Fω0 (φ) and Jωχ (φ) are defined by: Z 1 Z ωφn Fω0 (φ) = − dt φ˙ t n! X 0 Jωχ (φ) =

Z

1

dt

0

Z

X

˙ ∧ φχ

ωφn−1 t (n − 1)!

Let’s now focus on the Fano case as in the beginning of this paper. (2) is equivalent to the following singular complex Monge-Amp`ere equation: ¯ n = e−βφ (ω + ∂ ∂φ)

Ω1 ksk2(1−β)

(8)

−1 has the with Ω1 = ehω ω n and s is a defining section of [Y ]. Note that the line bundle [Y ] = KX Hermitian metric k · k such that the curvature is ω. −1 We have D = (1 − β)Y . Since [Y ] = KX , we can assume χD = (1 − β)ω, V ol((1 − β)D) = n(1 − β)V ol(X). Then (7) becomes Z  ωφn νω,D = νω (φ) + (1 − β) nFω0 (φ) + Jωω (φ) + (1 − β) log ksk2 n! X Z n ω φ = νω (φ) + (1 − β)(Iω − Jω ) + (1 − β) log ksk2 n! X Z Z ωφn ωφn log = log ksk2 − β(Iω − Jω ) + (1 − β) Ω1 n! X X

We have used the well known formula for K-energy [12]: Z ωφn − (Iω − Jω )(φ) log νω (φ) = Ω1 X where Iω (φ) =

Z

X

φ(ω n − ωφn )/n! 5

Jω (φ) =

Fω0 (φ)

+

Z

φ

X

And it’s easy to verify that

nFω0 (φ) + Jωω (φ) = (Iω − Jω )(φ) = −

ωn n!

Z

X

 φωφn + Fω0 (φ)

From above formula, we see that, in Fano case, the log-K-energy coincides with Berman’s free energy associated with (8) ([1]) Z  Z ωφn ωφn n 0 log νω,D = +β φωφ + Fω (φ) Ω1 /ksk2(1−β) n! X X

4

Log-K-stability

We imitate the definition of K-stability to define log-K-stability. First we recall the definition of test configuration [3] or special degeneration [14] of a polarized projective variety (X, L). Definition 3. A test configuration of (X, L), consists of 1. a scheme X with a C∗ -action; 2. a C∗ -equivariant line bundle L → X 3. a flat C∗ -equivariant map π : X → C, where C∗ acts on C by multiplication in the standard way; such that any fibre Xt = π −1 (t) for t 6= 0 is isomorphic to X and (X, L) is isomorphic to (Xt , L|Xt ). Any test configuration can be equivariantly embedded into PN × C∗ where the C∗ action on P is given by a 1 parameter subgroup of SL(N + 1, C). If Y is any subvariety of X, the test configuration of (X, L) also induces a test configuration (Y, L|Y ) of (Y, L|Y ) . Let dk , d˜k be the dimensions of H 0 (X, Lk ), H 0 (Y, L|Yk ), and wk , w ˜k be the weights of C∗ action 0 k 0 k on H (X0 , L|X0 ), H (Y0 , L|Y0 ), respectively. Then we have expansions: N

wk = a0 k n+1 + a1 k n + O(k n−1 ),

dk = b0 k n + b1 k n−1 + O(k n−2 ) d˜k = ˜b0 k n−1 + O(k n−2 )

w ˜k = a ˜0 k n + O(k n−1 ),

If the central fibre X0 is smooth, we can use equivariant differential forms to calculate the coefficients by [3]. Let ω be a smooth K¨ahler form in c1 (L), and θv = Lv − ∇v , then Z Z ωn 1 ωn θv a0 = − ; a1 = − (9) θv S(ω) n! 2 X n! X Z Z 1 ωn ωn b0 = = V ol(X); b1 = (10) S(ω) 2 X n! X n! Z Z ω n−1 ω n−1 ˜ ; b0 = = V ol(Y0 ) (11) θv a ˜0 = − (n − 1)! Y0 (n − 1)! Y0 Remark 3. To see the signs of coefficients and give an example, we consider the case where X = P1 , L = OP1 (k). C∗ acts on P1 by multiplication: t · z = tz. A general D ∈ |L| consists of k points. As t → 0, t · D → k{0}. D is the zero set of a general degree k homogeneous polynomial Pk (z0 , z1 ) and k{0} is the zero set of z1k . C∗ acts on H 0 (P1 , O(k)) by t · z0i z1j = t−j z0i z1j 6

so that √ limt→0 [t · Pk (z0 , z1 )] =√[z1k ], where [Pk ] ∈ P(H 0 (P1 , O(k))). Take the Fubini-Study metric ∂ log(1+|z|2 ) |z|2 −1 ¯ −1 dz∧d¯ z ωF S = 2π ∂ ∂ log(1 + |z|2 ) = 2π = 1+|z| 2 . So (1+|z|2 )2 , then θv = ∂ log |z|2

While

+∞

1 r2 2rdr = 2 3 (1 + r ) 2 P1 0 Z Z 1 1 −a1 = θv ω F S = S(ωF S )θv ωF S = 2 P1 2 1 P

−a0 =

Z

θv ω F S =

Z

1 1 wk = −(1 + · · · + k) = − k 2 − k 2 2

which gives exactly a0 = a1 = − 21 . Comparing (6), (9)-(11), we can define the algebraic log-Futaki invariant of the given test configuration to be F (X , Y, L)

= =

˜b0 2(a1 b0 − a0 b1 ) + (−˜ a0 + a0 ) b0 b0 ˜ (2a1 − a ˜0 )b0 − a0 (2b1 − b0 ) b0

(12)

Definition 4. (X, Y, L) is log-K-stable along the test configuration (X , L) if F (X , Y, L) ≤ 0, and equality holds if and only if (X , Y, L) is a product configuration. (X, Y, L) is semi-log-K-stable along (X , L) if F (X , Y, L) ≤ 0. Otherwise, it’s unstable. (X, Y, L) is log-K-stable (semi-log-K-stable) if, for any integer r > 0, (X, Y, Lr ) is log-K-stable (semi-log-K-stable) along any test configuration of (X, Y, Lr ). Remark 4. When Y is empty, then definition of log-K-stability becomes the definition of Kstability. ([14], [3]) Remark 5. In applications, we sometimes meet the following situation. Let λ(t) : C∗ → SL(N + 1, C) be a 1 parameter subgroup. As t → ∞, λ(t) will move X, Y ⊂ PN to the limit scheme X0 , Y0 . Then stability condition is equivalent to the other opposite sign condition F (X0 , Y0 , v) ≥ 0. This is of course related to the above definition by transformation t → t−1 . Pr Example 1 (Orbifold). Assume X is smooth. Y = i=1 (1 − n1i )Di is a normal crossing divisor, where ni > 0 are integers. The conic K¨ ahler metric on (X, Y ) is just the orbifold K¨ ahler metric on the orbifold (X, Y ). Orbifold behaves similarly as smooth variety, but in the calculation, we need to use orbifold canonical bundle Korb = KX + Y . For example, think L as an orbifold line bundle on X, then the orbifold Riemann-Roch says that 0 dimHorb ((X, Y ), L) =

=

Ln n 1 −(KX + Y ) · Ln n−1 k + k + O(k n−2 ) n! 2 (n − 1)! 1 b0 k n + (2b1 − ˜b0 )k n−1 + O(k n−2 ) 2

0 For the C∗ -weight of Horb ((X, Y ), L), we have expansion: n+1 n n−1 wkorb = aorb + aorb ) 0 k 1 k + O(k

By orbifold equivariant Riemann-Roch, we have the formula: Z Z ω ˆn ωn orb ˆ θv θv a0 = = = a0 n! n! X X 7

aorb 1

=

n

ω ˆ θˆv S(ˆ ω) n! X

Z

To calculate the second coefficient aorb ˆ , then by (9): 1 , we choose an orbifold metric ω Z ω ˆ n−1 1 θˆv n Ric(ˆ ω) ∧ a1 = − 2 X n! Z ω ˆ n−1 1 θˆv n(Ric(Ω) − c1 ([D], h) + {D}) ∧ = − 2 X n! Z Z n−1 n ω ˆ ω ˆ 1 1 θˆv S(ˆ ω) θˆv − = − 2 X n! 2 D (n − 1)! Z 1 ω n−1 1 orb = aorb ˜0 = a1 − θv 1 + a 2 D (n − 1! 2 So

1 (2a1 − a ˜0 ) (13) 2 Comparing (12), we see that the log-Futaki invariant recovers the orbifold Futaki invariant, and similarly log-K-stability recovers orbifold K-stability. Orbifold Futaki and orbifold K-stability were studied by Ross-Thomas [9]. Pr Example 2. X = P1 , L = KP−1 1 = OP1 (2), Y = i=1 αi pi . For any i ∈ {1, · · · , r}, we choose the 1 coordinate z on P , such that z(pi ) = 0. Then consider the holomorphic vector field v = z∂z . v generates the 1 parameter subgroup λ(t) : λ(t) · z = t · z. As t → ∞, λ(t) degenerate (X, Y ) into P −|z|−2 +|z|2 the pair (P1 , αi {0} + j6=i αj {∞}). We take θv = |z| −2 +1+|z|2 . Then it’s easy to get the log-Futaki invariant of the degeneration determined by λ: aorb = 1

1

F (P ,

r X i=1

If (P1 ,

Pr

i=1

αi pi , OP1 (2))(λ) =

αi pi ) is log-K-stable, by Remark 5, we have X αj − αi > 0

X j6=i

αj − αi

(14)

j6=i

P Equivalently, if we let t → 0, we get αi − j6=i αj < 0 from log-K-stability. Let’s consider the problem of constructing singular Riemannian metric g of constant scalar curvature on P1 which has conic angle 2π(1 − αi ) at pi and is smooth elsewhere. Assume pi 6= ∞ for any i = 1, . . . , r. Under conformal coordinate z of C ⊂ P1 , g = e2u |dz|2 . u is a smooth function in the punctured complex plane C − {p1 , . . . , pr } so that near each pi , u(z) = −2αi log |z − pi |+a continuous function, where αi ∈ (0, 1) and u = −2 log |z|+ a continuous function near infinity. We call such function is of conic type. The condition of constant scalar curvature corresponds to the following Liouville equations. 1. ∆u = −e2u 2. ∆u = 0 3. ∆u = e2u which correspond to scalar curvature=1, 0, -1 case respectively. For such equations, we have the following nice theorem due to Troyanov, McOwen, Thurston, Luo-Tian.

8

Theorem 3 (See [7] and the reference there). 1. For equation 1, it has a solution of conic type if and only if Pr (a) i=1 αi < 2, and P (b) j6=i αj − αi > 0, for all i = 1, . . . , n. Pr 2. For equation 2, it has a solution of conic type if and only if (a): i=1 αi = 2. P In this case, (a) implies the condition: (b) j6=i αj − αi > 0, for all i = 1, . . . , r. Pr 3. For equation 3, it has a solution of conic type if and only if (a): i=1 αi > 2. P Again in this case, (a) implies the condition: (b) j6=i αj − αi > 0, for all i = 1, . . . , r. Moreover, the above solutions are all unique. Pr Pr Note that deg(−(KP1 + i=1 αi pi )) = 2 − i=1 αi , so by (4), conditions (a) in above theorem correspond to the cohomological conditions for the scalar curvature to be positive, zero, negative rePr 1 So by the above theorem, if (P , α p spectively. While the condition (b) is the same as (14). i=1 i i ) Pr 1 is log-K-stable, then there is a conic metric on (P , α p ) with constant curvature whose sign i=1 i i P is the same as that of 2 − i αi . This example clearly suggests

Conjecture 3 (Logarithmic version of Tian-Yau-Donaldson conjecture). There is a constant scalar curvature conic K¨ ahler metric on (X, Y ) if and only if (X, Y ) is log-K-stable.

5 5.1

Toric Fano case Log-Futaki invariant for 1psg on toric Fano variety

For a reflexive lattice polytope △ in Rn = Λ ⊗Z R, we have a Fano toric manifold (C∗ )n ⊂ X△ with a (C∗ )n action. In the following, we will sometimes just write X for X△ for simplicity. Let (S 1 )n ⊂ (C∗ )n be the standard real maximal torus. Let {zi } be the standard coordinates of the dense orbit (C∗ )n , and xi = log |zi |2 . We have Lemma 1. Any (S 1 )n invariant K¨ ahler metric ω on X has a potential u = u(x) on (C∗ )n , i.e. √ −1 ¯ ω = 2π ∂ ∂u. u is a proper convex function on Rn , and satisfies the momentum map condition: Du(Rn ) = △ Also, dz1 z1

¯ n /n! (∂ ∂u) z1 dzn ∧ d¯ z¯1 · · · ∧ zn ∧

d¯ zn z¯n

= det



∂2u ∂xi ∂xj



(15)

Let {pα ; α = 1, · · · , N } be all the lattice points of △. Each pα corresponds to a holomorphic −1 section sα ∈ H 0 (X△ , KX ). We can embed X△ into PN using {sα }. Define u to be the potential △ √ ¯ = ωF S ): on (C∗ )n for the pull back of Fubini-Study metric (i.e. −1 ∂ ∂u 2π

u = log

N X

e

α=1

!

+C

C is some constant determined by normalization condition: Z Z c1 (X△ )n 1 −u ωn = e dx = V ol(△) = n! X△ n! Rn 9

(16)

By the above normalization of u, it’s easy to see that e hω =

| · |2F S = 1 | · |2ωn ω n /( dz z1 ∧

e−u d¯ z1 z¯1

···∧

dzn zn



d¯ zn ) z¯n

So hω = − log det(uij ) − u

(17) ∗ n

Now let’s calculate the log-Futaki invariant for any 1-parameter subgroup in (C ) . Each 1parameter subgroup in (C∗ )n is determined by some λ ∈ Rn such that the generating holomorphic vector field is n X ∂ λi zi vλ = ∂z i i=1 A general Calabi-Yau hypersurface Y ∈ | − KX | is a hyperplane section given by the equation: s :=

N X

b(pα )z pα = 0

α=1

By abuse of notation, we denote λ(t) to be the 1 parameter subgroup generated by vλ , then λ(t) · s =

N X

b(pα )t−hpα ,λi z pα

(18)

α=1

Let W (λ) = maxp∈△ hp, λi

n

Then Hλ = {p ∈ R , hp, λi = W (λ)} is a supporting plane of △, and Fλ := {p ∈ △; hp, λi = W (λ)} = Hλ ∩ △ is a face of △. i h P We have limt→0 [s] = s0 := pα ∈Fλ b(pα )z pα , and by (18), the C ∗ -weight of s0 is −W (λ).

−1 Proposition 2. Let F (KX , βY )(λ) denote the Futaki invariant of the test configuration associated with the 1 parameter subgroup generated by vλ . We have −1 F (KX , βY )(λ) = − (βhPc , λi + (1 − β)W (λ)) V ol(△)

(19)

Proof. We will use the algebraic definition of log-Futaki invariant (12) to do the calculation. −1 −1 Note that (X, Y, KX ) degenerates to (X, Y0 , KX ) under λ. −1 0 Y0 is a hyperplane section of X, and s0 ∈ H (X, KX ) is the defining section, i.e. Y0 = {s0 = 0}. Then −(k−1)

−k −1 k ∼ )/(s0 ⊗ H 0 (X, KX H 0 (Y0 , KX |Y0 ) = H 0 (X, KX

So w ˜k = wk − (wk−1 − W (λ)dk−1 ) Plugging the expansions, we get a ˜0 = (n + 1)a0 + W (λ)b0 Note that ˜b0 = nb0 = nV ol(△), we have −˜ a0 +

˜b0 a0 = −a0 − W (λ)b0 b0 10

))

where −a0 =

Z

X

θv

ωn = n!

Z

Rn

X

λi ui det(uij )dx =



i

By (17), the ordinary Futaki invariant is given by F (c1 (X))(vλ )

Z

v(hω )

= −

Z X

=

Z X

X



i

ωn =− n!

Z

λi yi dy = V ol(△)hPc , λi

i

n X

Rn i=1

λi

∂u det(uij )dx ∂xi

λi yi dy = −V ol(△)hPc , λi

Substituting these into (12), we get −1 F (KX , βY )(λ)

= =

−V ol(△)hPc , λi + (1 − β)(V ol(△)hPc , λi − W (λ)V ol(△))

−(βhPc , λi + (1 − β)W (λ))V ol(△)

Proof of Theorem 1. Note that for any Pλ ∈ Fλ ⊂ ∂△, W (λ) = hPλ , λi. By Theorem 2, we have   β 1 − R(X) −1 F (KX , βY )(λ) = hQ, λi − W (λ) (1 − β)V ol(△) 1 − β R(X) = hQβ − Pλ , λi β 1−R(X) where Qβ = 1−β R(X) Q. Note that λ is a outward normal vector of Hλ . By convexity of △, it’s easy to see that (see the picture after Example 2)

• β < R(X): Qβ ∈ △◦ . For any λ ∈ Rn , hQβ − Pλ , λi < 0. • β = R(X): Qβ = Q ∈ ∂△. For any λ ∈ Rn , hQβ − Pλ , λi ≤ 0. Equality holds if and only if hQ, λi = W (λ), i.e. Hλ is a supporting plane of △ at point Q. • β > R(X): Qβ ∈ / △. There exists λ ∈ Rn such that hQβ − Pλ , λi > 0

5.2

Example

1. X△ = Blp P2 . See the picture in Introduction. Pc = 14 ( 13 , 31 ), Q = −6Pc ∈ ∂△, so R(X) = 67 . If we take λ = h−1, −1i, then W (λ) = 1. So by (19) −1 F (KX , βY )(λ) =

2 β − 4(1 − β) 3

−1 So F (KX , βY )(λ) ≤ 0 if and only if β ≤ 76 , and equality holds exactly when β = 67 .

2. X△ = Blp,q P2 , Pc = 27 (− 13 , − 13 ), Q = − 21 4 Pc ∈ ∂△, so R(X△ ) = If we take λ1 = h1, 1i, then W (λ1 ) = 1. By (19), −1 F (KX , βY )(λ1 ) = −1 F (KX , βY )(λ1 ) ≤ 0 if and only if β ≤

7 2 β − (1 − β) 3 2

21 25 .

This is essentially the same as Donaldson’s calculation in [4]. 11

21 25 .

If we take λ3 = h−1, 2i, then W (λ3 ) = h−1, 2i · h−1, 1i = 3. By (19) −1 F (KX , βY )(λ3 ) = −1 So F (KX , βY )(λ3 ) ≤ 0 if and only if β ≤ 63 λ3 when β ≤ 21 25 < 65 .

AAλ3 H3 K Pr3  A 

63 65

21 1 β − (1 − β) 3 2

which means that (X, βY ) is log-K-stable along

q λ1 Q 21 @ r r > 25 [email protected] 1 s Q 21 r [email protected] 25 @ < 25 @q sq

q Pc q

Pr2 λ ?2

q

References [1] Berman, R.: A thermodynamic formalism for Monge-Amp`ere euations, Moser-Trudinger inequalities and K¨ ahler-Einstein metrics arXiv:1011.3976 [2] Calabi, E.: Extremal K¨ ahler metrics II, in ”Differential geometry and complex analysis”, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 95-114. [3] Donaldson, S.K.: Scalar curvature and stability of toric varieties, Jour. Differential Geometry 62 289-349 (2002) [4] Donaldson, S.K.: K¨ ahler metrics with cone singularities along a divisor, arXiv:1102.1196 [5] Futaki, A.: An obstruction to the existence of Einstein K¨ahler metrics, Invent. Math., 73, 437-443 (1983) [6] Li, C.: Greatest lower bounds on Ricci curvature for toric Fano manifolds, arXiv:0909.3443 [7] Luo, F. and Tian, G.: Liouville equation and spherical convex polytopes, Proceedings of the American Mathematical Society, Vol 116, No. 4 (1992) 1119-1129 [8] Oda, T.: Convex bodies and algebraic geometry-an introduction to the theory of toric varieties, Springer-Vergla, 1988 [9] Ross, J., and Thomas, R.: Weighted projective embeddings, stability of orbifolds and constant scalar curvature K¨ ahler metrics arXiv:0907.5214 [10] Sz´ekelyhidi, G.: Greatest lower bounds on the Ricci curvature of Fano manifolds, arXiv:0903.5504 [11] Tian, G.: K¨ ahler-Einstein metrics on algebraic manifolds. Lecture notes in Mathematics, 1996, Volume 1646/1996, 143-185. 12

[12] Tian, G.: Canonical Metrics on K¨ ahler Manifolds, Birkhauser, 1999 [13] Tian, G.: On stability of the tangent bundles of Fano varieties. Internat. J. Math. 3, 3(1992), 401-413 [14] Tian, G.: K¨ ahler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1-39 [15] Wang, X.J. and Zhu, X.H.: K¨ ahler-Ricci solitons on toric manifolds with positive first Chern class. Advances in Math. 188 (2004) 87-103 [16] Yau, S.T.: On the Ricci curvature of a compact K¨ahler manifold and the complex MongeAmp`ere equation, I, Comm. Pure Appl. Math. 31 (1978) 339-441. Department of Mathematics, Princeton University, Princeton, NJ 08544, USA E-mail address: [email protected]

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2011-04-03
English