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Tian’s \(\alpha _{m,k}^{{\hat{K}}}\)-invariants on group compactifications

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Abstract

In this paper, we compute Tian’s \(\alpha _{m,k}^{K\times K}\)-invariant on a polarized G-group compactification, where K denotes a maximal compact subgroup of a connected complex reductive group G. We prove that Tian’s conjecture (see Conjecture 1.1 below) is true for \(\alpha _{m,k}^{K\times K}\)-invariant on such manifolds when \(k=1\), but it fails in general by producing counter-examples when \(k\ge 2\).

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Notes

  1. We need not assume that the G-group compactification is Fano.

  2. When \(G=T^{\mathbb {C}}\), we set \({\mathfrak {a}}_+={\mathfrak {a}}\).

  3. When G is not semi-simple, \(\psi \) is determined up to an affine function \(l_\xi (x)=\xi _ix^i\) whose gradient \(\xi \) lie in the center \({\mathfrak {z}}({\mathfrak {g}})\). We may choose \(\xi \) such that the image of \(\nabla \psi \) is 2P.

  4. We would like to thank Professor D. A. Timashëv for telling us his paper [31].

  5. \(u_A\) denotes the outer norm in [19] but here is the inner one

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Acknowledgements

The authors would like to thank Professor Gang Tian for inspiring conversations on the paper. They would also like to thank referee for many valuable comments, especially on Lemma 3.1.

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Correspondence to Yan Li.

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Y. Li: Partially supported by China Post-doctoral Grant BX20180010. X. Zhu: Partially supported by NSFC Grants 11771019 and BJSF Grants Z180004.

Appendix: a direct proof of Lemma 3.1

Appendix: a direct proof of Lemma 3.1

In this Appendix we give a direct proof of (3.7) by an argument in [19, Section 5]. First we need the following lemma which is essentially a corollary of [31, Theorem 9].Footnote 4

Lemma 6.1

Suppose that \(\alpha \in \Phi _+\) is a simple root and denote by \(W_\alpha \) the Weyl wall

$$\begin{aligned} W_\alpha =\{y|~\alpha (y)=0\}. \end{aligned}$$

Let F be any facet of P, which is not orthogonal to \(W_\alpha \) such that

$$\begin{aligned} F\cap W_\alpha \not =\emptyset ,~F\cap {\mathfrak {a}}_+\not =\emptyset . \end{aligned}$$

Then the prime inner norm of F satisfies

$$\begin{aligned} \alpha (u)=-1. \end{aligned}$$
(6.1)

Proof

By assumption, there is a vertex \(p_0\) of F that lies in \(W_\alpha \). Let \({\mathfrak {L}}\) be the Levi group of the parabolic subgroup \(P(p_0)\subseteq G\) corresponding to \(p_0\), and \({\mathfrak {Z}}\) be the closure of \({\mathfrak {L}}\) (called the transversal slice) defined in [31, Section 8]. Then there is a neighbourhood \({\mathcal {U}}\) of \(p_0\), such that we can realize \(\text {Span}_{{\mathbb {R}}_+}\{({\mathfrak {a}}_+\cap {\mathcal {U}})-p_0\}\) as the positive Weyl chamber \({\mathfrak {a}}_{{\mathfrak {L}},+}\) of \({\mathfrak {L}}\), and

$$\begin{aligned} {\mathfrak {C}}:=\text {Span}_{{\mathbb {R}}_+}\{P-p_0\}\cap {\mathfrak {M}} \end{aligned}$$

as the cone generated by the weights of the \({\mathfrak {L}}\)-linear representation \({\mathbb {V}}_0(-p_0)\) defining \({\mathfrak {Z}}\), respectively (cf. [31, Proposition 6]).

The smoothness criterion [31, Theorem 9] implies that \({\mathfrak {L}}\) is a product of general linear groups

$$\begin{aligned} {\mathfrak {L}}=\prod _{k=1}^{m_0}GL_{n_k}({\mathbb {C}}), \end{aligned}$$

and \({\mathfrak {Z}}\) is the product of the corresponding matrix algebras

$$\begin{aligned} {\mathfrak {Z}}=\prod _{k=1}^{m_0}\text {Mat}_{n_k\times n_k}({\mathbb {C}}). \end{aligned}$$

Thus \({\mathfrak {C}}\) is generated by all weights of \(\cup _k\{\epsilon ^k_1,...,\epsilon ^k_{n_k}\}\), where \(\{\epsilon ^k_1,...,\epsilon ^k_{n_k}\}\) are the weights of the natural \(GL_{n_k}({\mathbb {C}})\)-action on \(\text {Mat}_{n_k\times n_k}({\mathbb {C}})\). Also the cone

$$\begin{aligned} {\mathfrak {C}}_+:={\mathfrak {a}}_{{\mathfrak {L}},+}\cap {\mathfrak {C}} \end{aligned}$$

is generated by the weights (cf. [31, Section 11])

$$\begin{aligned} \{\sum _{j=l}^l\epsilon ^k_j|1\le j\le n_k\}. \end{aligned}$$

On the other hand, by our assumption, we see that

$$\begin{aligned} {\mathfrak {C}}_{F}:=\text {Span}_{{\mathbb {R}}_+}\{F-p_0\} \end{aligned}$$

is a facet of \({\mathfrak {C}}\), which lies in the half-space \(\{y|\alpha (y)\ge 0\}\) and intersects \({\mathfrak {a}}_{{\mathfrak {L}},+}\). The only possibility is that \({\mathfrak {C}}_{F}\) is generated by

$$\begin{aligned} \cup _k\{\epsilon ^k_1,...,\epsilon ^k_{n_k}\}\setminus \{\epsilon ^{k_0}_{n_{k_0}}\} \end{aligned}$$

for some \(k_0\in \{1,...,m_0\}\). In this way

$$\begin{aligned} \alpha =\epsilon ^{k_0}_{n_{k_0}-1}-\epsilon ^{k_0}_{n_{k_0}} \text { and } u=(\epsilon ^{k_0}_{n_{k_0}})^\vee . \end{aligned}$$

Thus (6.1) is true and the lemma is proved. \(\square \)

Proof of Lemma 3.1

Let \(l_A(y)\) be the linear functions as in (3.1) and

$$\begin{aligned} {\hat{u}}_0(y)=\sum _Al_A\left( \frac{1}{2}y\right) \log l_A\left( \frac{1}{2}y\right) ,~y\in 2P. \end{aligned}$$

Let \({\hat{u}}\) be the Legendre function of u, defined by

$$\begin{aligned} {\hat{u}}(y(x))=\sum _i y_i x^i-u(x), \end{aligned}$$

where

$$\begin{aligned} y_i= \frac{\partial }{\partial x^i}u(x). \end{aligned}$$

Then by [1], \({\hat{u}}\) can be regarded as a function on \(\overline{2P}\) and \(({\hat{u}}-{\hat{u}}_0)\in C^\infty (\overline{2P})\). Thus one can show that

$$\begin{aligned} \frac{\partial }{\partial y_i} {\hat{u}}&=\frac{1}{2}\sum _{A=1}^{d_0}(u_A^i)(1+\log l_A)(y)+O(1) \end{aligned}$$
(6.2)

and

$$\begin{aligned} \frac{\partial ^2}{\partial y_i\partial y_j}{\hat{u}}_{,ij}=\frac{1}{2}\sum _{A=1}^{d_0}\frac{u_{A}^i u_{A}^j}{l_A(y)}+O(1). \end{aligned}$$
(6.3)

Define

$$\begin{aligned} h=2\Upsilon _{K^{-1}_M|_Z}(-x)+\log {\mathbf {J}}(x)-\log \det (\nabla ^2u)-\log \prod _{\alpha \in \Phi _+}\langle \alpha ,\nabla u\rangle ^2(x). \end{aligned}$$

It suffices to prove that h is uniformly bounded on \(\overline{{\mathfrak {a}}_+}\). Taking Legendre transformation, we have

$$\begin{aligned} h=\log \det (\nabla ^2{\hat{u}})+2\Upsilon _{K^{-1}_M|_Z}(-\nabla {\hat{u}}(y))-\log \pi (y)+\log {\mathbf {J}}(\nabla {\hat{u}}(y)), \end{aligned}$$
(6.4)

where \(\pi (y)=\prod _{\alpha \in \Phi _+}\alpha ^2(y)\). Since \({\hat{u}}\) is smooth in Int(2P), h is locally bounded in the interior of \(2P_+\). It returns to prove that h is bounded near each \(y_0\in \partial (2P_+)\). Suppose that there is a sequence such that

$$\begin{aligned} y_{(k)}\rightarrow y_0\in \partial (2P_+),~k\rightarrow +\infty . \end{aligned}$$

By replacing \(\{y_{(k)}\}_k\) by a subsequence, we may assume that

$$\begin{aligned} x_{(k)}=\nabla {\hat{u}}(y_{(k)})\in (-\sigma _p),~k=1,2,... \end{aligned}$$

for some fixed vertex p of 2P which lies in \(\overline{{\mathfrak {a}}_+}\).

Recall the fact that on \((-\sigma _p)\),

$$\begin{aligned} \Upsilon |_{K^{-1}_M}(-x)=-{\hat{p}}(x), \end{aligned}$$

where \({\hat{p}}\) is determined by (3.28). Then by (6.2), we have

$$\begin{aligned} 2\Upsilon _{K^{-1}_M|_Z}(-\nabla {\hat{u}}(y))=\sum _{A=1}^{d_0}(1-2\rho _A(u_A))\log l_A(y)+O(1). \end{aligned}$$
(6.5)

Thus plugging (6.5) into (6.4) together with (6.2)-(6.3), we get (cf. [19, Section 5] for details),Footnote 5

$$\begin{aligned} h=&-2\sum _{A=1}^{d_0}\rho _A(u_A)\log l_A+2\sum _{\beta \in \Phi _+}\log \sinh (\frac{1}{2}\sum _{A=1}^{d_0}\beta (u_A)\log l_A)\\&-2\sum _{\beta \in \Phi _+}\log \beta (y)+O(1)\\ =&\sum _{\beta \in \Phi _+}\left[ \sum _{A=1}^{d_0}|\beta (u_A)|\log l_A+2\log \sinh (\frac{1}{2}\sum _{A=1}^{d_0}\beta (u_A)\log l_A)\right. \\&\left. -2\log \beta (y)\right] +O(1),~\text {as }y\rightarrow y_0. \end{aligned}$$

Denote

$$\begin{aligned} I_\beta (y)=\sum _{A=1}^{d_0}|\beta (u_A)|\log l_A+2\log \sinh (\frac{1}{2}\sum _{A=1}^{d_0}\beta (u_A)\log l_A)-2\log \beta (y). \end{aligned}$$

We need to estimate each \(I_\beta \). As in [19, Section 5], we divide \(y_0\) into the following three cases:

Case-1. \(y_0\in {\mathfrak {a}}_+\) and is away from any Weyl walls;

Case-2. \(y_0\in \partial (2P_+)\setminus \partial (2P)\). That is \(y_0\) lie on at least one Weyl wall but not on the boundary of 2P;

Case-3. \(y_0\in \partial (2P)\) and lies on a Weyl wall \(W_\alpha \).

figure c

In the first two cases, it is easy to show that

$$\begin{aligned} I_\beta (y)=O(1),\text { as }y\rightarrow y_0,~\forall ~\beta \in \Phi _+. \end{aligned}$$
(6.6)

In fact, such two cases correspond to Case-1 and Case-2 in [19, Section 5], respectibvely. Thus h is bounded near \(y_0\).

In Case-3, it can be direct to check that (6.6) holds for any \(\alpha '\in \Phi _+\) such that \(y_0\not \in W_{\alpha '}\) as in Case-2. Thus it suffices to estimate \(I_\alpha \). Suppose that there are p facets \(\{F_{A'}\}_{{A'}=1}^p\) of 2P passing through \(y_0\). We have two subcases:

Case-3.1. All of the p facets are orthogonal to \(W_\alpha \);

Case-3.2. There is at least one pair of facets \(\{F_1,F_2\}\subseteq \{F_{A'}\}_{{A'}=1}^p\) such that

$$\begin{aligned} F_2=s_\alpha (F_1), \end{aligned}$$

which are not orthogonal to \(W_\alpha \).

figure d

In Case-3.1, (6.6) can be obtained as in Case-3.1 [19, Section 5].

In Case-3.2, we first show there is no more such pair of facets, which are not orthogonal to \(W_\alpha \). In fact, if \(\{F_{A'}\}_{{A'}=1}^{2p_1}\subseteq \{F_{A'}\}_{{A'}=1}^p\) are \(2p_1\) facets passing through \(y_0\), which are not orthogonal to \(W_\alpha \) such that

$$\begin{aligned} F_{2k}=s_\alpha (F_{2k-1}),~k=1,...,p_1. \end{aligned}$$

Then for the set of norms \(\{u_{A'}\}\), we have

$$\begin{aligned} \text {rank}(u_1,...,u_p)=p-(p_1-1). \end{aligned}$$

Thus \(p_1=1\) by the Delzant condition.

Next we suppose that \(F_2\cap {\mathfrak {a}}_+\not =\emptyset \). As in Case-3.2 [19, Section 5], we see that

$$\begin{aligned} I_{\alpha }(y)=2(-\alpha (u_2)-1)\log l_2+O(1),~y\rightarrow y_0. \end{aligned}$$

On the other hand, since M is smooth, by Lemma 6.1, we have

$$\begin{aligned} \alpha (u_2)=-1. \end{aligned}$$

Thus,

$$\begin{aligned} I_\alpha (y)=O(1),\text { as }y\rightarrow y_0. \end{aligned}$$

Hence (6.6) holds for any \(\beta \in \Phi _+\). The proof of lemma is complete. \(\square \)

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Li, Y., Zhu, X. Tian’s \(\alpha _{m,k}^{{\hat{K}}}\)-invariants on group compactifications. Math. Z. 298, 231–259 (2021). https://doi.org/10.1007/s00209-020-02591-9

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