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

TL;DR

This paper offers a straightforward method to recover a known criterion for K-stability in Fano reductive group compactifications, simplifying the understanding of stability conditions in this geometric context.

Contribution

It introduces an elementary approach to derive the K-stability criterion for Fano reductive group compactifications, enhancing conceptual clarity.

Findings

Simplified derivation of K-stability criterion

Clarification of stability conditions for Fano compactifications

Potential for broader application in geometric stability analysis

Abstract

We present an elementary way of recovering a well-known criterion of K-stability for Fano reductive group compactifications.

Equations

$$\dim (\text{mbox}End(E_{\lambda }))=(\dim (E_{\lambda }){)}^2=H_d(\lambda )+H_{d-1}(\lambda )+\dots$$

$$R_P=\underset{\lambda \in C(P^{+})\cap (M\times \mathbb{Z})}{⨁}\text{mbox}End(E_{\lambda }).$$

$$P=\{x\in M_{\mathbb{R}}|⟨a_i,x⟩\ge ⟨a_i,2\rho ⟩-1\}.$$

$$-F_{1}(f)=\frac{1}{2\int_{P^{+}}H_{d}d\mu}\Big{(}\int_{\partial P^{+}}fH_{d}d\sigma+2\int_{P^{+}}fH_{d-1}d\mu-a\int_{P^{+}}fH_{d}d\mu\Big{)},$$

$$a=\frac{{\int }_{\partial P^{+}}{H}_{d}d\sigma +2{\int }_{P^{+}}{H}_{d-1}d\mu }{{\int }_{P^{+}}{H}_{d}d\mu }.$$

$$H_d(x)=\frac{1}{c}\prod _{i=1}^r⟨{\alpha }_i,x{⟩}^2,$$

$$H_{d-1}(x)=\frac{1}{c}\sum _{j=1}^{r}2⟨{\alpha }_j,x⟩⟨{\alpha }_j,\rho ⟩⟨{\alpha }_1,x{⟩}^2\dots {\widehat{⟨{\alpha }_j,x⟩}}^2\dots ⟨{\alpha }_r,x{⟩}^2,$$

$$\nabla H_{d}(x)=\frac{1}{c}\sum_{j=1}^{n}\big{(}\sum_{i=1}^{n}2\langle\alpha_{i},x\rangle\langle\alpha_{i},e_{j}\rangle\langle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{j},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\big{)}e_{j},$$

$$div((x-2\rho )f{H}_d)=⟨\nabla f,x-2\rho ⟩H_d+(2r+n)f{H}_d-2f{H}_{d-1}.$$

$$\dim (E_x)=\frac{{\prod }_{i=1}^r⟨{\alpha }_i,x+\rho ⟩}{{\prod }_{i=1}^r⟨{\alpha }_i,\rho ⟩}.$$

$$\dim (E_x{)}^2=\frac{1}{c}\prod _{i=1}^r(⟨{\alpha }_i,x{⟩}^2+2⟨{\alpha }_i,x⟩⟨{\alpha }_i,\rho ⟩+⟨{\alpha }_i,\rho {⟩}^2),$$

$$H_d(x)=\frac{1}{c}\prod _{i=1}^r⟨{\alpha }_i,x{⟩}^2,$$

$$H_{d-1}(x)=\frac{1}{c}\sum _{j=1}^{r}2⟨{\alpha }_j,x⟩⟨{\alpha }_j,\rho ⟩⟨{\alpha }_1,x{⟩}^2\dots {\widehat{⟨{\alpha }_j,x⟩}}^2\dots ⟨{\alpha }_r,x{⟩}^2.$$

$$\frac{\partial }{\partial x_j}{H}_d(x)=\frac{1}{c}\sum _{i=1}^{r}2⟨{\alpha }_i,x⟩⟨{\alpha }_i,e_j⟩⟨{\alpha }_1,x{⟩}^2\dots {\widehat{⟨{\alpha }_i,x⟩}}^2\dots ⟨{\alpha }_r,x{⟩}^2,$$

$$\nabla H_{d}(x)=\sum_{j=1}^{r}\frac{\partial}{\partial x_{j}}H_{d}(x)e_{j}\\ =\frac{1}{c}\sum_{j=1}^{n}\big{(}\sum_{i=1}^{n}2\langle\alpha_{i},x\rangle\langle\alpha_{i},e_{j}\rangle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{j},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\big{)}e_{j}.$$

$$\begin{split}\nabla H_{d}(x)&=\frac{1}{c}\sum_{j=1}^{n}\big{(}\sum_{i=1}^{n}2\langle\alpha_{i},x\rangle\langle\alpha_{i},e_{j}\rangle\langle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{j},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\big{)}e_{j}\\ &=\sum_{i=1}^{r}\big{(}\frac{1}{c}2\langle\alpha_{i},x\rangle\langle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{i},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\big{)}\sum_{j=1}^{n}\langle\alpha_{i},e_{j}\rangle e_{j}\\ &=\sum_{i=1}^{r}\big{(}\frac{1}{c}2\langle\alpha_{i},x\rangle\langle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{i},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\big{)}\alpha_{i}\end{split}$$

$$⟨\nabla H_d(x),\rho ⟩=\sum _{i=1}^r\frac{1}{c}2⟨{\alpha }_i,x⟩⟨{\alpha }_i,\rho ⟩⟨{\alpha }_1,x{⟩}^2\dots {\widehat{⟨{\alpha }_i,x⟩}}^2\dots ⟨{\alpha }_r,x{⟩}^2=H_{d-1}(x).$$

$$\nabla H_{d}(x)=\sum_{i=1}^{r}\big{(}\frac{1}{c}2\langle\alpha_{i},x\rangle\langle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{i},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\big{)}\alpha_{i},$$

$$\big{\langle}\frac{1}{c}2\langle\alpha_{i},x\rangle\langle\alpha_{1},x\rangle^{2}\dots\widehat{\langle\alpha_{i},x\rangle}^{2}\dots\langle\alpha_{r},x\rangle^{2}\alpha_{i},x\big{\rangle}=2\big{(}\frac{1}{c}\langle\alpha_{1},x\rangle^{2}\dots\dots\langle\alpha_{r},x\rangle^{2}\big{)},$$

$$\langle\nabla H_{d}(x),x\rangle=2r\Big{(}\frac{1}{c}\prod_{i=1}^{r}\langle\alpha_{i},x\rangle^{2}\Big{)}=2rH_{d}(x).$$

$$\begin{array}{rl}div((x-2\rho )f{H}_d)& =⟨\nabla (f{H}_d),x-2\rho ⟩+div(x-2\rho )f{H}_d\\ & =⟨\nabla f,x-2\rho ⟩H_d+⟨\nabla H_d,x-2\rho ⟩f+nf{H}_d\\ & =⟨\nabla f,x-2\rho ⟩H_d+⟨\nabla H_d,x⟩f-2⟨\nabla H_d,\rho ⟩f+nf{H}_d\\ & =⟨\nabla f,x-2\rho ⟩H_d+(2r+n)f{H}_d-2f{H}_{d-1}.\end{array}$$

$$-F_1(f)=\frac{1}{2Vo{l}_{DH}(P^{+})}{\int }_{P^{+}}⟨\nabla f,x-2\rho ⟩H_{d}d\mu =\frac{1}{2}⟨ba{r}_{DH}(P^{+})-2\rho ,\nabla f⟩,$$

$$\begin{split}\langle(x-2\rho)fH_{d},-\frac{a_{i}}{\|a_{i}\|}\rangle&=-H_{d}f\Big{(}\Big{\langle}x-2\rho,\frac{a_{i}}{\|a_{i}\|}\Big{\rangle}\Big{)}\\ &=\frac{-H_{d}(x)f}{\|a_{i}\|}\Big{(}\langle x,a_{i}\rangle-\langle 2\rho,a_{i}\rangle\Big{)}\\ &=\frac{H_{d}(x)f}{\|a_{i}\|}\Big{(}\langle 2\rho,a_{i}\rangle-c_{i}\Big{)}\\ &=\frac{H_{d}(x)f}{\|a_{i}\|}.\end{split}$$

$${\int }_{P^{+}}div((x-2\rho )f{H}_d)d\mu =\sum _{i=1}^m{\int }_{\partial P_i^{+}}\frac{H_{d}f}{\Vert a_i\Vert }d{\sigma }_i+\sum _{i=m+1}^k{\int }_{\partial P_i^{+}}⟨(x-2\rho )f{H}_d,-\frac{a_i}{\Vert a_i\Vert }⟩d{\sigma }_i,$$

$${\int }_{\partial P_i^{+}}⟨(x-2\rho )f{H}_d,\frac{a_i}{\Vert a_i\Vert }⟩d{\sigma }_i=0.$$

$${\int }_{P^{+}}div((x-2\rho )f{H}_d)d\mu =\sum _{i=1}^m{\int }_{\partial P_i^{+}}\frac{H_d(x)f}{\Vert a_i\Vert }d{\sigma }_i$$

$${\int }_{\partial P^{+}}f{H}_{d}d\sigma .$$

$$div((x-2\rho )H_d)=(2r+n)H_d-2H_{d-1}.$$

$${\int }_{\partial P^{+}}{H}_{d}d\sigma =(2r+n){\int }_{P^{+}}{H}_{d}d\mu -2{\int }_{P^{+}}{H}_{d-1}d\mu .$$

$$a=\frac{{\int }_{\partial P^{+}}{H}_{d}d\sigma +2{\int }_{P^{+}}{H}_{d-1}d\mu }{{\int }_{P^{+}}{H}_{d}d\mu }=2r+n.$$

Full text

Key Words: Donaldson-Futaki invariant, K-stability, reductive varieties

Reductive group compactifications from polytopes. Let $G$ be a reductive group and $T\subset G$ be a maximal torus with character lattice $M,$ Lie algebra $\mathfrak{t},$ and dual Lie algebra $\mathfrak{t}^{*}\simeq M_{\mathbb{R}}:=M\otimes\mathbb{R}.$ Let $W$ be the Weyl group of $(G,T),$ and let $\Phi$ denote the root system of $(G,T)$ with a fixed choice of positive roots $\Phi^{+}.$ We declare $2\rho$ to be the sum of the positive roots. The positive Weyl chamber is $M_{\mathbb{R}}^{+}:=\{x\in M_{\mathbb{R}}|\langle\alpha,x\rangle\geq 0\mbox{ for all }\alpha\in\Phi^{+}\}.$ There is a one-to-one correspondence between lattice points $\lambda\in M_{\mathbb{R}}^{+}$ and irreducible $G-$representations $E_{\lambda}.$ Furthermore, to a lattice point $\lambda\in M_{\mathbb{R}}^{+}$ corresponds a $G\times G-$representation ${\mbox{End}}(E_{\lambda}).$ The dimension of ${\mbox{End}}(E_{\lambda})$ is a polynomial

[TABLE]

in $\lambda,$ and here $H_{d}$ stands for the degree $d$ homogeneous part of the polynomial $\dim({\mbox{End}}(E_{\lambda})),$ $H_{d-1}$ stands for the degree $d-1$ part, and so on.

Let $P^{+}:=P\cap M^{+}_{\mathbb{R}},$ $C(P^{+})\subseteq M_{\mathbb{R}}\times\mathbb{R}$ be the cone over $(P^{+},1),$ and consider the finitely generated algebra

[TABLE]

To any $W-$invariant lattice polytope $P\subseteq M_{\mathbb{R}},$ we can associate a polarized reductive group compactification $(X_{P},L_{P}),$ where $X_{P}=Proj(R_{P})$ and $L_{P}=\mathcal{O}(1).$

The Fano condition. At the polytope level, Fano is the condition that the distance between $2\rho$ 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 $X_{P}$ by $Z,$ which is a toric subvariety of $X_{P}.$ When $X_{P}$ is Fano, the support function $v$ of $P$ is of the form $v=v_{K_{\mathbb{C}}}+v_{Z},$ where $v_{K_{\mathbb{C}}}(x)=\langle 2\rho,x\rangle$ for all $x$ in the positive Weyl chamber, $v_{K_{\mathbb{C}}}(wx)=v_{K_{\mathbb{C}}}(x)$ for all $w\in W,$ and $v_{Z}(x)=-g_{-K_{Z}}(-x),$ where $-g_{-K_{Z}}$ is the support function of the anti-canonical line bundle of the toric subvariety $Z\subset X_{P}.$ Since $P$ is also the polytope of $Z,$ the associated fan $\Sigma_{P}$ gives rise to the toric subvariety $Z.$ From the theory of toric varieties, $-K_{Z}=\sum_{\rho\in\Sigma(1)}D_{\rho},$ where $\Sigma(1)$ is the set of $1$-dimensional cones of $\Sigma_{P}$ and $D_{\rho}$ is a prime torus invariant divisor on $Z.$ The support function $g_{-K_{Z}}$ has the property that $g_{-K_{Z}}(u_{\rho})=-1$ for all $\rho\in\Sigma(1),$ where $u_{\rho}$ is the minimal generator of the ray $\rho.$ In particular, if $a_{i}$ is the inward pointing normal to the $i$-th codimension one face of $P,$ $g_{-K_{Z}}(a_{i})=-g_{-K_{Z}}(-a_{i})=-1.$ Then, $v(a_{i})=\langle a_{i},2\rho\rangle-1,$ and so the facet presentation of the polytope is

[TABLE]

As a consequence, the equation that defines the $i$-th boundary face of $P$ is $f_{i}(x)=\langle a_{i},x-2\rho\rangle+1$ so that $f_{i}(2\rho)=1.$

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

[TABLE]

where

[TABLE]

Here $d\mu$ is the Lebesgue measure restricted to $P,$ and the boundary measure $d\sigma$ is a positive measure on $\partial P$ that is normalized so that on each codimension one face, which is defined by an equation $l(x):=\langle a,x\rangle=c,$ $d\sigma\wedge dl=\pm d\mu$ holds.

Choose once and for all an isomorphism $M_{\mathbb{R}}\simeq\mathbb{R}^{n}$ so that $P$ can be viewed as though contained in $\mathbb{R}^{n}.$

[TABLE] 2. 2.

[TABLE] 3. 3.

[TABLE] 4. 4.

$\langle\nabla H_{d}(x),\rho\rangle=H_{d-1}(x),$** 5. 5.

$\langle\nabla H_{d}(x),x\rangle=2rH_{d}(x),$* and* 6. 6.

for any smooth function $f:P\rightarrow\mathbb{R},$

[TABLE]

[TABLE]

From the expression

[TABLE]

it follows that if $d$ is the highest degree homogeneous part of the polynomial $\dim(E_{x})^{2},$ then

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and the $(d-1)-$degree homogeneous part of $\dim(E_{x})^{2}$ is

[TABLE]

For 3., note that $\frac{\partial}{\partial x_{j}}\langle\alpha_{i},x\rangle=\langle\alpha_{i},e_{j}\rangle$ so that

[TABLE]

and hence

[TABLE]

For 4., notice that

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and then

[TABLE]

For 5., observe that since

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and since for each $i,$

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indeed we have that

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The above identities now imply the last point. Namely,

[TABLE]

∎

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

[TABLE]

where $bar_{DH}(P^{+})=\frac{1}{Vol_{DH}(P^{+})}\int_{P^{+}}xH_{d}d\mu$ and $Vol_{DH}(P^{+})=\int_{P^{+}}H_{d}d\mu$ are the barycenter and respectively the volume of $P^{+}$ with respect to the Duistermaat-Heckman (DH) measure.

[TABLE]

The divergence theorem implies that

[TABLE]

where $d\sigma_{i}$ is the standard Lebesgue measure on $\partial P$ with domain restricted to $\partial P_{i}.$ When $i=m+1,\dots,k,$ $\partial P_{i}^{+}$ is in the boundary of the positive Weyl chamber, and

[TABLE]

Then

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and the right hand side is the definition of

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

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Then, by the divergence theorem,

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Hence,

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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

[TABLE]

Suppose that $f$ on $P^{+}$ is given as $f(x)=\sum_{j=1}^{n}b_{j}x_{j}+k.$ Put $b=(b_{1},\dots,b_{n}),$ $x=(x_{1},\dots,x_{n})$ and $2\rho=(2\rho_{1},\dots,2\rho_{n}),$ and let $e_{j}$ be the $j-$th standard basis vector of $\mathbb{R}^{n}.$ Then $\langle\nabla f,x-2\rho\rangle=\sum_{j=1}^{n}b_{j}(x_{j}-2\rho_{j})$ and it follows that

[TABLE]

∎

Acknowledgment. I wrote up this note while receiving funding from the European Research Council, grant project “Algebraic and Kähler geometry” (ALKAGE, no. 670846). The note 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.