A technical remark on the Donaldson-Futaki invariant for Fano reductive group compactifications

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A technical remark on the Donaldson-Futaki invariant for Fano reductive group compactifications

Gabriella Clemente

To cite this version:

Gabriella Clemente. A technical remark on the Donaldson-Futaki invariant for Fano reductive group

compactifications. 2019. �hal-02267071�

A technical remark on the

Donaldson-Futaki invariant for Fano reductive group compactifications

Gabriella Clemente

Abstract

We present an elementary way of computing the Donaldson-Futaki invariant associated to a test-configuration of an anti-canonically po- larized Fano reductive group compactification.

Reductive group compactifications from polytopes. Let G be a re- ductive group and T ⊂ G be a maximal torus with character lattice M, Lie algebra t , and dual Lie algebra t

≃ M

:= M ⊗ R . Let W be the Weyl group of (G, T ), and let Φ denote the root system of (G, T ) with a fixed choice of positive roots Φ

. We declare 2ρ to be the sum of the positive roots.

The positive Weyl chamber is M

:= { x ∈ M

|h α, x i ≥ 0 for all α ∈ Φ

} . There is a one-to-one correspondence between lattice points λ ∈ M

and irreducible G −representations E

. Furthermore, to a lattice point λ ∈ M

corresponds a G × G −representation End(E

). The dimension of End(E

) is a polynomial

dim(End(E

)) = (dim(E

))

= H

(λ) + H

(λ) + . . .

in λ, and here H

stands for the degree d homogeneous part of the poly- nomial dim(End(E

)), H

stands for the degree d − 1 part, and so on.

Let P

:= P ∩ M

, C(P

) ⊆ M

× R be the cone over (P

, 1), and consider the finitely generated algebra

R

= M

λ∈C(P⁺)∩(M×Z)

End(E

).

To any W −invariant lattice polytope P ⊆ M

, we can associate a polarized reductive group compactification (X

, L

), where X

= Proj(R

) and L

= O(1).

The Fano condition. At the polytope level, Fano is the condition that the distance between 2ρ and any codimension one face of P

that does not meet the boundary of the positive Weyl chamber is equal to one. This is a result that can be found in [3], and which we recapitulate below.

Denote the Zariski closure of T in X

by Z, which is a toric subvariety of X

. When X

is Fano, the support function v of P is of the form v = v

+ v

, where v

(x) = h2ρ, x i for all x in the positive Weyl chamber, v

(wx) = v

(x) for all w ∈ W , and v

(x) = − g

(− x), where − g

is the support function of the anti-canonical line bundle of the toric subvariety Z ⊂ X

. Since P is also the polytope of Z, the associated fan Σ

gives rise to the toric subvariety Z. From the theory of toric varieties, − K

= P

ρ∈Σ(1)

D

, where Σ (1) is the set of 1-dimensional cones of Σ

and D

is a prime torus invariant divisor on Z. The support function g

has the property that g

(u

) = −1 for all ρ ∈ Σ (1), where u

is the minimal generator of the ray ρ. In particular, if a

is the inward pointing normal to the i -th codimension one face of P, g

(a

) = − g

(− a

) = −1. Then, v(a

) = h a

, 2ρ i − 1, and so the facet presentation of the polytope is

P = { x ∈ M

|h a

, x i ≥ h a

, 2ρ i − 1} .

As a consequence, the equation that defines the i -th boundary face of P is f

(x) = ha

, x − 2ρi + 1 so that f

(2ρ) = 1.

Calculation of the Donaldson-Futaki invariant. In the sequel, we ob- tain a number of identities that together with the Fano condition will allow us to simplify Alexeev’s and Katzarkov’s Donaldson-Futaki (DF) invariant:

Theorem. (Theorem 3.3, [1]) Let f be a convex rational W −invariant piece- wise linear function on P. Then the DF invariant of the corresponding test- configuration is given by the formula

−F

(f ) = 1 2 R

P⁺

H

dµ Z

∂P⁺

f H

dσ + 2 Z

P⁺

f H

dµ − a Z

P⁺

f H

dµ ,

where

a = R

∂P⁺

H

dσ + 2 R

P⁺

H

dµ

R H

dµ .

Here dµ is the Lebesgue measure restricted to P, and the boundary measure dσ is a positive measure on ∂P that is normalized so that on each codimension one face, which is defined by an equation l (x) := h a, x i = c, dσ ∧ dl = ± dµ holds.

Choose once and for all an isomorphism M

≃ R

so that P can be viewed as though contained in R

.

Claim. Let Φ

= { α

, . . . , α

} , c = Q

i=1

h α

, ρ i

, where ρ =

P

i=1

α

, and let { e

}

be the standard basis of R

. Then,

1.

H

(x) = 1 c

Y

h α

, x i

,

2.

H

(x) = 1 c

X

2h α

, x ih α

, ρ ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

, 3.

∇ H

(x) = 1 c

X

X

i=1

2h α

, x ih α

, e

ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

e

,

4. h∇ H

(x), ρ i = H

(x), 5. h∇H

(x), xi = 2rH

(x), and

6. for any smooth function f : P → R ,

div((x − 2ρ)f H

) = h∇f , x − 2ρiH

+ (2r + n)f H

− 2f H

. Proof. Let E

be an irreducible representation with highest weight x. To prove 1. and 2., we make use of the Weyl dimension formula

dim(E

) = Q

i=1

h α

, x + ρ i Q

i=1

h α

, ρ i .

From the expression

dim(E

)

= 1 c

Y

(hα

, xi

+ 2hα

, xihα

, ρi + hα

, ρi

),

it follows that if d is the highest degree homogeneous part of the polyno- mial dim(E

)

, then

H

(x) = 1 c

Y

h α

, x i

,

and the (d − 1)−degree homogeneous part of dim(E

)

is H

(x) = 1

c X

j=1

2h α

, x ih α

, ρ ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

.

For 3., note that

j

hα

, xi = hα

, e

i so that

∂

∂x

H

(x) = 1 c

X

2h α

, x ih α

, e

ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

, and hence

∇ H

(x) = X

j=1

∂

∂x

H

(x)e

= 1 c

X

X

i=1

2h α

, x ih α

, e

i α

, x i

. . . h [ α

, x i

. . . h α

, x i

e

.

For 4., notice that

∇H

(x) = 1 c

X

X

i=1

2hα

, xihα

, e

ihα

, xi

. . . hα [

, xi

. . . hα

, xi

e

= X

i=1

1

c 2hα

, xihα

, xi

. . . hα [

, xi

. . . hα

, xi

X

j=1

hα

, e

ie

= X

i=1

1

c 2h α

, x ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

α

and then

h∇H

(x), ρi = X

i=1

1

c 2hα

, xihα

, ρihα

, xi

. . . hα [

, xi

. . . hα

, xi

= H

(x).

For 5., observe that since

∇ H

(x) = X

i=1

1

c 2h α

, x ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

α

,

and since for each i, D 1

c 2h α

, x ih α

, x i

. . . h [ α

, x i

. . . h α

, x i

α

, x E

= 2 1

c h α

, x i

. . . . h α

, x i

, indeed we have that

h∇ H

(x), x i = 2r 1

c Y

i=1

h α

, x i

= 2rH

(x).

The above identities now imply the last point. Namely, div((x − 2ρ)f H

) = h∇(f H

), x − 2ρ i + div(x − 2ρ)f H

= h∇f , x − 2ρiH

+ h∇H

, x − 2ρif + nf H

= h∇f , x − 2ρiH

+ h∇H

, xif − 2h∇H

, ρif + nf H

= h∇f , x − 2ρiH

+ (2r + n)f H

− 2f H

.

The following is analogous to Theorem C in [2].

Proposition. Suppose that P satisfies the Fano condition. Let f : P → R be a function as in the theorem that is affine linear on P

. Then, the DF invariant of the test-configuration associated to f is given by

−F

(f ) = 1 2V ol

(P

)

Z

P⁺

h∇f , x − 2ρiH

dµ = 1

2 hbar

(P

) − 2ρ, ∇f i, where bar

(P

) =

DH(P⁺)

R

P⁺

xH

dµ and V ol

(P

) = R

P⁺

H

dµ are the

barycenter and respectively the volume of P

with respect to the Duistermaat-

Heckman (DH) measure.

Proof. Suppose that ∂P

has k codimension one faces ∂P

. Let { ∂P

: i = 1, . . . , m } be the set of all codimension one faces of P

that do not intersect the boundary of the positive Weyl chamber. Suppose that ∂P

is defined by h a

, x i − c

= 0 and set f

(x) := h a

, x i − c

. The (inward) unit normal vector field to ∂P

is −

ik

= −

ik

. Since P

satisfies the Fano condition, for x ∈

∂P

, we have that

h(x − 2ρ)f H

, − a

ka

k i = − H

f

x − 2ρ, a

ka

k

= −H

(x)f k a

k

h x, a

i − h2ρ, a

i

= H

(x)f k a

k

h2ρ, a

i − c

= H

(x)f k a

k . The divergence theorem implies that Z

P⁺

div((x −2ρ)f H

)dµ = X

i=1

Z

∂P_i⁺

H

f ka

k dσ

+

X

Z

∂P_i⁺

h(x −2ρ)f H

, − a

ka

k i dσ

, where dσ

is the standard Lebesgue measure on ∂P with domain restricted to ∂P

. When i = m + 1, . . . , k, ∂P

is in the boundary of the positive Weyl chamber, and

Z

∂P_i⁺

h(x − 2ρ)f H

, a

ka

k i dσ

= 0.

Then

Z

P⁺

div((x − 2ρ)f H

)dµ = X

i=1

Z

∂P_i⁺

H

(x)f k a

k dσ

and the right hand side is the definition of

Z

∂P⁺

f H

dσ.

By 6. of the claim, taking f = 1, we obtain that

div((x − 2ρ)H

) = (2r + n)H

− 2H

.

Then, by the divergence theorem, Z

∂P⁺

H

dσ = (2r + n) Z

P⁺

H

dµ − 2 Z

P⁺

H

dµ.

Hence,

a = R

∂P⁺

H

dσ + 2 R

P⁺

H

dµ R

P⁺

H

dµ = 2r + n.

Upon substituting the above calculations into Alexeev’s and Katzarkov’s DF invariant (cf. Theorem), again using 6. of the claim to rewrite the first integral, we find that

− F

(f ) = 1 2 R

P⁺

H

dµ Z

P⁺

h∇ f , x − 2ρ i H

dµ.

Suppose that f on P

is given as f (x) = P

j=1

b

x

+ k. Put b = (b

, . . . , b

), x = (x

, . . . , x

) and 2ρ = (2ρ

, . . . , 2ρ

), and let e

be the j −th standard basis vector of R

. Then h∇f , x − 2ρi = P

j=1

b

(x

− 2ρ

) and it follows that

−F

(f ) = 1 2V ol

(P

)

Z

P⁺

h∇f , x − 2ρiH

dµ

= 1

2V ol

(P

) X

j=1

b

Z

P⁺

x

H

dµ − X

j=1

b

(2ρ

)V ol

(P

)

= 1 2

h bar

(P

), X

j=1

b

e

i − X

j=1

b

(2ρ

)

= 1 2

h bar

(P

), b i − h b, 2ρ i

= 1

2 h bar

(P

) − 2ρ, ∇ f i .

Acknowledgment. I thank Jean-Pierre Demailly, my PhD supervisor,

for his encouragement to make this computation available. I thank the

European Research Council for financial support in the form of a PhD

grant from the project “Algebraic and K¨ahler geometry” (ALKAGE, no.

670846). The present work is an excerpt of my Master’s degree project, which I completed in May of 2017, under the supervision of Richard Hind.

I thank him and Mark Behrens, who funded me during the final stage of my Master’s degree. I also thank those faculty members of the Notre Dame mathematics department who were supportive of my work.

References

[1] V. Alexeev and L. Katzarkov, On K-stability of reductive varieties, G.

Geom. Funct. Anal. 15 (2005), no. 2, 297-310.

[2] T. Delcroix, K-stability of Fano spherical varieties, arXiv:1608.01852.

[3] A. Ruzzi, Fano symmetric varieties with low rank, Publ. Res. Inst. Math.

Sci. 48 (2012), no. 2, 235-278.

Gabriella Clemente

Universit´e Grenoble Alpes, Institut Fourier, UMR 5582 du CNRS, 100 rue des Maths, 38610 Gi`eres, France

email: [email protected]