Generalized optimal degenerations of Fano varieties

Linsheng Wang Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China [email protected]

Abstract.

We prove a generalization of the algebraic version of Tian conjecture. Precisely, for any smooth strictly increasing function $g:\mathbb{R}\to\mathbb{R}_{>0}$ with ${\rm log}\circ g$ convex, we define the $\mathbf{H}^{g}$ -invariant on a Fano variety $X$ generalizing the $\mathbf{H}$ -invariant introduced by Tian-Zhang-Zhang-Zhu, and show that $\mathbf{H}^{g}$ admits a unique minimizer. Such a minimizer will induce the $g$ -optimal degeneration of the Fano variety $X$ , whose limit space admits a $g^{\prime}$ -soliton. We present an example of Fano threefold which has the same $g$ -optimal degenerations for any $g$ .

1. Introduction

As predicted by [Tia97, Conjecture 9.1], a normalized Kähler-Ricci flow $\omega_{t}$ on a Fano manifold $M$ will converge in the Cheeger-Gromov-Hausdorff topology to $(M_{\infty},\omega_{\infty})$ with mild singularities, where $\omega_{\infty}$ is a Kähler-Einstein metric or a Kähler-Ricci soliton on the smooth part of $M_{\infty}$ . This conjecture was widely studied, and has been solved now, see [TZ16, Bam18, CW20, WZ21]. The limit $M_{\infty}$ is called the optimal degeneration of the Fano manifold $M$ .

There is an algebraic version of the above conjecture, which is closely related to the $\mathbf{H}$ -invariant introduced by [TZZZ13]. By [BLXZ23, HL24], for any log Fano pair $(X,\Delta)$ , the $\mathbf{H}$ -invariant is strictly convex along geodesics and admits a unique quasi-monomial valuation $v_{0}$ as its minimizer, whose associated graded ring is finitely generated, hence inducing a multistep special degeneration of $(X,\Delta)$ to some weighted K-semistable log Fano triple $(X_{0},\Delta_{0},\xi_{0})$ . Moreover, $(X_{0},\Delta_{0},\xi_{0})$ will specially degenerate to a weighted K-polystable log Fano triple $(Y,\Delta_{Y},\xi_{0})$ , which admits a Kähler-Ricci soliton by [HL23, BLXZ23].

In the second step of the above degenerations, [HL23, BLXZ23] work not only for Kähler-Ricci solitons, but also $g$ -solitons. Precisely, they showed that for any smooth function $g:{\mathbb{R}}\to{\mathbb{R}}_{>0}$ , any $g$ -weighted K-semistable log Fano triple $(X,\Delta,\xi_{0})$ will specially degenerate to a $g$ -weighted K-polystable log Fano triple $(Y,\Delta_{Y},\xi_{0})$ , which is $g$ -weighted reduced uniformly K-stable by [BLXZ23], hence admits a $g$ -soliton by [HL23]. Motivated by this step, one may ask whether there is an associated first step degeneration in the algebraic version of Tian conjecture or not.

In this paper, we give a generalization of the $\mathbf{H}$ -invariant, namely, the $\mathbf{H}^{g}$ -invariant for some

(1)

smooth strictly increasing function

g:{\mathbb{R}}\to{\mathbb{R}}_{>0}

with

{\rm log}\circ g

convex.

This will lead to the first step degeneration asked in the previous paragraph. We aim to prove the following generalized version of Tian conjecture.

Theorem 1.1 (Generalized Tian conjecture).

Let $(X,\Delta)$ be a log Fano pair, and $g:{\mathbb{R}}\to{\mathbb{R}}_{>0}$ be a smooth strictly increasing function with ${\rm log}\circ g$ convex. Then the $\mathbf{H}^{g}$ -invariant (Definition 3.1) of $(X,\Delta)$ admits a unique minimizer $v_{0}$ , which is a special valuation (Theorem 2.12), such that the central fiber $({\mathcal{X}}_{0},\Delta_{{\mathcal{X}}_{0}},\xi_{0})$ of the multistep special degeneration $({\mathcal{X}},\Delta_{\mathcal{X}},\xi_{0})$ of $(X,\Delta)$ induced by $v_{0}$ is $g^{\prime}$ -weighted K-semistable. Moreover $({\mathcal{X}}_{0},\Delta_{{\mathcal{X}}_{0}},\xi_{0})$ has a unique $g^{\prime}$ -weighted K-polystable special degeneration $(Y,\Delta_{Y},\xi_{0})$ , which admits a $g^{\prime}$ -soliton.

We say that $(Y,\Delta_{Y},\xi_{0})$ is the $g$ -optimal degeneration of $(X,\Delta)$ . The last statement of the theorem has been established by [BLXZ23, HL24]. We aim to prove the first part of the theorem.

Remark 1.2.

In the setting of $g$ -optimal degenerations, the correct weighted stability notion is the $g^{\prime}$ -weighted K-stability, where $g^{\prime}$ is the first order derivative of the function $g$ . See Lemma 3.11 and Theorem 4.14 for details. If we choose $g(x)=e^{x}$ , then it reveals the ordinary optimal degeneration. In this case $g^{\prime}(x)=g(x)$ .

The following theorem is an analog of [HL24, Theorem 5.3], which is the key ingredient in finding $g$ -optimal degenerations.

Theorem 1.3 (Theorem 4.14).

Let $v_{0}$ be a quasi-monomial valuation over $X$ with finitely generated associated graded ring ${\rm gr}_{v_{0}}R$ , which induces a multistep special degeneration $({\mathcal{X}},\Delta_{{\mathcal{X}}},\xi_{0})$ with klt central fiber. Then $v_{0}$ minimizes $\mathbf{H}^{g}$ if and only if $({\mathcal{X}}_{0},\Delta_{{\mathcal{X}},0},\xi_{0})$ is $g^{\prime}$ -weighted K-semistable.

If Theorem 1.1 is established, then it’s natural to ask what is the relationship between the $g$ -optimal degenerations of a log Fano pair $(X,\Delta)$ for different functions $g$ .

Question 1.4.

Let $(X,\Delta)$ be a log Fano pair and $g,\bar{g}$ be functions satisfying (1). Let $(Y,\Delta_{Y},\xi_{0})$ , $(\overline{Y},\Delta_{\overline{Y}},\bar{\xi}_{0})$ be the $g$ -, $\bar{g}$ -optimal degenerations of $(X,\Delta)$ respectively. When do we have

(2)

\displaystyle(Y,\Delta_{Y})\cong(\overline{Y},\Delta_{\overline{Y}})?

If $(X,\Delta)$ is a toric log Fano pair, then the isomorphism (2) always holds since $(X,\Delta)$ $g_{0}$ -weighted K-polystable for any weight function $g_{0}:\mathbf{P}\to{\mathbb{R}}_{>0}$ (see Corollary 5.1 for details). We have the following non-trivial examples given by [Wan24, Example 5.5 and 5.7].

Theorem 1.5.

For any Fano threefold in families №2.28, №3.14 and №2.23(a) of Mori-Mukai’s list, the isomorphism (2) always holds.

The paper is organized as follows. In Section 2 we recall some basic notions in K-stability theory that we will use. We define the generalized $\mathbf{H}$ -invariant $\mathbf{H}^{g}$ for polarized klt pairs $(X,\Delta;L)$ in Section 3 and study the basic properties of it. In Section 4, we show the existence of the $\mathbf{H}^{g}$ -minimizer and its finite generation property in the log Fano case. Finally, we give some examples of $g$ -optimal degenerations in Section 5.

Acknowledgments. I would like to thank my advisor Gang Tian for his constant support and guidance. I thank Jiyuan Han for many helpful comments in this paper. I thank Thibaut Delcroix for informing me to consider the ${\mathbb{G}}$ -equivariant $g$ -optimal degenerations. I also thank Minghao Miao, Lu Qi, Kewei Zhang and Shengxuan Zhou for helpful discussions.

2. Preliminaries

We work over an algebraically closed field ${\mathbbm{k}}$ of characteristic $0$ . A pair $(X,\Delta)$ consists of a normal variety $X$ and an effective ${\mathbb{Q}}$ -divisor $\Delta$ on $X$ such that $K_{X}+\Delta$ is ${\mathbb{Q}}$ -Cartier. A polarized pair $(X,\Delta;L)$ consists of a projective pair $(X,\Delta)$ and a ${\mathbb{Q}}$ -Cartier ample divisor $L$ on $X$ . It is called log Fano if $L=-(K_{X}+\Delta)$ . Fix an integer $l_{0}>0$ such that $l_{0}L$ is Cartier. We denote by $R:=R(X;L):=\oplus_{m\in l_{0}{\mathbb{N}}}R_{m}$ the section ring of $L$ where $R_{m}:=H^{0}(X,mL)$ .

2.1. Filtrations, concave transforms and DH measures

Let $(X,\Delta;L)$ be a polarized pair of dimension $n$ . Following [BJ20, 2.1], a graded linear series $V_{\bullet}=\{V_{m}\}$ of $L$ is a sequence of subspaces $V_{m}\subseteq R_{m}$ such that $V_{0}={\mathbbm{k}}$ and $V_{m}\cdot V_{m^{\prime}}\subseteq V_{m+m^{\prime}}$ . We assume that $V_{\bullet}$ contains an ample series, that is, $H^{0}(X,mA)\subseteq V_{m}$ for $m\gg 0$ , where $A$ is an ample ${\mathbb{Q}}$ -divisor such that $|L-A|_{\mathbb{Q}}\neq\varnothing$ . Then

\displaystyle{\rm vol}(V_{\bullet})=\mathop{{\rm lim}}_{m\to\infty}\frac{\dim V_{m}}{m^{n}/n!}>0.

For such a graded linear series $V_{\bullet}$ , we may construct a convex body $\mathbf{O}=\mathbf{O}(V_{\bullet})\subseteq{\mathbb{R}}^{n}$ called the Okounkov body by choosing an admissible flag on $X$ , such that ${\rm vol}(\mathbf{O}(V_{\bullet}))=\frac{1}{n!}{\rm vol}(V_{\bullet})$ . See for example [JM12]. Note that the section ring $R_{\bullet}=R(X;L)$ is a graded linear series containing an ample series.

Definition 2.1.

A filtration ${\mathcal{F}}$ on $V_{\bullet}$ is a collection of subspaces ${\mathcal{F}}^{\lambda}V_{m}\subseteq V_{m}$ for each $\lambda\in{\mathbb{R}}$ and $m\geq 0$ such that

•

Decreasing. ${\mathcal{F}}^{\lambda}V_{m}\supseteq{\mathcal{F}}^{\lambda^{\prime}}V_{m}$ for $\lambda\leq\lambda^{\prime}$ ;
•

Left-continuous. ${\mathcal{F}}^{\lambda}V_{m}={\mathcal{F}}^{\lambda-\epsilon}V_{m}$ for $0<\epsilon\ll 1$ ;
•

Bounded. ${\mathcal{F}}^{\lambda}V_{m}=V_{m}$ for $\lambda\ll 0$ and ${\mathcal{F}}^{\lambda}V_{m}=0$ for $\lambda\gg 0$ ;
•

Multiplicative. ${\mathcal{F}}^{\lambda}V_{m}\cdot{\mathcal{F}}^{\lambda^{\prime}}V_{m^{\prime}}\subseteq{\mathcal{F}}^{\lambda+\lambda^{\prime}}V_{m+m^{\prime}}$ .

For any $s\in V_{m}$ , we set ${\rm ord}_{\mathcal{F}}(s)={\rm max}\{\lambda:s\in{\mathcal{F}}^{\lambda}V_{m}\}.$ The filtration is called linearly bounded if there is a constant $C>0$ such that ${\mathcal{F}}^{-mC}V_{m}=V_{m}$ and ${\mathcal{F}}^{mC}V_{m}=0$ for all $m$ . In this case, the sequence of numbers $\lambda^{(m)}_{\rm max}={\rm max}\{\lambda\in{\mathbb{R}}:{\mathcal{F}}^{\lambda}R_{m}\neq 0\}$ is linearly bounded, that is,

\displaystyle\lambda_{\rm max}(V_{\bullet};{\mathcal{F}}):=\mathop{{\rm sup}}_{m\in{\mathbb{N}}}\frac{\lambda^{(m)}_{\rm max}}{m}=\mathop{{\rm lim}}_{m\rightarrow\infty}\frac{\lambda^{(m)}_{\rm max}}{m}<+\infty.

A basis $\{s_{i}\}$ of $V_{m}$ is called compatible with ${\mathcal{F}}$ if ${\mathcal{F}}^{\lambda}V_{m}$ is generated by $\{s_{i}:{\rm ord}_{\mathcal{F}}(s_{i})\geq\lambda\}$ .

For example, if $v$ is a valuation over $X$ , then ${\mathcal{F}}_{v}^{\lambda}V_{m}:=\{s\in V_{m}:v(s)\geq\lambda\}$ defines a filtration on $V_{\bullet}$ . It is linearly bounded if $A_{X,\Delta}(v)<+\infty$ , which holds for quasi-monomial valuations over $X$ , see [JM12].

For any filtration ${\mathcal{F}}$ on $V_{\bullet}$ and $a\in{\mathbb{R}}_{>0},b\in{\mathbb{R}}$ , we define the $a$ -rescaling and $b$ -shift of ${\mathcal{F}}$ by

\displaystyle(a{\mathcal{F}})^{\lambda}V_{m}:={\mathcal{F}}^{\lambda/a}V_{m},\,\,{\mathcal{F}}(b)^{\lambda}V_{m}:={\mathcal{F}}^{\lambda-bm}V_{m},

and we also denote by $a{\mathcal{F}}(b):=(a{\mathcal{F}})(b)$ , that is $(a{\mathcal{F}}(b))^{\lambda}V_{m}={\mathcal{F}}^{\frac{\lambda-bm}{a}}V_{m}$ .

Definition 2.2.

Let ${\mathcal{F}}$ be a linearly bounded filtration on $V_{\bullet}$ . Then for any $t\in{\mathbb{R}}$ , we have a graded linear subseries ${\mathcal{F}}^{(t)}V_{\bullet}\subseteq V_{\bullet}$ defined by $({\mathcal{F}}^{(t)}V)_{m}={\mathcal{F}}^{mt}V_{m}$ . Note that ${\mathcal{F}}^{(t)}V_{\bullet}$ is linearly bounded and contains an ample series since $V_{\bullet}$ does. We denote the Okounkov body of ${\mathcal{F}}^{(t)}V_{\bullet}$ by $\mathbf{O}^{(t)}$ , and let $\mathbf{O}=\mathbf{O}(V_{\bullet})$ . Then $\mathbf{O}^{(t)}\subseteq\mathbf{O}$ is a descending collection of convex bodies. The concave transform of ${\mathcal{F}}$ is the function on ${\mathbb{R}}^{n}$ defined by

\displaystyle G_{\mathcal{F}}(y)={\rm sup}\{t\in{\mathbb{R}}:y\in\mathbf{O}^{(t)}\}.

Note that $G_{\mathcal{F}}$ is concave and upper-semicontinuous. The linear boundedness of ${\mathcal{F}}$ guarantees that $\mathbf{O}^{(-C)}=\mathbf{O}$ and $\mathbf{O}^{(C)}=0$ . In other word, $\mathbf{O}$ is contained in the level set $\{-C\leq G_{\mathcal{F}}\leq C\}\subseteq{\mathbb{R}}^{n}$ .

Lemma 2.3.

For any $a\in{\mathbb{R}}_{>0},b\in{\mathbb{R}}$ , we have $G_{a{\mathcal{F}}(b)}=aG_{\mathcal{F}}+b$ .

Definition 2.4.

Let ${\mathcal{F}}$ be a linearly bounded filtration on $V_{\bullet}$ . We have the following discrete measure,

\displaystyle{\rm DH}_{{\mathcal{F}},m}=\sum_{\lambda}\delta_{\frac{\lambda}{m}}\cdot\frac{\dim{\rm gr}_{\mathcal{F}}^{\lambda}V_{m}}{\dim V_{m}}=-\frac{{\rm d}}{{\rm d}t}\frac{\dim{\mathcal{F}}^{mt}V_{m}}{\dim V_{m}}

on ${\mathbb{R}}$ , where $\delta_{\frac{\lambda}{m}}$ is the Dirac measure at $\frac{\lambda}{m}\in{\mathbb{R}}$ . By [BC11, BHJ17], ${\rm DH}_{{\mathcal{F}},m}\to{\rm DH}_{\mathcal{F}}$ converges weakly as $m\to\infty$ , where

\displaystyle{\rm DH}_{\mathcal{F}}=-\frac{{\rm d}}{{\rm d}t}\frac{{\rm vol}({\mathcal{F}}^{(t)}V_{\bullet})}{{\rm vol}(V_{\bullet})}

is called the Duistermaat-Heckman (DH) measure of ${\mathcal{F}}$ .

Let ${\mathcal{G}}$ be another linearly bounded filtration on $V_{\bullet}$ . By [BLXZ23, 3.1.3], we define

\displaystyle{\rm DH}_{{\mathcal{F}},{\mathcal{G}},m}=\sum_{\lambda}\delta_{(\frac{\lambda}{m},\frac{\mu}{m})}\cdot\frac{\dim{\rm gr}_{\mathcal{F}}^{\lambda}{\rm gr}_{\mathcal{G}}^{\mu}V_{m}}{\dim V_{m}}=-\frac{\partial^{2}}{\partial x\partial y}\frac{\dim{\mathcal{F}}^{mx}V_{m}\cap{\mathcal{G}}^{my}V_{m}}{\dim V_{m}}

on ${\mathbb{R}}^{2}$ , which also converges weakly to

\displaystyle{\rm DH}_{{\mathcal{F}},{\mathcal{G}}}=-\frac{\partial^{2}}{\partial x\partial y}\frac{{\rm vol}({\mathcal{F}}^{(x)}{\mathcal{G}}^{(y)}V_{\bullet})}{{\rm vol}(V_{\bullet})}

as $m\to\infty$ by [BLXZ23, Theorem 3.3], where ${\mathcal{F}}^{(x)}{\mathcal{G}}^{(y)}V_{\bullet}$ is the graded linear series defined by

\displaystyle({\mathcal{F}}^{(x)}{\mathcal{G}}^{(y)}V_{\bullet})_{m}:={\mathcal{F}}^{mx}V_{m}\cap{\mathcal{G}}^{my}V_{m}.

This measure is called the DH measure compatible with both ${\mathcal{F}}$ and ${\mathcal{G}}$ .

The two measures defined above both have compact support since ${\mathcal{F}}$ and ${\mathcal{G}}$ are linearly bounded. Let $f$ be a continuous function on ${\mathbb{R}}$ , then

\displaystyle\int_{{\mathbb{R}}^{2}}f(x){\rm DH}_{{\mathcal{F}},{\mathcal{G}}}({\rm d}x{\rm d}y)=\int_{\mathbb{R}}f(x){\rm DH}_{{\mathcal{F}}}({\rm d}x).

By [BJ20, 2.5], we also have

\displaystyle{\rm DH}_{\mathcal{F}}=G_{{\mathcal{F}},*}{\rm LE},

where ${\rm LE}$ is the Lebesgue measure on the Okounkov body $\mathbf{O}=\mathbf{O}(V_{\bullet})$ .

We define the $L^{1}$ -distance of ${\mathcal{F}}$ and ${\mathcal{G}}$ by

\displaystyle d_{1}({\mathcal{F}},{\mathcal{G}}):=\int_{{\mathbb{R}}^{2}}|x-y|{\rm DH}_{{\mathcal{F}},{\mathcal{G}}}({\rm d}x{\rm d}y),

and say that ${\mathcal{F}},{\mathcal{G}}$ are equivalent if $d_{1}({\mathcal{F}},{\mathcal{G}})=0$ . Let $v,w$ be valuations over $X$ , if ${\mathcal{F}}_{v}$ and ${\mathcal{F}}_{w}$ are equivalent, then $v=w$ by [HL24, Proposition 2.27], see also [BLXZ23, Lemma 3.16].

2.2. Log canonical slopes and $\mathbf{L}$ -functionals

Definition 2.5.

Let $(X,\Delta;L)$ be a polarized klt pair and ${\mathcal{F}}$ be a linearly bounded filtration on $R=R(X;L)$ . The base ideal sequence $I^{(t)}_{\bullet}=\{I_{m,mt}\}_{m\in l_{0}{\mathbb{N}}}$ of ${\mathcal{F}}$ is defined by

\displaystyle I_{m,mt}\,\,\,=\,\,\,I_{m,mt}(L;{\mathcal{F}})\,\,\,:=\,\,\,{\rm im}\Big{(}{\mathcal{F}}^{mt}H^{0}(X,mL)\otimes{\mathcal{O}}(-mL)\to{\mathcal{O}}\Big{)},

for any $m\in l_{0}{\mathbb{N}}$ and $t\in{\mathbb{R}}$ . The log canonical slope of ${\mathcal{F}}$ is defined by

\displaystyle\mu({\mathcal{F}})\,\,\,=\,\,\,\mu_{X,\Delta;L}({\mathcal{F}})\,\,\,:=\,\,\,{\rm sup}\Big{\{}t:{\rm lct}(X,\Delta;I^{(t)}_{\bullet})\geq 1\Big{\}}.

Note that $I^{(t)}_{\bullet}=0$ (hence ${\rm lct}(X,\Delta;I^{(t)}_{\bullet})=0$ ) when $t>\lambda_{\rm max}$ . We have $\mu({\mathcal{F}})\leq\lambda_{\rm max}$ .

Lemma 2.6.

For any $a\in{\mathbb{R}}_{>0},b\in{\mathbb{R}}$ , we have $\mu(a{\mathcal{F}}(b))=a\mu({\mathcal{F}})+b$ .

By [JM12], for any valuation $v$ on $X$ , we have

\displaystyle v(I^{(t)}_{\bullet})=\mathop{{\rm inf}}_{m\in{\mathbb{N}}}\frac{v(I_{m,mt})}{m}=\mathop{{\rm lim}}_{m\to\infty}\frac{v(I_{m,mt})}{m}.

Consider the following function of $t\in{\mathbb{R}}$ in the definition of $\mu({\mathcal{F}})$ ,

\displaystyle f(t)={\rm lct}(X,\Delta;I^{(t)}_{\bullet})={\rm inf}_{v}\frac{A_{X,\Delta}(v)}{v(I^{(t)}_{\bullet})},

where the infimum runs over all the valuations over $X$ . We have the following useful lemma in computing log canonical slope.

Lemma 2.7.

[Xu24, Proposition 3.46] The function $f(t)$ is continuous non-increasing on $(-\infty,\lambda_{\rm max})$ . If we set $\mu_{+\infty}={\rm sup}\{t:{\rm lct}(X,\Delta;I^{(t)}_{\bullet})=+\infty\}$ , then $f(t)$ is strictly decreasing on $[\mu_{+\infty},\lambda_{\rm max})$ .

As a consequence, we have

(3)

\displaystyle\mu_{X,\Delta;L}({\mathcal{F}}_{v})\leq A_{X,\Delta}(v),

for any valuation $v$ over $X$ . Indeed, we only need to prove the inequality when $A_{X,\Delta}(v)<\lambda_{\rm max}$ since $\mu({\mathcal{F}}_{v})\leq\lambda_{\rm max}$ . By definition, we have $v(I^{(t)}_{\bullet})\geq t$ . Hence for any $t\geq A_{X,\Delta}(v)$ , we have ${\rm lct}(X,\Delta;I^{(t)}_{\bullet})\leq\frac{A_{X,\Delta}(v)}{v(I^{(t)}_{\bullet})}\leq 1$ . So $\mu({\mathcal{F}}_{v})\leq A_{X,\Delta}(v)$ by Lemma 2.7.

Lemma 2.8.

If there exists $\Gamma\in|L|_{\mathbb{Q}}$ such that $(X,\Delta+\Gamma)$ is lc, and $v$ is an lc place of $(X,\Delta+\Gamma)$ . Then $\mu_{X,\Delta;L}({\mathcal{F}}_{v})=A_{X,\Delta}(v)$ .

Proof.

Assume that $\Gamma\in\frac{1}{m}|mL|$ . Since $v(\Gamma)=A_{X,\Delta}(v)$ , we have $\Gamma\in\frac{1}{m}|{\mathcal{F}}_{v}^{mA_{X,\Delta}(v)}R_{m}|$ and

\displaystyle{\rm lct}(X,\Delta;I^{(A_{X,\Delta}(v))}_{\bullet})\geq{\rm lct}(X,\Delta;\Gamma)\geq 1.

Hence $\mu({\mathcal{F}}_{v})\geq A_{X,\Delta}(v)$ . We conclude by (3). ∎

Remark 2.9.

If ${\rm gr}_{v}R=\oplus_{m,\lambda}{\mathcal{F}}_{v}^{\lambda}R_{m}/{\mathcal{F}}_{v}^{>\lambda}R_{m}$ is finitely generated, then the converse of this lemma also holds. Indeed, for sufficiently divisible $m$ we have

\displaystyle 1={\rm lct}(X,\Delta;I_{\bullet}^{(A_{X,\Delta}(v))})={\rm lct}(X,\Delta;I_{m,mA_{X,\Delta}(v)}^{1/m}).

This means that there exists $D\in\frac{1}{m}|mL|$ with $v(D)\geq A_{X,\Delta}(v)$ and $(X,\Delta+D)$ is lc. Thus $v$ is an lc place of $(X,\Delta+D)$ . The condition holds if $v$ is induced by some weakly special test configuration, see [Xu24, Theorem 4.24].

Definition 2.10.

Let ${\mathcal{F}}$ be a linearly bounded filtration on $R$ , and $e_{-},e_{+}\in{\mathbb{Z}}$ such that ${\mathcal{F}}^{me_{-}}R_{m}=R_{m}$ and ${\mathcal{F}}^{me_{+}}R_{m}=0$ for any $m\in l_{0}{\mathbb{N}}$ . Recall that $I_{m,\lambda}$ is the base ideal sequence of ${\mathcal{F}}$ (Definition 2.5). We denote by

	$\displaystyle{\mathcal{I}}_{m}(e_{+},e_{-})$	$\displaystyle=$	$\displaystyle{\mathcal{I}}_{m}({\mathcal{F}};e_{+},e_{-})$
		$\displaystyle:=$	$\displaystyle I_{m,me_{-}}\cdot s^{-me_{-}+me_{+}}+I_{m,me_{-}+1}\cdot s^{-(me_{-}+1)+me_{+}}+\cdots+I_{m,me_{+}}\cdot s^{0}\subseteq{\mathcal{O}}_{X}[s].$

Since $I_{m,me_{-}}={\mathcal{O}}_{X},I_{m,me_{+}}=0$ and ${\mathcal{O}}_{X}\cdot s^{-(me_{-}-1)}\subseteq{\mathcal{O}}_{X}\cdot s^{-me_{-}}$ , we see that ${\mathcal{I}}(e_{+}+a,e_{-}-b)={\mathcal{I}}(e_{+},e_{-})s^{ma}$ for any $a,b\in{\mathbb{N}}$ . Hence ${\mathcal{I}}_{m}(e_{+}):={\mathcal{I}}_{m}(e_{+},e_{-})$ is independent of the choice of $e_{-}$ and

\displaystyle{\mathcal{I}}_{m}:={\mathcal{I}}_{m}(e_{+})\cdot s^{-me_{+}}\subseteq{\mathcal{O}}_{X}[s,s^{-1}]

is independent of the choice of $e_{+}$ . The $\mathbf{L}$ -functional of ${\mathcal{F}}$ is defined by

\displaystyle\mathbf{L}({\mathcal{F}})=\mathbf{L}_{X,\Delta;L}({\mathcal{F}}):=\mathop{{\rm lim}}_{m\to\infty}{\rm lct}(X_{{\mathbb{A}}^{1}},\Delta_{{\mathbb{A}}^{1}}+{\mathcal{I}}_{m}^{\frac{1}{m}};X_{0})-1,

where the limit exists by [Xu24, Lemma 1.49].

Lemma 2.11.

[Xu24, Theorem 3.55] For any linearly bounded filtration ${\mathcal{F}}$ on $R$ , we have

\displaystyle\mu({\mathcal{F}})=\mathbf{L}({\mathcal{F}}).

2.3. Multistep special degenerations and higher rank finite generation

Let $(X,\Delta)$ be a log Fano pair, and ${\mathcal{F}}$ be a filtration on $R=R(X,\Delta)$ such that ${\rm gr}_{\mathcal{F}}R$ is finitely generated. Assume that ${\mathcal{F}}$ is of rational rank $r$ . Then the Rees construction gives a ${\mathbb{G}}_{m}^{r}$ -equivariant family ${\mathcal{X}}_{\mathcal{F}}={\rm Proj}_{A}{\rm Rees}_{\mathcal{F}}R\to{\mathbb{A}}^{r}$ , where $A={\mathbbm{k}}[t_{1},\cdots,t_{r}]$ and

\displaystyle{\rm Rees}_{\mathcal{F}}R:=\bigoplus_{m\in{\mathbb{N}}}\bigoplus_{\lambda\in\Gamma_{m}({\mathcal{F}})}t^{-\lambda}{\mathcal{F}}^{\lambda}R_{m}.

We denote by $\Delta_{{\mathcal{X}}_{\mathcal{F}}}$ the closure of $\Delta\times({\mathbb{A}}^{1}\setminus\{0\})^{r}$ in ${\mathcal{X}}_{\mathcal{F}}$ and say that $({\mathcal{X}}_{\mathcal{F}},\Delta_{{\mathcal{X}}_{\mathcal{F}}})$ is the multistep special degeneration induced by ${\mathcal{F}}$ . If ${\mathcal{F}}={\mathcal{F}}_{v}$ for some valuation $v$ over $X$ , we simply denote the multistep special degeneration by $({\mathcal{X}}_{v},\Delta_{{\mathcal{X}}_{v}})$ and the central fiber by $(X_{v},\Delta_{v})$ . We have the following deep theorem of higher rank finite generation developed by [LXZ22, XZ22, Xu24].

Theorem 2.12.

Let $(X,\Delta)$ be a log Fano pair, and $v$ be a quasi-monomial valuation over $X$ . The following statements are all equivalent.

(a)

The associated graded ring ${\rm gr}_{v}R$ is finitely generated, and the central fiber $(X_{v},\Delta_{v})$ of the induced degeneration is klt.
(b)

There exists a special ${\mathbb{Q}}$ -complement $\Gamma$ of $(X,\Delta)$ with respect to some toroidal model $\pi:(Y,E)\to(X,\Delta)$ such that $v\in{\rm QM}(Y,E)\cap{\rm LC}(X,\Delta+\Gamma)$ .
(c)

There exists a qdlt Fano type model $\pi:(Y,E)\to(X,\Delta)$ such that $v\in{\rm QM}(Y,E)$ .

In this case, the valuation $v$ is called special with respect to $(X,\Delta)$ .

Motivated by [LX18, Lemma 2.7] and [Che24, Lemma 4.2], we have the following characterization of weakly special valuations.

Theorem 2.13.

Let $(X,\Delta)$ be a log Fano pair, and $v$ be a quasi-monomial valuation over $X$ . The following statements are all equivalent.

(a)

$\mu({\mathcal{F}}_{v})=A_{X,\Delta}(v)$ .
(b)

There exists a ${\mathbb{Q}}$ -complement $\Gamma$ of $(X,\Delta)$ such that $v\in{\rm LC}(X,\Delta+\Gamma)$ .
(c)

There exists a qdlt model $(Y,E)\to(X,\Delta)$ and a birational contraction $(Y,E)\dashrightarrow({\overline{Y}},{\overline{E}})$ which is an isomorphism at any stratum of $E$ , such that $-(K_{\overline{Y}}+{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}})$ is semiample and ${\rm QM}(Y,E)$ is a minimal simplex containing $v$ .

In this case, the valuation $v$ is called weakly special with respect to $(X,\Delta)$ .

Proof.

By Lemme 2.8, we have (b) $\Rightarrow$ (a). Now we prove (a) $\Rightarrow$ (c). By [HMX14], there exists $\varepsilon>0$ depending only on $\dim X$ and coefficients of $\Delta$ such that, for any birational morphism $\pi:Y\dashrightarrow X$ and any reduced divisor $E$ on $Y$ , the pair $(Y,\pi_{*}^{-1}\Delta+(1-\varepsilon)E)$ is lc if and only if $(Y,\pi_{*}^{-1}\Delta+E)$ is.

Let $\mu=\mu({\mathcal{F}}_{v})=A_{X,\Delta}(v)$ . This is equivalent to $v$ computing ${\rm lct}(X,\Delta;I^{(\mu)}_{\bullet})=1$ . Since $v$ is a quasi-monomial valuation over $X$ , there exists a quasi-monomial simplicial cone $\sigma\subseteq{\rm Val}_{X}$ containing $v$ . The functions $w\mapsto A_{X,\Delta}(w)$ and $w\mapsto w(\mathfrak{a}_{\bullet}^{c})$ are linear and concave on $\sigma$ respectively. Hence the function $A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(-):\sigma\to{\mathbb{R}}$ ,

(4)

\displaystyle w\mapsto A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(w)=A_{X,\Delta}(w)-w(\mathfrak{a}_{\bullet}^{c})

is convex on $\sigma$ . In particular, it is Lipschitz on $\sigma$ . Hence there exists a constant $C>0$ such that

\displaystyle|A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(w)-A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(v)|\leq C|w-v|.

On the other hand, $A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(w)\geq 0$ for any $w\in\sigma$ since $v$ compute ${\rm lct}(X,\Delta;I^{(\mu)}_{\bullet})=1$ . Hence

(5)

\displaystyle 0\leq A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(w)=|A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(w)-A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(v)|\leq C|w-v|.

By Diophantine approximation [LX18, Lemma 2.7], there exist divisorial valuations $v_{1},\cdots,v_{r}$ and positive integers $q_{1},\cdots,q_{r},c_{1},\cdots,c_{r}$ such that

•

$\{v_{1},\cdots,v_{r}\}$ spans a quasi-monomial simplicial cone in ${\rm Val}_{X}$ containing $v$ ;
•

for any $1\leq i\leq r$ , there exists a prime divisor $E_{i}$ over $X$ such that $q_{i}v_{i}=c_{i}{\rm ord}_{E_{i}}$ ;
•

$|v_{i}-v|<\frac{\varepsilon}{2Cq_{i}}$ for any $1\leq i\leq r$ .

In particular,

(6)

\displaystyle A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(E_{i})=\frac{q_{i}}{c_{i}}\cdot A_{X,\Delta+\mathfrak{a}_{\bullet}^{c}}(v_{i})\leq\frac{q_{i}}{c_{i}}\cdot C|v_{i}-v|<\frac{q_{i}}{c_{i}}\cdot C\cdot\frac{\varepsilon}{2Cq_{i}}\leq\frac{\varepsilon}{2}.

Choose $0<\varepsilon^{\prime}<\varepsilon/2{\rm ord}_{E_{i}}(I_{\bullet}^{(\mu)})$ . Then for $m\gg 0$ and general $D_{m}\in\frac{1}{m}|{\mathcal{F}}^{m\mu}R_{m}|$ , we have

\displaystyle{\rm lct}(X,\Delta;(1-\varepsilon^{\prime})D_{m})={\rm lct}(X,\Delta;I_{m,m\mu}^{(1-\varepsilon^{\prime})/m})>1,

and ${\rm ord}_{E_{i}}(D_{m})=\frac{1}{m}{\rm ord}_{E_{i}}(I_{m,m\mu})$ for any $i$ . Hence

	$\displaystyle a_{i}\,\,\,:=\,\,\,A_{X,\Delta+(1-\varepsilon^{\prime})D_{m}}(E_{i})$	$\displaystyle=$	$\displaystyle(1-\varepsilon^{\prime})\Big{(}{\rm ord}_{E_{i}}(I_{\bullet}^{(\mu)})-\frac{1}{m}{\rm ord}_{E_{i}}(I_{m,m\mu})\Big{)}$
			$\displaystyle+\,\varepsilon^{\prime}\cdot{\rm ord}_{E_{i}}(I_{\bullet}^{(\mu)})+A_{X,\Delta+I_{\bullet}^{(\mu)}}(E_{i})\,\,\,\leq\,\,\,\varepsilon,$

since ${\rm ord}_{E_{i}}(\mathfrak{a}_{\bullet})\leq\frac{1}{m}{\rm ord}_{E_{i}}(\mathfrak{a}_{m})$ for any graded ideal sequence $\mathfrak{a}_{\bullet}$ .

By [BCHM10, Corollary 1.4.3], there exists a ${\mathbb{Q}}$ -factorial model $\pi:Y\to X$ extracts precisely $E_{1},\cdots,E_{r}$ . Then

(7)

\displaystyle K_{Y}+\pi_{*}^{-1}(\Delta+(1-\varepsilon^{\prime})D_{m})+\sum_{i=1}^{r}(1-a_{i})E_{i}=\pi^{*}(K_{X}+\Delta+(1-\varepsilon^{\prime})D_{m}).

In particular, $\pi^{*}(K_{X}+\Delta+(1-\varepsilon^{\prime})D_{m})\geq K_{Y}+\pi_{*}^{-1}\Delta+(1-\varepsilon)E$ . Since ${\rm lct}(X,\Delta;(1-\varepsilon^{\prime})D_{m})>1$ , the pair $(Y,\pi_{*}^{-1}\Delta+(1-\varepsilon)E)$ is lc. Hence $(Y,\pi_{*}^{-1}\Delta+E)$ is also lc by our choice of $\varepsilon$ . Since $Y$ is ${\mathbb{Q}}$ -factorial, $(Y,\pi_{*}^{-1}\Delta+E)$ is indeed qdlt by [Xu24, Lemma 5.3]. So we get a qdlt model $\pi:(Y,E)\to(X,\Delta)$ with $v\in{\rm QM}(Y,E)$ .

Since ${\rm lct}(X,\Delta;(1-\varepsilon^{\prime})D_{m})>1$ , we see that $(X,\Delta+(1-\varepsilon^{\prime})D_{m})$ is an lc Fano pair. Hence $Y$ is of Fano type by (7). We may run $-(K_{Y}+\pi_{*}^{-1}\Delta+E)$ -MMP and get a ${\mathbb{Q}}$ -factorial good minimal model $\phi:Y\dashrightarrow{\overline{Y}}$ with induced birational map ${\overline{\pi}}:{\overline{Y}}\dashrightarrow X$ . Then $-(K_{{\overline{Y}}}+{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}})$ is nef, hence semiample since ${\overline{Y}}$ is of Fano type, where ${\overline{E}}=\phi_{*}E$ . With the same argument in the previous paragraph, we see that $({\overline{Y}},{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}})$ is also lc. On the other hand, for any prime divisor $F$ over $Y$ , we have

\displaystyle A_{Y,\pi_{*}^{-1}\Delta+E}(F)\geq A_{{\overline{Y}},{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}}}(F),

and the equality holds if and only if $\phi$ is an isomorphism at the generic point of $C_{Y}(F)$ . Hence $\phi$ is an isomorphism at the generic point of each lc center of $(Y,\pi_{*}^{-1}\Delta+E)$ . In particular, $\phi$ is an isomorphism at any stratum of $E$ . The proof of (a) $\Rightarrow$ (c) is finished.

Finally we prove (c) $\Rightarrow$ (b). Since $\phi$ is an isomorphism at any stratum of $E$ , we have $K_{Y}+\pi_{*}^{-1}\Delta+E\leq\phi^{*}(K_{\overline{Y}}+{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}})$ . It suffices to show that $({\overline{Y}},{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}})$ admits a ${\mathbb{Q}}$ -complement, which follows from Bertini theorem since $-(K_{\overline{Y}}+{\overline{\pi}}_{*}^{-1}\Delta+{\overline{E}})$ is semiample. ∎

3. Generalized $\mathbf{H}$ -invariants

Fix a polarized klt pair $(X,\Delta;L)$ . In this section, we will define the generalized $\mathbf{H}$ -invariant $\mathbf{H}^{g}$ of $(X,\Delta;L)$ for any function $g$ satisfying (1), and study the basic properties of it. Some existence results will be established for log Fano pairs in the next section. We fix an Okounkov body $\mathbf{O}$ of $L$ with respect to some admissible flag in the following.

Definition 3.1 ( $\mathbf{H}^{g}$ -invariants).

For any linearly bounded filtration ${\mathcal{F}}$ on $R=R(X;L)$ , we define

$\displaystyle\mathbf{H}^{g}({\mathcal{F}})\,\,\,=\,\,\,\mathbf{H}^{g}_{X,\Delta;L}({\mathcal{F}})$	$\displaystyle:=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbb{R}}g(\mu({\mathcal{F}})-t){\rm DH}_{\mathcal{F}}({\rm d}t)\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g(\mu({\mathcal{F}})-G_{\mathcal{F}}(y)){\rm d}y\Big{)},$
$\displaystyle h^{g}(X,\Delta;L)$	$\displaystyle:=$	$\displaystyle{\rm inf}_{\mathcal{F}}\,\,\mathbf{H}^{g}({\mathcal{F}}),$

where the infimum runs over all the linearly bounded filtrations ${\mathcal{F}}$ on $R$ .

Remark 3.2.

If we choose $g(x)=e^{x}$ , then $\mathbf{H}^{g}$ reveals the original $\mathbf{H}$ -invariant as [TZZZ13, DS20, HL24], see also [MW24, Definition 2.7]. It’s well-known that $\mu({\mathcal{F}})$ and $G_{\mathcal{F}}$ are affine with respect to shifting, we have $\mathbf{H}^{g}({\mathcal{F}}(b))=\mathbf{H}^{g}({\mathcal{F}})$ for any $b\in{\mathbb{R}}$ .

3.1. Convexity

We study the global behavior of $\mathbf{H}^{g}$ in the rest of this section. Following [BLXZ23, Theorem 3.7], we prove the convexity of the $\mathbf{H}^{g}$ -invariants, which mainly relies on our choice of $g$ . As a consequence, we prove the uniqueness of valuative minimizer of $\mathbf{H}^{g}$ . Let ${\mathcal{F}}_{0},{\mathcal{F}}_{1}$ be linearly bounded filtrations on $R$ . The geodesic connecting ${\mathcal{F}}_{0}$ and ${\mathcal{F}}_{1}$ is defined by

(8)

\displaystyle{\mathcal{F}}^{\lambda}_{t}R_{m}=\sum_{(1-t)\mu+t\nu\geq\lambda}{\mathcal{F}}_{0}^{\mu}R_{m}\cap{\mathcal{F}}_{1}^{\nu}R_{m}.

Theorem 3.3.

The functional $\mathbf{H}^{g}$ is convex along geodesics. More precisely, for any $0\leq t\leq 1$ , we have $\mathbf{H}^{g}({\mathcal{F}}_{t})\leq(1-t)\mathbf{H}^{g}({\mathcal{F}}_{0})+t\mathbf{H}^{g}({\mathcal{F}}_{1}).$

Proof.

By [BLXZ23, Proposition 3.12], we know that

\displaystyle\mu({\mathcal{F}}_{t})\leq(1-t)\mu({\mathcal{F}}_{0})+t\mu({\mathcal{F}}_{1}).

Hence

$\displaystyle\mathbf{H}^{g}({\mathcal{F}}_{t})$	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbb{R}}g(\mu({\mathcal{F}}_{t})-s){\rm DH}_{{\mathcal{F}}_{t}}({\rm d}s)\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{{\mathbb{R}}^{2}}g(\mu({\mathcal{F}}_{t})-(1-t)x-ty){\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}({\rm d}x{\rm d}y)\Big{)}$
	$\displaystyle\leq$	$\displaystyle{\rm log}\Big{(}\int_{{\mathbb{R}}^{2}}g((1-t)(\mu({\mathcal{F}}_{0})-x)+t(\mu({\mathcal{F}}_{1})-y)){\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}({\rm d}x{\rm d}y)\Big{)}$
	$\displaystyle\leq$	$\displaystyle{\rm log}\Big{(}\int_{{\mathbb{R}}^{2}}g(\mu({\mathcal{F}}_{0})-x)^{1-t}\cdot g(\mu({\mathcal{F}}_{1})-y)^{t}\cdot{\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}({\rm d}x{\rm d}y)\Big{)}$
	$\displaystyle\leq$	$\displaystyle(1-t){\rm log}\Big{(}\int_{\mathbb{R}}g(\mu({\mathcal{F}}_{0})-x){\rm DH}_{{\mathcal{F}}_{0}}({\rm d}x)\Big{)}+t{\rm log}\Big{(}\int_{\mathbb{R}}g(\mu({\mathcal{F}}_{1})-y){\rm DH}_{{\mathcal{F}}_{1}}({\rm d}y)\Big{)}$
	$\displaystyle=$	$\displaystyle(1-t)\mathbf{H}^{g}({\mathcal{F}}_{0})+t\mathbf{H}^{g}({\mathcal{F}}_{1}),$

where the first inequality follows from (8) and $g$ being increasing, the second one follows from the log concavity of $g$ , and the third one follows from Hölder’s inequality. ∎

Corollary 3.4.

Let $v,w$ be valuations over $X$ . If $\mathbf{H}^{g}({\mathcal{F}}_{v})=\mathbf{H}^{g}({\mathcal{F}}_{w})=h^{g}(X,\Delta;L)$ , then $v=w$ .

Proof.

The proof is slightly different from [BLXZ23, Proposition 3.14], which relies on the linearity of ${\rm log}\circ g$ . Let ${\mathcal{F}}_{0}={\mathcal{F}}_{v}$ and ${\mathcal{F}}_{1}={\mathcal{F}}_{w}$ , and ${\mathcal{F}}_{t}$ be the geodesic connecting them. Then

\displaystyle\mathbf{H}^{g}({\mathcal{F}}_{t})\leq(1-t)\mathbf{H}^{g}({\mathcal{F}}_{0})+t\mathbf{H}^{g}({\mathcal{F}}_{1})=h^{g}(X,\Delta;L).

So the equality holds, hence do those in the proof of Theorem 3.3. Then since we used Hölder’s inequality, we have $g(\mu({\mathcal{F}}_{0})-x)=c\cdot g(\mu({\mathcal{F}}_{1})-y)$ almost everywhere on ${\mathbb{R}}^{2}$ with respect to the measure ${\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}$ for some $c>0$ . On the other hand, since $\mathbf{H}^{g}({\mathcal{F}}_{0})=\mathbf{H}^{g}({\mathcal{F}}_{1})$ , we have $c=1$ . Hence $\mu({\mathcal{F}}_{0})-x=\mu({\mathcal{F}}_{1})-y$ almost everywhere on ${\mathbb{R}}^{2}$ with respect to the measure ${\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}$ since $g$ is continuous and strictly increasing, that is,

\displaystyle 0=\int_{{\mathbb{R}}^{2}}|x-y-d|{\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}({\rm d}x{\rm d}y)=d_{1}({\mathcal{F}}_{0},{\mathcal{F}}_{1}(d)),

where $d=\mu({\mathcal{F}}_{0})-\mu({\mathcal{F}}_{1})$ . Then ${\mathcal{F}}_{0}$ and ${\mathcal{F}}_{1}(d)$ are equivalent, so they have the same $\lambda_{\rm min}$ , and $d=0$ by [BLXZ23, Lemma 2.5]. We conclude that $v=w$ by [HL24, Proposition 2.27] or [BLXZ23, Lemma 3.16]. ∎

Another corollary is the behavior of $\mathbf{H}^{g}$ on a quasi-monomial simplicial cone $\sigma={\rm QM}_{\eta}(Y,E)$ , where $(Y,E)\to(X,\Delta)$ is a log smooth model and $\eta$ is the generic point of some stratum of $E$ . In this case, the geodesic connecting $v,w\in\sigma$ is the obvious line segment in $\sigma$ .

Theorem 3.5.

The function $v\mapsto\mathbf{H}^{g}({\mathcal{F}}_{v})$ on $\sigma$ is strictly convex. In particular, it is continuous and admits a unique minimizer $v_{0}\in\sigma$ .

Proof.

With the same argument as Corollary 3.4, The function $\mathbf{H}^{g}:\sigma\to{\mathbb{R}}_{>0}$ is strictly convex and admits at most one minimizer. To see the existence, it suffice to show that for any $v\in\sigma\setminus\{0\}$ , $\mathbf{H}^{g}(a{\mathcal{F}}_{v})\to+\infty$ as $a\to+\infty$ , which holds since $g$ is strictly increasing. ∎

3.2. Approximation by valuations

Definition 3.6 ( $\tilde{\beta}^{g}$ -invariants).

For any valuation $v$ over $X$ , we define

\displaystyle\tilde{\beta}^{g}(v)\,\,\,=\,\,\,\tilde{\beta}^{g}_{X,\Delta;L}(v)\,\,\,:=\,\,\,{\rm log}\Big{(}\int_{\mathbb{R}}g(A_{X,\Delta}(v)-t){\rm DH}_{{\mathcal{F}}_{v}}({\rm d}t)\Big{)}.

Remark 3.7.

Since $\mu_{X,\Delta;L}({\mathcal{F}}_{v})\leq A_{X,\Delta}(v)$ , we have naturally $\mathbf{H}^{g}({\mathcal{F}}_{v})\leq\tilde{\beta}^{g}(v)$ . The equality holds if $v$ is an lc place of $(X,\Delta+\Gamma)$ by Lemma 2.8, where $\Gamma\in|L|_{\mathbb{Q}}$ such that $(X,\Delta+\Gamma)$ is lc.

We have shown that the $\mathbf{H}^{g}$ -invariants admit at most one valuative minimizer. For the existence, we prove the following theorem as preparation.

Theorem 3.8.

$h^{g}(X,\Delta;L)={\rm inf}_{v\in{\rm Val}_{X}}\,\tilde{\beta}^{g}(v).$

Proof.

We need to show that for any linearly bounded filtration ${\mathcal{F}}$ on $R_{\bullet}$ , there exists a valuation $v$ over $X$ such that $\mathbf{H}^{g}({\mathcal{F}})\geq\tilde{\beta}^{g}(v)$ .

Just assume that $\mu=\mu({\mathcal{F}})<\lambda_{\rm max}({\mathcal{F}})$ . Then we have ${\rm lct}(X,\Delta;I^{(\mu)}_{\bullet})\leq 1$ . There exists a valuation $v$ on $X$ computing ${\rm lct}(X,\Delta;I^{(\mu)}_{\bullet})$ by [JM12]. Hence $v(I^{(\mu)}_{\bullet})\leq A_{X,\Delta}(v)$ . We denote by $f_{v}(t)=v(I^{(t)}_{\bullet})$ , which is a convex function on ${\mathbb{R}}$ . Rescale $v$ such that the first order left-derivative at $\mu\in{\mathbb{R}}$ equals to one, that is, $f^{\prime}_{v,-}(\mu)=1$ . Then we have

(9)

\displaystyle f_{v}(t)\geq t+f_{v}(\mu)-\mu\geq t+A_{X,\Delta}(v)-\mu.

We claim that ${\mathcal{F}}^{\prime}:={\mathcal{F}}(A_{X,\Delta}(v)-\mu)\subseteq{\mathcal{F}}_{v}$ , hence $G_{{\mathcal{F}}^{\prime}}\leq G_{{\mathcal{F}}_{v}}$ . Indeed, for any $\lambda\in{\mathbb{R}}$ and $s\in{\mathcal{F}}^{m(\lambda-A_{X,\Delta}(v)+\mu)}R_{m}$ ,

\displaystyle\frac{1}{m}v(s)\geq\frac{1}{m}v(I_{m,m(\lambda-A_{X,\Delta}(v)+\mu)})\geq f_{v}(\lambda-A_{X,\Delta}(v)+\mu)\geq\lambda,

where the third inequality follows from (9) with $t=\lambda-A_{X,\Delta}(v)+\mu$ . Hence $s\in{\mathcal{F}}^{m\lambda}_{v}R_{m}$ . Recall that the functional $\mu({\mathcal{F}})$ and measure ${\rm DH}_{{\mathcal{F}}}$ are affine with respect to shift of filtrations, that is, $\mu({\mathcal{F}}(b))=\mu({\mathcal{F}})+b$ and $\int_{\mathbb{R}}f(s){\rm DH}_{{\mathcal{F}}(b)}({\rm d}s)=\int_{\mathbb{R}}f(s+b){\rm DH}_{{\mathcal{F}}}({\rm d}s)$ for any $b\in{\mathbb{R}}$ . Hence $\mathbf{H}^{g}({\mathcal{F}})=\mathbf{H}^{g}({\mathcal{F}}(b))$ . We conclude that

$\displaystyle\mathbf{H}^{g}({\mathcal{F}})\,\,\,=\,\,\,\mathbf{H}^{g}({\mathcal{F}}^{\prime})$	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g(\mu({\mathcal{F}}^{\prime})-G_{{\mathcal{F}}^{\prime}}(y)){\rm d}y\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g(A_{X,\Delta}(v)-G_{{\mathcal{F}}^{\prime}}(y)){\rm d}y\Big{)}$
	$\displaystyle\geq$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g(A_{X,\Delta}(v)-G_{{\mathcal{F}}_{v}}(y)){\rm d}y\Big{)}\,\,\,=\,\,\,\tilde{\beta}^{g}(v).$

The proof is finished. ∎

Remark 3.9.

In the theorem $v\in{\rm Val}_{X}$ can be replaced by $v$ being quasi-monomial valuations over $X$ . Indeed, in the proof we can choose a quasi-monomial minimizer of ${\rm lct}(X,\Delta;I^{(\mu)}_{\bullet})$ by [Xu20].

3.3. Weighted delta invariants

By [BLXZ23, Definition 4.1], we define the following version of weighted delta invariants. This is one of the key ingredient in the proof of speciality of $\mathbf{H}^{g}$ -minimizer in the next section.

Let $g^{\prime}:{\mathbb{R}}\to{\mathbb{R}}_{>0}$ be the first order derivative of $g$ , and $N_{m}=\dim R_{m}$ .

Definition 3.10.

Let ${\mathcal{F}}_{0},{\mathcal{F}}$ be linearly bounded filtrations on $R$ , and $\mu_{0}=\mu({\mathcal{F}}_{0})$ , we define

	$\displaystyle N^{g^{\prime},{\mathcal{F}}_{0}}_{m}$	$\displaystyle:=$	$\displaystyle\sum_{i=1}^{N_{m}}g^{\prime}\Big{(}\mu_{0}-\frac{{\rm ord}_{{\mathcal{F}}_{0}}(s_{i})}{m}\Big{)},$
	$\displaystyle S^{g^{\prime},{\mathcal{F}}_{0}}_{m}({\mathcal{F}})\,\,\,=\,\,\,S^{g^{\prime},{\mathcal{F}}_{0}}_{m}(L;{\mathcal{F}})$	$\displaystyle:=$	$\displaystyle\frac{1}{N^{g^{\prime},{\mathcal{F}}_{0}}_{m}}\sum_{i=1}^{N_{m}}g^{\prime}\Big{(}\mu_{0}-\frac{{\rm ord}_{{\mathcal{F}}_{0}}(s_{i})}{m}\Big{)}\cdot\frac{{\rm ord}_{\mathcal{F}}(s_{i})}{m},$

where $\{s_{i}\}$ is a basis of $R_{m}$ which is compatible with both ${\mathcal{F}}_{0}$ and ${\mathcal{F}}$ . It’s clear that $S^{g^{\prime},{\mathcal{F}}_{0}}_{m}(L;{\mathcal{F}})$ does not depend on the choice of $\{s_{i}\}$ . Let

\displaystyle S^{g^{\prime},{\mathcal{F}}_{0}}({\mathcal{F}})\,\,\,=\,\,\,S^{g^{\prime},{\mathcal{F}}_{0}}(L;{\mathcal{F}})\,\,\,:=\,\,\,\mathop{{\rm lim}}_{m\to\infty}S^{g^{\prime},{\mathcal{F}}_{0}}_{m}(L;{\mathcal{F}})\,\,\,=\,\,\,\frac{\int_{{\mathbb{R}}^{2}}g^{\prime}(\mu_{0}-x)y\cdot{\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}}({\rm d}x{\rm d}y)}{\int_{{\mathbb{R}}}g^{\prime}(\mu_{0}-x)\cdot{\rm DH}_{{\mathcal{F}}_{0}}({\rm d}x)},

Finally let

\displaystyle\delta^{g^{\prime},{\mathcal{F}}_{0}}_{m}(X,\Delta;L)\,\,\,:=\,\,\,{\rm inf}_{v}\frac{A_{X,\Delta}(v)}{S^{g^{\prime},{\mathcal{F}}_{0}}_{m}(L;v)},\qquad\delta^{g^{\prime},{\mathcal{F}}_{0}}(X,\Delta;L)\,\,\,:=\,\,\,{\rm inf}_{v}\frac{A_{X,\Delta}(v)}{S^{g^{\prime},{\mathcal{F}}_{0}}(L;v)},

where the infimum runs over all the valuations $v$ over $X$ .

We have the following generalization of [BLXZ23, Theorem 5.1].

Lemma 3.11.

Let ${\mathcal{F}}_{0}$ be a linearly bounded filtration on $R=R(X;L)$ with $\mu_{0}=\mu({\mathcal{F}}_{0})$ and $v_{0}$ be a valuation minimizing ${\rm lct}(X,\Delta;I_{\bullet}^{(\mu_{0})})$ . By shifting ${\mathcal{F}}_{0}$ , we may assume that $\mu_{0}=A_{X,\Delta}(v_{0})$ .

Then ${\mathcal{F}}_{0}$ minimizes $\mathbf{H}^{g}$ if and only if $\delta^{g^{\prime},{\mathcal{F}}_{0}}(X,\Delta;L)=\frac{A_{X,\Delta}(v_{0})}{S^{g^{\prime},{\mathcal{F}}_{0}}(L;v_{0})}=1$ and $\mathbf{H}^{g}({\mathcal{F}}_{0})=\tilde{\beta}^{g}(v_{0})$ .

Proof.

The proof follows from [BLXZ23, Theorem 5.1]. We first prove the “if” part. By Theorem 3.8, it suffices to show $\tilde{\beta}^{g}(v)\geq\mathbf{H}^{g}({\mathcal{F}}_{0})$ for any valuation $v$ over $X$ .

By the proof of Theorem 3.8, we know that ${\mathcal{F}}_{0}\subseteq{\mathcal{F}}_{v_{0}}$ , hence $G_{{\mathcal{F}}_{0}}\leq G_{{\mathcal{F}}_{v_{0}}}$ . The assumptions $\mu_{0}=A_{X,\Delta}(v_{0})$ and $\mathbf{H}^{g}({\mathcal{F}}_{0})=\tilde{\beta}^{g}(v_{0})$ imply that $G_{{\mathcal{F}}_{0}}=G_{{\mathcal{F}}_{v_{0}}}$ almost everywhere on $\mathbf{O}$ . Hence

(10)

\displaystyle S^{g^{\prime},{\mathcal{F}}_{0}}({\mathcal{F}}_{0})=S^{g^{\prime},{\mathcal{F}}_{0}}(v_{0}).

Let ${\mathcal{F}}_{t}$ be the geodesic connecting ${\mathcal{F}}_{0}$ and ${\mathcal{F}}_{1}:={\mathcal{F}}_{v}$ . We define the following analog of $\mathbf{H}^{g}({\mathcal{F}}_{t})$ ,

\displaystyle f(t)

\displaystyle:=

\displaystyle{\rm log}\Big{(}\int_{{\mathbb{R}}^{2}}g((1-t)(\mu_{0}-x)+t(A_{X,\Delta}(v)-y)){\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}({\rm d}x{\rm d}y)\Big{)}.

Then similar argument of Theorem 3.3 shows that $f$ is convex. We have

$\displaystyle f^{\prime}(0)$	$\displaystyle=$	$\displaystyle e^{-f(0)}\cdot\int_{{\mathbb{R}}^{2}}\big{(}(A_{X,\Delta}(v)-y)-(\mu_{0}-x)\big{)}g^{\prime}(\mu_{0}-x){\rm DH}_{{\mathcal{F}}_{0},{\mathcal{F}}_{1}}({\rm d}x{\rm d}y),$
	$\displaystyle=$	$\displaystyle e^{-f(0)}\mathbf{v}^{g^{\prime},{\mathcal{F}}_{0}}\cdot\Big{(}(A_{X,\Delta}(v)-S^{g^{\prime},{\mathcal{F}}_{0}}(v))-(\mu_{0}-S^{g^{\prime},{\mathcal{F}}_{0}}({\mathcal{F}}_{0}))\Big{)},$
	$\displaystyle=$	$\displaystyle e^{-f(0)}\mathbf{v}^{g^{\prime},{\mathcal{F}}_{0}}\cdot(A_{X,\Delta}(v)-S^{g^{\prime},{\mathcal{F}}_{0}}(v))\,\,\,\geq\,\,\,0,$

where $\mathbf{v}^{g^{\prime},{\mathcal{F}}_{0}}=\int_{{\mathbb{R}}}g^{\prime}(\mu_{0}-x){\rm DH}_{{\mathcal{F}}_{0}}({\rm d}x)$ and the third equality follows from (10). Hence

\displaystyle\mathbf{H}^{g}({\mathcal{F}}_{0})=f(0)\leq f(1)=\tilde{\beta}^{g}(v).

Next, we prove the “only if” part. By Theorem 3.8, we know that $\mathbf{H}^{g}({\mathcal{F}}_{0})\geq\tilde{\beta}^{g}(v_{0})\geq\mathbf{H}^{g}({\mathcal{F}}_{v_{0}})$ . Hence both the equalities hold since ${\mathcal{F}}_{0}$ minimizes $\mathbf{H}^{g}$ , and we also have (10).

For any valuation $v$ over $X$ , let ${\mathcal{F}}_{t}$ and $f$ be the same as above. Since $\mu({\mathcal{F}}_{v})\leq A_{X,\Delta}(v)$ , we have

\displaystyle\mu({\mathcal{F}}_{t})\leq(1-t)\mu({\mathcal{F}}_{0})+t\mu({\mathcal{F}}_{1})\leq(1-t)\mu_{0}+tA_{X,\Delta}(v).

Hence $f(0)=\mathbf{H}^{g}({\mathcal{F}}_{0})\leq\mathbf{H}^{g}({\mathcal{F}}_{t})\leq f(t)$ for any $0\leq t\leq 1$ . We conclude that $f^{\prime}(0)\geq 0$ since $f$ is convex, that is,

\displaystyle A_{X,\Delta}(v)-S^{g^{\prime},{\mathcal{F}}_{0}}(v)\geq\mu_{0}-S^{g^{\prime},{\mathcal{F}}_{0}}({\mathcal{F}}_{0})=A_{X,\Delta}(v_{0})-S^{g^{\prime},{\mathcal{F}}_{0}}(v_{0}),

by the assumption and (10). If $v=\lambda v_{0}$ , we see that

(\lambda-1)(A_{X,\Delta}(v_{0})-S^{g^{\prime},{\mathcal{F}}_{0}}(v_{0}))\geq 0,

for any $\lambda>0$ . Hence $A_{X,\Delta}(v_{0})-S^{g^{\prime},{\mathcal{F}}_{0}}(v_{0})=0$ . The proof of Lemma 3.11 is finished. ∎

4. Existence of $\mathbf{H}^{g}$ -minimizers and finite generation

In this section, let $(X,\Delta)$ be a log Fano pair and $L=-(K_{X}+\Delta)$ .

4.1. Approximation by test configurations

Recall that a normal test configuration (TC) of $(X,\Delta)$ is a collection $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ consisting of

•

A normal variety ${\mathcal{X}}$ with a ${\mathbb{G}}_{m}$ -action generated by $\eta\in{\rm Hom}({\mathbb{G}}_{m},{\rm Aut}({\mathcal{X}}))$ ;
•

A ${\mathbb{G}}_{m}$ -equivariant morphism $\pi:{\mathcal{X}}\to{\mathbb{A}}^{1}$ , where the ${\mathbb{G}}_{m}$ -action on ${\mathbb{A}}^{1}$ is standard;
•

A ${\mathbb{G}}_{m}$ -equivariant $\pi$ -semiample ${\mathbb{Q}}$ -Cartier divisor ${\mathcal{L}}$ on ${\mathcal{X}}$ ;
•

A ${\mathbb{G}}_{m}$ -equivariant trivialization over the punctured plane $i_{\eta}:({\mathcal{X}},{\mathcal{L}})|_{\pi^{-1}({\mathbb{G}}_{m})}\cong(X,L)\times{\mathbb{G}}_{m}$ , which is compatible with $\pi$ and ${\rm pr}_{1}$ . And $\Delta_{\mathcal{X}}$ is the closure of $i_{\eta}^{-1}(\Delta\times{\mathbb{G}}_{m})$ in ${\mathcal{X}}$ .

The TC $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ is called (weakly) special if $({\mathcal{X}},{\mathcal{X}}_{0}+\Delta_{\mathcal{X}})$ is (lc) plt, and ${\mathcal{L}}=-K_{{\mathcal{X}}/{\mathbb{A}}^{1}}-\Delta_{\mathcal{X}}+c{\mathcal{X}}_{0}$ for some $c\in{\mathbb{Q}}$ . Note by adjunction that $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ being special is equivalent that the central fiber $({\mathcal{X}}_{0},\Delta_{{\mathcal{X}},0})$ is a log Fano pair.

For any test configuration $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ of $(X,\Delta)$ , we have the following ${\mathbb{Z}}$ -filtration ${\mathcal{F}}={\mathcal{F}}_{(X,\Delta_{\mathcal{X}};{\mathcal{L}},\eta)}$ on the anti-canonical ring $R=R(X,\Delta)$ ,

(11)

\displaystyle{\mathcal{F}}^{\lambda}R_{m}

\displaystyle:=

\displaystyle\{f\in H^{0}(X,mL):t^{-\lambda}\bar{f}\in H^{0}({\mathcal{X}},m{\mathcal{L}})\},

where $t$ is the parameter on ${\mathbb{A}}^{1}$ , and $\bar{f}$ is the ${\mathbb{G}}_{m}$ -extension of $f$ on ${\mathcal{X}}\setminus{\mathcal{X}}_{0}$ and viewed as a rational section of $m{\mathcal{L}}$ . We simply denote $\mathbf{F}({\mathcal{F}}_{(X,\Delta_{\mathcal{X}};{\mathcal{L}},\eta)})$ by $\mathbf{F}(X,\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ for $\mathbf{F}=\mathbf{L}$ or $\mathbf{H}^{g}$ . We have

(12)

\displaystyle\mathbf{L}(X,\Delta_{\mathcal{X}};{\mathcal{L}},\eta):={\rm lct}(X,\Delta_{\mathcal{X}}+{\mathcal{D}};{\mathcal{X}}_{0})-1,

where ${\mathcal{D}}\sim_{\mathbb{Q}}-(K_{\mathcal{X}}+\Delta_{\mathcal{X}})-{\mathcal{L}}$ is supported on ${\mathcal{X}}_{0}$ , see for example [Xu24, Theorem 3.66].

Conversely, for any linearly bounded filtration ${\mathcal{F}}$ on $R$ , one may construct a sequence of TC $({\mathcal{X}}_{m};{\mathcal{L}}_{m})$ approximating it, see for example [Xu24, Definition 3.65]. We shortly recall the construction. Recall that ${\mathcal{I}}_{m}(e_{+})\subseteq{\mathcal{O}}_{X}[s]$ is the graded ideal sequence associated to ${\mathcal{F}}$ in Definition 2.10. Let $\pi_{m}:{\mathcal{X}}_{m}\to X_{{\mathbb{A}}^{1}}$ be the normalized blowup along ${\mathcal{I}}_{m}(e_{+})$ with exceptional divisor ${\mathcal{E}}_{m}$ , and $\Delta_{{\mathcal{X}}_{m}}=\pi_{m,*}^{-1}\Delta_{{\mathbb{A}}^{1}}$ . Then ${\mathcal{L}}_{m}=\pi_{m}^{*}L_{{\mathbb{A}}^{1}}-\frac{1}{m}{\mathcal{E}}_{m}$ is semiample by [Xu24, Lemma 3.64]. Hence $({\mathcal{X}}_{m},\Delta_{{\mathcal{X}}_{m}};{\mathcal{L}}_{m},\eta_{m})$ is a normal TC of $(X,\Delta)$ and is called the $m$ -th approximating TC of ${\mathcal{F}}$ . We remark that the definition depends on the choice of $e_{+}$ .

Lemma 4.1.

[HL24, Proposition 2.16 and 2.28]

(13)		$\displaystyle\mathbf{L}({\mathcal{F}})$	$\displaystyle\geq$	$\displaystyle\mathop{{\rm lim}}_{m\to\infty}\mathbf{L}({\mathcal{X}}_{m},\Delta_{{\mathcal{X}}_{m}};{\mathcal{L}}_{m},\eta_{m}),$
(14)		$\displaystyle{\rm DH}_{\mathcal{F}}$	$\displaystyle=$	$\displaystyle\mathop{{\rm lim}}_{m\to\infty}{\rm DH}_{({\mathcal{X}}_{m},\Delta_{{\mathcal{X}}_{m}};{\mathcal{L}}_{m},\eta_{m})}.$

We remark that (13) only holds for Fano varieties, but (14) holds for polarized varieties.

Corollary 4.2.

(15)

\displaystyle\mathbf{H}^{g}({\mathcal{F}})\geq\mathop{{\rm lim}}_{m\to\infty}\mathbf{H}^{g}({\mathcal{X}}_{m},\Delta_{{\mathcal{X}}_{m}};{\mathcal{L}}_{m},\eta_{m}),

Theorem 4.3.

For any log Fano pair $(X,\Delta)$ , we have

(16)

\displaystyle h^{g}(X,\Delta)=\mathop{{\rm inf}}_{({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)}\mathbf{H}^{g}({\mathcal{X}},\Delta_{{\mathcal{X}}};{\mathcal{L}},\eta),

where the infimum runs over all the normal test configurations $({\mathcal{X}},\Delta_{{\mathcal{X}}};{\mathcal{L}},\eta)$ of $(X,\Delta)$ .

For any TC $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ of $(X,\Delta)$ , we denote by $-{\mathcal{D}}={\mathcal{L}}+(K_{\mathcal{X}}+\Delta_{\mathcal{X}})=\sum_{i}e_{i}E_{i}$ and ${\mathcal{X}}_{0}=\sum_{i}b_{i}E_{i}$ , where $E_{i}\subseteq{\mathcal{X}}$ are irreducible components of ${\mathcal{X}}_{0}$ . Let $v_{i}={\rm ord}_{E_{i}}|_{{\mathcal{X}}_{1}}$ be the corresponding divisorial valuations over $X={\mathcal{X}}_{1}$ . We have the following description of the filtration ${\mathcal{F}}={\mathcal{F}}_{({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)}$ induced by $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ .

Lemma 4.4.

\displaystyle{\mathcal{F}}_{({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)}\,\,\,=\,\,\,\bigcap_{i}b_{i}^{-1}\Big{(}{\mathcal{F}}_{v_{i}}(e_{i}+1-b_{i}-A_{X,\Delta}(v_{i}))\Big{)}.

Proof.

Let ${\mathcal{Y}}$ be the graph of the birational map ${\mathcal{X}}\dashrightarrow X_{{\mathbb{A}}^{1}}$ , and $\pi:{\mathcal{Y}}\to{\mathcal{X}}$ , $\tau:{\mathcal{Y}}\to X_{{\mathbb{A}}^{1}}$ be the corresponding morphisms.

By [BHJ17, Lemma 5.17] (whose notation is $v_{E_{i}}=b_{i}^{-1}v_{i}$ ), for any $\lambda$ and $m$ , we have

\displaystyle{\mathcal{F}}_{({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)}^{\lambda}R_{m}\,\,\,=\,\,\,\bigcap_{i}{\mathcal{F}}_{v_{i}}^{b_{i}\lambda-m\cdot{\rm ord}_{E_{i}}(D)}R_{m},

where $D=\pi^{*}{\mathcal{L}}-\tau^{*}L_{{\mathbb{A}}^{1}}$ is supported on ${\mathcal{Y}}_{0}$ . It suffices to prove ${\rm ord}_{E_{i}}(D)=e_{i}+1-b_{i}-A_{X,\Delta}(v_{i})$ . Since

\displaystyle D=\pi^{*}({\mathcal{L}}+K_{\mathcal{X}}+\Delta_{\mathcal{X}})+(-\pi^{*}(K_{\mathcal{X}}+\Delta_{\mathcal{X}})-\tau^{*}L_{{\mathbb{A}}^{1}})=\sum_{i}e_{i}E_{i}+B,

where $B=-\pi^{*}(K_{\mathcal{X}}+\Delta_{\mathcal{X}})+\tau^{*}(K_{X_{{\mathbb{A}}^{1}}}+\Delta_{{\mathbb{A}}^{1}})$ is supported on ${\mathcal{Y}}_{0}$ . By Lemma 4.5, we have

\displaystyle{\rm ord}_{E_{i}}(B)=A_{{\mathcal{X}},\Delta_{\mathcal{X}}}(E_{i})-A_{X_{{\mathbb{A}}^{1}},\Delta_{{\mathbb{A}}^{1}}}(E_{i})=1-(b_{i}+A_{X_{{\mathbb{A}}^{1}},\Delta_{{\mathbb{A}}^{1}}+X_{0}}(E_{i}))=1-b_{i}-A_{X,\Delta}(v_{i}),

where the second and third equalities follows from ${\rm ord}_{E_{i}}(X_{0})=b_{i}$ and adjunction respectively. ∎

Lemma 4.5.

Let $\pi:Z\to(X,\Delta_{X})$ and $\tau:Z\to(Y,\Delta_{Y})$ be birational morphisms of ${\mathbb{Q}}$ -Gorenstein families over a curve $C$ , which are isomorphisms away from $0\in C$ , and ${\rm Supp}(\Delta_{X}),{\rm Supp}(\Delta_{X})$ do not contain any fiber of the families. Then for any irreducible component $E$ of $Z_{0}\subseteq Z$ , we have

{\rm ord}_{E}(-\pi^{*}(K_{X}+\Delta_{X})+\tau^{*}(K_{Y}+\Delta_{Y}))=A_{X,\Delta_{X}}(E)-A_{Y,\Delta_{Y}}(E).

Proof.

Note that

	$\displaystyle\pi^{}(K_{X}+\Delta_{X})=K_{Z}+\pi^{-1}_{}\Delta_{X}+(1-A_{X,\Delta_{X}}(E))E+F,$
	$\displaystyle\tau^{}(K_{Y}+\Delta_{Y})=K_{Z}+\tau^{-1}_{}\Delta_{Y}+(1-A_{Y,\Delta_{Y}}(E))E+F^{\prime},$

where $F,F^{\prime}\subseteq Z_{0}$ are ${\mathbb{Q}}$ -divisors that do not contain $E$ as a component. By assumption, we have $\pi^{-1}_{*}\Delta_{X}=\tau^{-1}_{*}\Delta_{Y}$ . Hence

\displaystyle B=-\pi^{*}(K_{X}+\Delta_{X})+\tau^{*}(K_{Y}+\Delta_{Y})=(A_{X,\Delta_{X}}(E)-A_{Y,\Delta_{Y}}(E))E+F^{\prime}-F,

is a ${\mathbb{Q}}$ -divisor supported in $Z_{0}$ . We conclude that ${\rm ord}_{E}(B)=A_{X,\Delta_{X}}(E)-A_{Y,\Delta_{Y}}(E)$ . ∎

4.2. Approximation by special test configurations

The following theorem is an analog of [HL24, Theorem 3.4], which depends on Li-Xu’s proof of Tian’s conjecture [LX14]. Different from Han-Li’s proof which relies on an analytic description of the $\mathbf{H}$ -invariants, we give a pure algebraic proof by considering the filtrations induced by test configurations.

Theorem 4.6.

For any normal TC $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},\eta)$ of $(X,\Delta)$ and $a\in{\mathbb{R}}_{>0}$ , there exists a special TC $({\mathcal{X}}^{s},\Delta_{{\mathcal{X}}^{s}};{\mathcal{L}}^{s},\eta^{s})$ and $a^{s}\in{\mathbb{R}}_{>0}$ such that

\displaystyle\mathbf{H}^{g}({\mathcal{X}}^{s},\Delta_{{\mathcal{X}}^{s}};{\mathcal{L}}^{s},a^{s}\eta^{s})\leq\mathbf{H}^{g}({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},a\eta).

Proof.

We follow the proof of [HL24, Theorem 3.4].

Step 1. (Semistable reduction ${\mathcal{X}}^{(d_{1})}$ ). By [LX14, Lemma 5], there exists a semistable reduction ${\mathcal{X}}^{(d_{1})}\to{\mathcal{X}}$ over ${\mathbb{A}}^{1}\to{\mathbb{A}}^{1},z\mapsto z^{d_{1}}$ , such that ${\mathcal{X}}^{(d_{1})}_{0}$ is reduced. Since the filtration

\displaystyle{\mathcal{F}}_{({\mathcal{X}}^{(d_{1})},\Delta_{{\mathcal{X}}^{(d_{1})}};{\mathcal{L}}^{(d_{1})},\frac{a}{d_{1}}\eta^{(d_{1})})}={\mathcal{F}}_{({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},a\eta)}

is not changed, the $\mathbf{H}^{g}$ -invariants are the same.

Step 2. (Lc modification ${\mathcal{X}}^{\rm lc}$ ). By [LX14, Theorem 2], which is proved by running a ${\mathbb{G}}_{m}$ -equivariant MMP on a log resolution of $({\mathcal{X}}^{(d_{1})},\Delta_{{\mathcal{X}}^{(d_{1})}}+{\mathcal{X}}^{(d_{1})}_{0})$ , there is a ${\mathbb{G}}_{m}$ -equivariant lc modification $\pi^{\rm lc}:{\mathcal{X}}^{\rm lc}\to{\mathcal{X}}^{(d_{1})}$ such that $({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}}+{\mathcal{X}}^{\rm lc}_{0})$ is lc and $K_{{\mathcal{X}}^{\rm lc}}+\Delta_{{\mathcal{X}}^{\rm lc}}$ is ample over ${\mathcal{X}}^{(d_{1})}$ .

Write $E={\mathcal{L}}^{(d_{1})}+K_{{\mathcal{X}}^{\rm lc}}+\Delta_{{\mathcal{X}}^{\rm lc}}=\sum_{i=1}^{l}e_{i}E_{i}$ with $e_{1}\leq e_{2}\leq\cdots\leq e_{l}$ , where $E_{i}$ are irreducible components of ${\mathcal{X}}^{\rm lc}_{0}$ . Let ${\mathcal{L}}^{\rm lc}_{\lambda}={\mathcal{L}}^{(d_{1})}+\lambda E=-(K_{{\mathcal{X}}^{\rm lc}}+\Delta_{{\mathcal{X}}^{\rm lc}})+(1+\lambda)E$ and ${\mathcal{F}}_{\lambda}:={\mathcal{F}}_{({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc}_{\lambda},\eta^{\rm lc})}$ . By Lemma 4.4, we have

	$\displaystyle\frac{a}{d_{1}}{\mathcal{F}}_{\lambda}$	$\displaystyle=$	$\displaystyle{\mathcal{F}}_{({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc}_{\lambda},\frac{a}{d_{1}}\eta^{\rm lc})}\,\,\,=\,\,\,\frac{a}{d_{1}}\bigcap_{i}\Big{(}{\mathcal{F}}_{v_{i}}((1+\lambda)e_{i}-A_{X,\Delta}(v_{i}))\Big{)},$
	$\displaystyle G_{{\mathcal{F}}_{\lambda}}(y)$	$\displaystyle=$	$\displaystyle{\rm min}_{i}\,\Big{(}G_{v_{i}}(y)+(1+\lambda)e_{i}-A_{X,\Delta}(v_{i})\Big{)},\,\,\,\forall y\in\mathbf{O}.$

On the other hand, by [HL24, Example 2.31] we have

\displaystyle\mathbf{L}({\mathcal{F}}_{\lambda})

\displaystyle=

\displaystyle\mathbf{L}({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc}_{\lambda},\eta^{\rm lc})\,\,\,=\,\,\,(1+\lambda)e_{1}.

If $\lambda=0$ , we have

\displaystyle\frac{a}{d_{1}}{\mathcal{F}}_{0}\,\,\,=\,\,\,{\mathcal{F}}_{({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc}_{0},\frac{a}{d_{1}}\eta^{\rm lc})}\,\,\,=\,\,\,{\mathcal{F}}_{({\mathcal{X}}^{(d_{1})},\Delta_{{\mathcal{X}}^{(d_{1})}};{\mathcal{L}}^{(d_{1})},\frac{a}{d_{1}}\eta^{(d_{1})})}.

We denote by $i(y)$ the minimizer of the above minimum for any $y\in\mathbf{O}$ . Then

$\displaystyle\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}}{\mathcal{F}}_{\lambda}\Big{)}$	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\Big{(}\frac{a}{d_{1}}\big{(}\mathbf{L}({\mathcal{F}}_{\lambda})-G_{{\mathcal{F}}_{\lambda}}(y)\big{)}\Big{)}{\rm d}y\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\Big{(}\frac{a}{d_{1}}{\rm max}_{i}\big{(}(1+\lambda)(e_{1}-e_{i})+A_{X,\Delta}(v_{i})-G_{v_{i}}(y)\big{)}\Big{)}{\rm d}y\Big{)},$
$\displaystyle\frac{{\rm d}}{{\rm d}\lambda}\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}}{\mathcal{F}}_{\lambda}\Big{)}$	$\displaystyle=$	$\displaystyle\frac{a}{d_{1}}\frac{\int_{\mathbf{O}}(e_{1}-e_{i(y)})\cdot g^{\prime}\circ f(\lambda,y){\rm d}y}{\int_{\mathbf{O}}g\circ f(\lambda,y){\rm d}y}\leq 0,$

where $f(\lambda,y)=\frac{a}{d_{1}}\big{(}\mathbf{L}({\mathcal{F}}_{\lambda})-G_{{\mathcal{F}}_{\lambda}}(y)\big{)}$ . Recall that $K_{{\mathcal{X}}^{\rm lc}}+\Delta_{{\mathcal{X}}^{\rm lc}}$ is ample over ${\mathcal{X}}^{(d_{1})}$ , so is $E={\mathcal{L}}^{(d_{1})}+K_{{\mathcal{X}}^{\rm lc}}+\Delta_{{\mathcal{X}}^{\rm lc}}$ . Hence ${\mathcal{L}}^{\rm lc}_{\lambda}$ is ample over ${\mathbb{A}}^{1}$ for $0<\lambda\ll 1$ . Fix a very small $\lambda>0$ and let ${\mathcal{L}}^{\rm lc}={\mathcal{L}}^{\rm lc}_{\lambda}$ . We get an ample TC $({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc},\frac{a}{d_{1}}\eta^{\rm lc})$ such that

\displaystyle\mathbf{H}^{g}({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc},\frac{a}{d_{1}}\eta^{\rm lc})\leq\mathbf{H}^{g}({\mathcal{X}}^{(d_{1})},\Delta_{{\mathcal{X}}^{(d_{1})}};{\mathcal{L}}^{(d_{1})},\frac{a}{d_{1}}\eta^{(d_{1})}).

Step 3. (Ample configuration ${\mathcal{X}}^{\rm ac}$ ). Choose $q\gg 1$ such that ${\mathcal{H}}^{\rm lc}={\mathcal{L}}^{\rm lc}-(1+q)^{-1}({\mathcal{L}}^{\rm lc}+K_{{\mathcal{X}}^{\rm lc}}+\Delta_{{\mathcal{X}}^{\rm lc}})$ is ample over ${\mathbb{A}}^{1}$ . Set ${\mathcal{X}}^{0}={\mathcal{X}}^{\rm lc};{\mathcal{L}}^{0}={\mathcal{L}}^{\rm lc},{\mathcal{H}}^{0}={\mathcal{H}}^{\rm lc}$ and $\lambda_{0}=1+q$ . Running a ${\mathbb{G}}_{m}$ -equivariant $(K_{{\mathcal{X}}^{0}}+\Delta_{{\mathcal{X}}^{0}})$ -MMP with scaling ${\mathcal{H}}^{0}$ , we get a sequence of birational maps

\displaystyle{\mathcal{X}}^{0}\dashrightarrow{\mathcal{X}}^{1}\dashrightarrow\cdots\dashrightarrow{\mathcal{X}}^{k}.

Let ${\mathcal{H}}^{j}$ be the pushforward of ${\mathcal{H}}^{0}$ to ${\mathcal{X}}^{j}$ , and $\lambda_{j+1}={\rm inf}\{\lambda:K_{X^{j}}+\lambda{\mathcal{H}}^{j}\text{ is nef over }{\mathbb{A}}^{1}\}$ be the nef threshold. Then ${\mathcal{X}}^{j}\dashrightarrow{\mathcal{X}}^{j+1}$ is the contraction of a $(K_{{\mathcal{X}}^{j}}+\Delta_{{\mathcal{X}}^{j}}+\lambda_{j+1}{\mathcal{H}}^{j})$ -trivial extremal ray. We have

\displaystyle 1+q=\lambda_{0}\geq\lambda_{1}\geq\cdots\geq\lambda_{k}>\lambda_{k+1}=1,

where the last equality follows from the fact that the pseudo-effective threshold of $K_{{\mathcal{X}}^{0}}+\Delta_{{\mathcal{X}}^{0}}$ with respect to ${\mathcal{H}}^{0}$ is $1$ . For any $\lambda>1$ , we denote by

\displaystyle{\mathcal{L}}_{\lambda}=(\lambda-1)^{-1}(K_{{\mathcal{X}}^{0}}+\Delta_{{\mathcal{X}}^{0}}+\lambda{\mathcal{H}}^{0}),\quad E=K_{{\mathcal{X}}^{0}}+\Delta_{{\mathcal{X}}^{0}}+{\mathcal{H}}^{0}=\sum_{i}e_{i}E_{i},

with $e_{1}\leq e_{2}\leq\cdots\leq e_{l}$ . Then

\displaystyle{\mathcal{L}}_{\lambda}+K_{{\mathcal{X}}^{0}}+\Delta_{{\mathcal{X}}^{0}}=\frac{\lambda}{\lambda-1}(K_{{\mathcal{X}}^{0}}+\Delta_{{\mathcal{X}}^{0}}+{\mathcal{H}}^{0})=\frac{\lambda}{\lambda-1}E.

Let ${\mathcal{L}}^{j}_{\lambda}$ and $E^{j}$ be the push-forward of ${\mathcal{L}}_{\lambda}$ and $E$ to ${\mathcal{X}}^{j}$ respectively. And we denote by ${\mathcal{F}}^{j}_{\lambda}={\mathcal{F}}^{j}_{({\mathcal{X}}^{j},\Delta_{{\mathcal{X}}^{j}};{\mathcal{L}}^{j}_{\lambda},\eta^{j})}$ . Then for any $\lambda_{j}\geq\lambda\geq\lambda_{j+1}$ , we have

$\displaystyle\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}}{\mathcal{F}}^{j}_{\lambda}\Big{)}$	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\Big{(}\frac{a}{d_{1}}\big{(}\mathbf{L}({\mathcal{F}}_{\lambda})-G_{{\mathcal{F}}_{\lambda}}(y)\big{)}\Big{)}{\rm d}y\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\Big{(}\frac{a}{d_{1}}{\rm max}_{i}\big{(}\frac{\lambda}{\lambda-1}(e_{1}-e_{i})+A_{X,\Delta}(v_{i})-G_{v_{i}}(y)\big{)}\Big{)}{\rm d}y\Big{)},$
$\displaystyle\frac{{\rm d}}{{\rm d}\lambda}\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}}{\mathcal{F}}^{j}_{\lambda}\Big{)}$	$\displaystyle=$	$\displaystyle\frac{a}{d_{1}}\frac{\int_{\mathbf{O}}(\lambda-1)^{-2}(e_{i(y)}-e_{1})\cdot g^{\prime}\circ f^{j}(\lambda,y){\rm d}y}{\int_{\mathbf{O}}g\circ f^{j}(\lambda,y){\rm d}y}\geq 0.$

where $f^{j}(\lambda,y)=\frac{a}{d_{1}}\big{(}\mathbf{L}({\mathcal{F}}^{j}_{\lambda})-G_{{\mathcal{F}}^{j}_{\lambda}}(y)\big{)}$ . On the other hand, the filtration is not changed under divisorial contractions and flips. Hence for any $0\leq j\leq k$ we have

\displaystyle\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}}{\mathcal{F}}^{j}_{\lambda_{j+1}}\Big{)}=\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}}{\mathcal{F}}^{j+1}_{\lambda_{j+1}}\Big{)}.

Recall that $K_{{\mathcal{X}}^{k}}+\Delta_{{\mathcal{X}}^{k}}+{\mathcal{H}}^{k}$ is nef over ${\mathbb{A}}^{1}$ . So is

\displaystyle K_{{\mathcal{X}}^{k}}+\Delta_{{\mathcal{X}}^{k}}+{\mathcal{L}}^{k}_{\lambda_{k}}=\frac{\lambda_{k}}{\lambda_{k}-1}(K_{{\mathcal{X}}^{k}}+\Delta_{{\mathcal{X}}^{k}}+{\mathcal{H}}^{k}).

By negativity lemma, we have $K_{{\mathcal{X}}^{k}}+\Delta_{{\mathcal{X}}^{k}}+{\mathcal{L}}^{k}_{\lambda_{k}}\sim_{{\mathbb{Q}},{\mathbb{A}}^{1}}0$ . Let ${\mathcal{X}}^{\rm ac}={\mathcal{X}}^{k}$ and ${\mathcal{L}}^{\rm ac}={\mathcal{L}}^{k}_{\lambda_{k}}$ . Now we get a TC $({\mathcal{X}}^{\rm ac},\Delta_{{\mathcal{X}}^{\rm ac}},{\mathcal{L}}^{\rm ac},\frac{a}{d_{1}}\eta^{\rm ac})$ with $-(K_{{\mathcal{X}}^{\rm ac}}+\Delta_{{\mathcal{X}}^{\rm ac}})\sim_{{\mathbb{Q}},{\mathbb{A}}^{1}}{\mathcal{L}}^{\rm ac}$ ample over ${\mathbb{A}}^{1}$ , such that

\displaystyle\mathbf{H}^{g}({\mathcal{X}}^{\rm ac},\Delta_{{\mathcal{X}}^{\rm ac}},{\mathcal{L}}^{\rm ac},\frac{a}{d_{1}}\eta^{\rm ac})\leq\mathbf{H}^{g}({\mathcal{X}}^{\rm lc},\Delta_{{\mathcal{X}}^{\rm lc}};{\mathcal{L}}^{\rm lc},\frac{a}{d_{1}}\eta^{\rm lc}).

Step 4. (Special test configuration ${\mathcal{X}}^{\rm s}$ ). By [LX14, Theorem 6], there exists a special TC ${\mathcal{X}}^{\rm s}$ birational to $({\mathcal{X}}^{\rm ac})^{(d_{2})}$ over ${\mathbb{A}}^{1}$ for some $d_{2}>0$ , such that ${\mathcal{X}}^{s}_{0}$ is an lc place of $(({\mathcal{X}}^{\rm ac})^{(d_{2})},\Delta_{({\mathcal{X}}^{\rm ac})^{(d_{2})}}+({\mathcal{X}}^{\rm ac})^{(d_{2})}_{0})$ . By [BCHM10, 1.4.3], there exists a ${\mathbb{G}}_{m}$ -equivariant birational morphism $\pi^{\prime}:{\mathcal{X}}^{\prime}\to({\mathcal{X}}^{\rm ac})^{(d_{2})}$ which precisely extracts ${\mathcal{X}}^{\rm s}_{0}$ . Hence $K_{{\mathcal{X}}^{\prime}}+\Delta_{{\mathcal{X}}^{\prime}}=\pi^{\prime*}(K_{({\mathcal{X}}^{\rm ac})^{(d_{2})}}+\Delta_{({\mathcal{X}}^{\rm ac})^{(d_{2})}})$ and

\displaystyle{\mathcal{F}}_{({\mathcal{X}}^{\prime},\Delta_{{\mathcal{X}}^{\prime}},-(K_{{\mathcal{X}}^{\prime}}+\Delta_{{\mathcal{X}}^{\prime}}),\frac{a}{d_{1}d_{2}}\eta^{\prime})}={\mathcal{F}}_{({\mathcal{X}}^{\rm ac},\Delta_{{\mathcal{X}}^{\rm ac}},-(K_{{\mathcal{X}}^{\rm ac}}+\Delta_{{\mathcal{X}}^{\rm ac}}),\frac{a}{d_{1}}\eta^{\rm ac})}.

Let $p:\hat{{\mathcal{X}}}\to({\mathcal{X}}^{\prime},\Delta_{{\mathcal{X}}^{\prime}})$ and $q:\hat{{\mathcal{X}}}\to({\mathcal{X}}^{\rm s},\Delta_{{\mathcal{X}}^{\rm s}})$ be a common log resolution, and $E=-q^{*}(K_{{\mathcal{X}}^{\prime}}+\Delta_{{\mathcal{X}}^{\prime}})+p^{*}(K_{{\mathcal{X}}^{\rm s}}+\Delta_{{\mathcal{X}}^{\rm s}})=\sum_{i}e_{i}E_{i}$ with $e_{1}\leq\cdots\leq e_{l}$ . We denote by ${\mathcal{L}}_{\lambda}=-q^{*}(K_{{\mathcal{X}}^{\prime}}+\Delta_{{\mathcal{X}}^{\prime}})+\lambda E$ and ${\mathcal{F}}_{\lambda}={\mathcal{F}}_{({\mathcal{X}}^{\prime},\Delta_{{\mathcal{X}}^{\prime}};{\mathcal{L}}^{\prime}_{\lambda},\eta^{\prime})}$ . Then

	$\displaystyle\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}d_{2}}{\mathcal{F}}_{\lambda}\Big{)}$	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\Big{(}\frac{a}{d_{1}d_{2}}{\rm max}_{i}\big{(}\lambda(e_{1}-e_{i})+A_{X,\Delta}(v_{i})-G_{v_{i}}(y)\big{)}\Big{)}{\rm d}y\Big{)},$
	$\displaystyle\frac{{\rm d}}{{\rm d}\lambda}\mathbf{H}^{g}\Big{(}\frac{a}{d_{1}d_{2}}{\mathcal{F}}_{\lambda}\Big{)}$	$\displaystyle=$	$\displaystyle\frac{a}{d_{1}d_{2}}\frac{\int_{\mathbf{O}}(e_{1}-e_{i(y)})\cdot g^{\prime}\circ f(\lambda,y){\rm d}y}{\int_{\mathbf{O}}g\circ f(\lambda,y){\rm d}y}\leq 0.$

We conclude that

\displaystyle\mathbf{H}^{g}\Big{(}{\mathcal{X}}^{\rm s},\Delta_{{\mathcal{X}}^{\rm s}},-(K_{{\mathcal{X}}^{\rm s}}+\Delta_{{\mathcal{X}}^{\rm s}}),\frac{a}{d_{1}d_{2}}\eta^{\rm s}\Big{)}\leq\mathbf{H}^{g}\Big{(}{\mathcal{X}}^{\prime},\Delta_{{\mathcal{X}}^{\prime}},-(K_{{\mathcal{X}}^{\prime}}+\Delta_{{\mathcal{X}}^{\prime}}),\frac{a}{d_{1}d_{2}}\eta^{\prime}\Big{)}.

∎

Remark 4.7.

If $(X,\Delta)$ admits a connected reductive group ${\mathbb{G}}$ -action, and $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}},a\eta)$ is a ${\mathbb{G}}$ -equivariant normal TC of $(X,\Delta)$ , then the special TC $({\mathcal{X}}^{s},\Delta_{{\mathcal{X}}^{s}};{\mathcal{L}}^{s},a^{s}\eta^{s})$ obtained above can also be ${\mathbb{G}}$ -equivariant as explained in [Li22, Theorem A.1].

Recall that a divisorial valuation $v$ over $(X,\Delta)$ is called special if there exists a ${\mathbb{Q}}$ -complement of $(X,\Delta)$ such that $v$ is the unique lc place of $(X,\Delta+\Gamma)$ . By the one-to-one correspondence of special test configurations and special divisorial valuations [Xu24, Theorem 4.27], we have the following corollary, which is a strengthening of Theorem 3.8 in the log Fano case.

Corollary 4.8.

For any log Fano pair $(X,\Delta)$ , we have

\displaystyle h^{g}(X,\Delta)\,\,\,=\,\,\,{\rm inf}_{v}\,\mathbf{H}^{g}({\mathcal{F}}_{v})\,\,\,=\,\,\,{\rm inf}_{v}\,\tilde{\beta}^{g}(v),

where $v$ runs over all the special divisorial valuations over $X$ .

The second equality follows easily from Remark 3.7.

4.3. Existence of $\mathbf{H}^{g}$ -minimizer

Theorem 4.9.

There exists a quasi-monomial valuation $v_{0}$ such that

\displaystyle h^{g}(X,\Delta)\,\,\,=\,\,\,\mathbf{H}^{g}({\mathcal{F}}_{v_{0}})\,\,\,=\,\,\,\tilde{\beta}^{g}(v_{0}).

Proof.

The proof is verbatim to [HL24, Theorem 4.9] with $h(X,\Delta)$ and $\tilde{\beta}$ replaced by $h^{g}(X,\Delta)$ and $\tilde{\beta}^{g}$ respectively. We shortly recall the argument. By [BLX22, Theorem A.2] (a variant of boundedness of complements [Bir19]), there exists an integer $N$ depending only on $\dim X$ and the coefficients of $\Delta$ , such that every ${\mathbb{Q}}$ -complement of $(X,\Delta)$ is a $N$ -complement.

Recall $L=-(K_{X}+\Delta)$ and $R_{m}=H^{0}(X,mL)$ . Let $W={\mathbb{P}}(R_{N})$ and $D$ be the universal ${\mathbb{Q}}$ -divisor on $X\times W$ parametrizing divisors in $\frac{1}{N}|NL|$ . By lower semicontinuity of lct, the subset $Z=\{w\in W:{\rm lct}(X,\Delta+D_{w})=1\}\subseteq W$ is locally closed. For any $z\in Z$ , we denote by

(17)

\displaystyle b_{z}:=\mathop{{\rm inf}}_{v\in{\rm LC}(X,\Delta+D_{z})}\tilde{\beta}^{g}(v).

Choose a log resolution $(Y_{z},E_{z})\to(X,\Delta+D_{z})$ . Then ${\rm LC}(X,\Delta+D_{z})\subseteq{\rm QM}(Y,E)$ . Hence the infimum in (17) is a minimum by Theorem 3.5, that is, $b_{z}=\tilde{\beta}^{g}(v_{z})$ for some $v_{z}\in{\rm LC}(X,\Delta+D_{z})$ .

Since $(X_{Z},\Delta_{Z}+D_{Z}):=(X\times Z,\Delta\times Z+D|_{X\times Z})\to Z$ is a ${\mathbb{Q}}$ -Gorenstein family of pairs, we can divide $Z$ into a disjoint union of finitely many locally closed subsets $Z=\sqcup_{j}Z_{j}$ such that, for each $j$ , $Z_{j}$ is smooth, and there exists an étale cover $Z_{j}^{\prime}\to Z_{j}$ such that the base change $(X_{Z_{j}^{\prime}},\Delta_{Z_{j}^{\prime}}+D_{Z_{j}^{\prime}})$ admits a fiberwise log resolution $(Y_{Z_{j}^{\prime}},E_{Z_{j}^{\prime}})$ over $Z_{j}^{\prime}$ . For any prime divisor $F\in{\rm QM}(Y_{Z_{j}^{\prime}},E_{Z_{j}^{\prime}})$ , by the proof of [BLX22, Theorem 4.2] (using invariance of plurigenera [HMX13]), we see that ${\rm DH}_{F_{z}}$ is constant for $z\in Z_{j}^{\prime}$ . Hence for any $v\in{\rm QM}(Y_{Z_{j}^{\prime}},E_{Z_{j}^{\prime}})$ , the DH measure ${\rm DH}_{v_{z}}$ is constant for $z\in Z_{j}^{\prime}$ . On the other hand, $A_{X,\Delta}(v_{z})$ is constant for $z\in Z_{j}^{\prime}$ since $(Y_{Z_{j}^{\prime}},E_{Z_{j}^{\prime}})$ is snc over $Z_{j}^{\prime}$ . We conclude that $b_{z}$ is constant for $z\in Z_{j}^{\prime}$ , and we denote this number by $b_{j}$ .

Finally, by Corollary 4.8 and by our choice of $N$ and $Z$ , we have $h^{g}(X,\Delta)={\rm inf}_{z\in Z}b_{z}={\rm min}_{j}b_{j}$ . Let $j_{0}$ be a minimizer. Then for any $z\in Z_{j_{0}}^{\prime}$ , the minimizer $v_{z}$ of $b_{z}$ in (17) is the desired quasi-monomial valuation minimizing $h^{g}(X,\Delta)$ . ∎

Theorem 4.10.

If $(X,\Delta)$ admits a connected reductive group ${\mathbb{G}}$ -action, then the $\mathbf{H}^{g}$ -minimizer $v_{0}$ is ${\mathbb{G}}$ -invariant.

Proof.

This follows from the similar argument of [Xu24, Theorem 4.63 (i)]. We use the same notions as in the above proof. By Remark 4.7 and Corollary 4.8, we see that $h^{g}(X,\Delta)$ is approximated by a series of ${\mathbb{G}}$ -invariant special divisorial valuations $E_{m}$ , which are lc places of $N$ -complements. Hence $E_{m}$ is an lc place of $(X,\Delta+{\rm Bs}|M_{m}|^{\frac{1}{N}})$ , where

\displaystyle M_{m}={\mathcal{F}}_{E_{m}}^{NA_{X,\Delta}(E)}R_{N}\subseteq R_{N},

is a ${\mathbb{G}}$ -invarant sublinear series. Let $W$ be the subvariety of $\cup_{i}{\rm Gr}(i,R_{N})$ parametrizing ${\mathbb{G}}$ -invariant sublinear series of $R_{N}$ , and $M\to W$ be the corresponding universal family. Also by lower semicontinuity of lct, we have locally closed subset $Z=\{w\in W:{\rm lct}(X,\Delta+{\rm Bs}|M_{w}|^{\frac{1}{N}})=1\}\subseteq W$ . For any $z\in Z$ , we define

(18)

\displaystyle b_{z}:=\mathop{{\rm inf}}_{v\in{\rm LC}^{\mathbb{G}}(X,\Delta+{\rm Bs}|M_{w}|^{\frac{1}{N}})}\tilde{\beta}^{g}(v),

where ${\rm LC}^{\mathbb{G}}(X,\Delta+{\rm Bs}|M_{w}|^{\frac{1}{N}})\subseteq{\rm LC}(X,\Delta+{\rm Bs}|M_{w}|^{\frac{1}{N}})$ consists of ${\mathbb{G}}$ -invariant valuations. Also by Theorem 3.5, we have $b_{z}=\tilde{\beta}^{g}(v_{z})$ for some $v_{z}\in{\rm LC}^{\mathbb{G}}(X,\Delta+{\rm Bs}|M_{w}|^{\frac{1}{N}})$ . Now the same argument of the last two paragraph of the above proof shows that $h^{g}(X,\Delta)=b_{z}$ for some $z\in Z$ , which is minimized by the ${\mathbb{G}}$ -invariant quasi-monomial valuation $v_{z}$ . ∎

4.4. Finite generation and weighted K-stability

Theorem 4.11.

The minimizer $v_{0}$ of $\mathbf{H}^{g}$ is special.

Proof.

By Lemma 3.11, $v_{0}$ is a minimizer of $\delta^{g^{\prime},v_{0}}(X,\Delta)=1$ . Hence it is a special valuation by [BLXZ23, Theorem 5.4]. ∎

By definition of special valuations Theorem 2.12, we see that the $\mathbf{H}^{g}$ -minimizer $v_{0}$ induces a multistep special degeneration $({\mathcal{X}},\Delta_{\mathcal{X}},\xi_{0})$ of $(X,\Delta)$ with klt central fiber. We call $({\mathcal{X}},\Delta_{\mathcal{X}},\xi_{0})$ the $g$ -optimal degeneration of $(X,\Delta)$ . Next we study this degeneration of $(X,\Delta)$ . We first recall some notions in the weighted K-stability theory.

Assume that $(X,\Delta)$ admits a torus ${\mathbb{T}}={\mathbb{G}}_{m}^{r}$ -action. Then the anti-canonical ring $R_{\bullet}=R(X,\Delta)=\oplus_{m\in l_{0}{\mathbb{N}}}R_{m}$ admits a canonical weight decomposition $R_{m}=\oplus_{\alpha\in M}R_{m,\alpha}$ , where $M={\rm Hom}({\mathbb{T}},{\mathbb{G}}_{m})\cong{\mathbb{Z}}^{r}$ is the weight lattice. Let $N=M^{\vee}$ be the coweight lattice. A filtration ${\mathcal{F}}$ is called ${\mathbb{T}}$ -invariant if ${\mathcal{F}}^{\lambda}R_{m}=\oplus_{\alpha}{\mathcal{F}}^{\lambda}R_{m,\alpha}$ .

For any $\xi\in N_{\mathbb{R}}$ and ${\mathbb{T}}$ -invariant filtration ${\mathcal{F}}$ , the $\xi$ -twist of ${\mathcal{F}}$ is defined by

\displaystyle{\mathcal{F}}_{\xi}^{\lambda}R_{m}=\oplus_{\alpha\in M}({\mathcal{F}}_{\xi}^{\lambda}R_{m})_{\alpha},\quad({\mathcal{F}}_{\xi}^{\lambda}R_{m})_{\alpha}:={\mathcal{F}}^{\lambda-\langle\alpha,\xi\rangle}R_{m,\alpha}.

We will simple denote the filtration ${\mathcal{F}}_{{\rm triv},\xi}^{\lambda}R_{m}=\oplus_{\langle\alpha,\xi\rangle\geq\lambda}R_{m,\alpha}$ by $\xi$ , then

\displaystyle\mu(\xi)=\mu({\mathcal{F}}_{{\rm triv},\xi})=\mu({\mathcal{F}}_{\rm triv})=0,

by the following lemma.

Lemma 4.12.

[Xu24, Lemma 6.24] For any ${\mathbb{T}}$ -invariant linearly bounded filtration ${\mathcal{F}}$ on $R$ , and any $\xi\in N_{\mathbb{R}}$ , we have $\mu({\mathcal{F}}_{\xi})=\mu({\mathcal{F}})$ .

Recall that $g^{\prime}:{\mathbb{R}}\to{\mathbb{R}}_{>0}$ is the first order derivative of $g$ . Then for any $\xi\in N_{\mathbb{R}}$ , we may define the $(g^{\prime},\xi)$ -weighted Ding invariants of $(X,\Delta)$ .

Definition 4.13.

For any ${\mathbb{T}}$ -invariant linearly bounded filtration ${\mathcal{F}}$ on $R$ , we define the $(g^{\prime},\xi)$ -weighted Ding invariant by

\displaystyle\mathbf{D}^{g^{\prime},\xi}({\mathcal{F}})\,\,\,=\,\,\,\mathbf{D}^{g^{\prime},\xi}_{X,\Delta}({\mathcal{F}})\,\,\,:=\,\,\,\mu_{X,\Delta}({\mathcal{F}})-S^{g^{\prime},\xi}({\mathcal{F}}).

The log Fano pair $(X,\Delta)$ is called ${\mathbb{T}}$ -equivariantly $(g^{\prime},\xi)$ -weighted Ding-semistable if $\mathbf{D}^{g^{\prime},\xi}({\mathcal{F}})\geq 0$ for any ${\mathbb{T}}$ -invariant linearly bounded filtration ${\mathcal{F}}$ on $R$ . If moreover, for any ${\mathbb{T}}$ -equivariant normal TC $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}})$ of $(X,\Delta)$ , $\mathbf{D}^{g^{\prime},\xi}({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}})=0$ implies that $({\mathcal{X}},\Delta_{\mathcal{X}};{\mathcal{L}})$ is a product TC, then $(X,\Delta)$ is called ${\mathbb{T}}$ -equivariantly $(g^{\prime},\xi)$ -weighted Ding-polystable.

The log Fano triple $(X,\Delta,\xi)$ is called $g^{\prime}$ -weighted K-(semi/poly)stable if $(X,\Delta)$ is ${\mathbb{T}}$ -equivariantly $(g^{\prime},\xi)$ -weighted Ding-(semi/poly)stable for some ${\mathbb{T}}$ -action. By [BLXZ23, Remark 5.10], the definition is independent of the choice of the ${\mathbb{T}}$ -action.

Theorem 4.14.

Proof.

We follow the proof of [HL24, Theorem 5.3]. First assume that $v_{0}$ minimizes $\mathbf{H}^{g}$ . Denote by $(W,\Delta_{W},\xi)=({\mathcal{X}}_{0},\Delta_{{\mathcal{X}},0},\xi_{0})$ and assume that it is $g^{\prime}$ -weighted K-unstable. Then by a variant of [LX14], there exists a special TC $({\mathcal{W}},\Delta_{{\mathcal{W}}},\eta)$ such that

\displaystyle\mathbf{D}^{g^{\prime},\xi}_{W,\Delta_{W}}({\mathcal{W}},\Delta_{{\mathcal{W}}},\eta)<0.

We denote by $(Y,\Delta_{Y},\eta)=({\mathcal{W}}_{0},\Delta_{{\mathcal{W}},0},\eta)$ , then

\displaystyle\mathbf{D}^{g^{\prime},\xi}_{Y,\Delta_{Y}}(\eta)=\mathbf{D}^{g^{\prime},\xi}_{W,\Delta_{W}}({\mathcal{W}},\Delta_{{\mathcal{W}}},\eta)<0.

Then we can construct a series of valuations $\{v_{\varepsilon}\}_{\varepsilon\in{\mathbb{R}}}$ as [LX18] inducing special degenerations of $(X,\Delta)$ with central fibers $(Y,\Delta_{Y},\xi+\varepsilon\eta)$ . Then $\mathbf{H}^{g}_{X,\Delta}(v_{\varepsilon})=\mathbf{H}^{g}_{Y,\Delta_{Y}}(\xi+\varepsilon\eta)$ . Since $\mu(\xi^{\prime})=0$ for any holomorphic vector field $\xi^{\prime}$ on $Y$ , we have

\displaystyle\mathbf{H}^{g}_{Y,\Delta_{Y}}(\xi+\varepsilon\eta)={\rm log}\big{(}\int_{\mathbf{P}}g(-\langle\alpha,\xi+\varepsilon\eta\rangle){\rm DH}_{\mathbf{P}}({\rm d}\alpha)\big{)}.

Hence

	$\displaystyle\frac{{\rm d}}{{\rm d}\varepsilon}\|_{\varepsilon=0}\,\,\mathbf{H}^{g}_{X,\Delta}(v_{\varepsilon})$	$\displaystyle=$	$\displaystyle\frac{\int_{\mathbf{P}}(-\langle\alpha,\eta\rangle)\cdot g^{\prime}(-\langle\alpha,\xi\rangle){\rm DH}_{\mathbf{P}}({\rm d}\alpha)}{\int_{\mathbf{P}}g(-\langle\alpha,\xi\rangle){\rm DH}_{\mathbf{P}}({\rm d}\alpha)}$
		$\displaystyle=$	$\displaystyle\frac{1}{\mathbf{v}^{g}}\int_{\mathbf{P}}(-\langle\alpha,\eta\rangle)\cdot g^{\prime}(-\langle\alpha,\xi\rangle){\rm DH}_{\mathbf{P}}({\rm d}\alpha)\,\,\,=\,\,\,\frac{\mathbf{v}^{g^{\prime}}}{\mathbf{v}^{g}}\cdot\mathbf{D}^{g^{\prime},\xi}_{Y,\Delta_{Y}}(\eta)\,\,\,<\,\,\,0,$

which contradicts that $v_{0}$ minimizes $\mathbf{H}^{g}_{X,\Delta}$ .

Conversely, assume that $(W,\Delta_{W},\xi)$ is $g^{\prime}$ -weighted K-semistable. Then for any linearly bounded filtration ${\mathcal{F}}$ on $R$ . We define its initial term degeneration ${\mathcal{F}}^{\prime}$ on ${\rm gr}_{v_{0}}R$ by

\displaystyle{\mathcal{F}}^{\prime\lambda}{\rm gr}_{v_{0}}R_{m}:=\langle\bar{s}_{i}:s_{i}\in{\mathcal{F}}^{\lambda}R_{m}\rangle,

where $\{s_{i}\}$ is a basis of $R_{m}$ which is compatible with both $v_{0}$ and ${\mathcal{F}}$ . Hence ${\rm DH}_{\mathcal{F}}={\rm DH}_{{\mathcal{F}}^{\prime}}$ . By lower semicontinuity of lct, we have $\mu_{X,\Delta}({\mathcal{F}})\geq\mu_{W,\Delta_{W}}({\mathcal{F}}^{\prime}).$ Hence

(19)

\displaystyle\mathbf{H}^{g}_{X,\Delta}({\mathcal{F}})\geq\mathbf{H}^{g}_{W,\Delta_{W}}({\mathcal{F}}^{\prime})\geq\mathbf{H}^{g}_{W,\Delta_{W}}(\xi)=\mathbf{H}^{g}_{X,\Delta}(v_{0}),

where the second inequality follows from the $g^{\prime}$ -weighted K-semistability of $(W,\Delta_{W},\xi)$ . Indeed, since $\mathbf{H}^{g}$ is strictly convex along geodesics, it suffices to show that the derivative of $\mathbf{H}^{g}_{X,\Delta}({\mathcal{F}}_{t})$ at $t=0$ is non-negative, where ${\mathcal{F}}_{t}$ is the geodesic connecting ${\mathcal{F}}_{0}={\mathcal{F}}_{{\rm wt}_{\xi}}$ and ${\mathcal{F}}_{1}={\mathcal{F}}^{\prime}$ . Note that

$\displaystyle{\mathcal{F}}^{\lambda}_{t}R_{m}$	$\displaystyle=$	$\displaystyle\sum_{(1-t)\mu+t\nu\geq\lambda}{\mathcal{F}}_{0}^{\mu}R_{m}\cap{\mathcal{F}}_{1}^{\nu}R_{m}$
	$\displaystyle=$	$\displaystyle\Big{\{}s\in R_{m}:(1-t){\rm ord}_{{\mathcal{F}}_{0}}(s)+t\,{\rm ord}_{{\mathcal{F}}_{1}}(s)\geq\lambda\Big{\}}$
	$\displaystyle=$	$\displaystyle\bigoplus_{\alpha\in M}\Big{\{}s\in R_{m,\alpha}:(1-t)\langle\alpha,\xi\rangle+t\,{\rm ord}_{{\mathcal{F}}^{\prime}}(s)\geq\lambda\Big{\}}$
	$\displaystyle=$	$\displaystyle\bigoplus_{\alpha\in M}\Big{\{}s\in R_{m,\alpha}:t\Big{(}{\rm ord}_{{\mathcal{F}}^{\prime}}(s)+\langle\alpha,\frac{1-t}{t}\xi\rangle\Big{)}\geq\lambda\Big{\}}$
	$\displaystyle=$	$\displaystyle\Big{\{}s\in R_{m}:{\rm ord}_{t{\mathcal{F}}^{\prime}_{\frac{1-t}{t}\xi}}(s)\geq\lambda\Big{\}}\,\,\,=\,\,\,(t{\mathcal{F}}^{\prime}_{\frac{1-t}{t}\xi})^{\lambda}R_{m}.$

Hence ${\mathcal{F}}_{t}=t{\mathcal{F}}^{\prime}_{\frac{1-t}{t}\xi}$ . Recall that $\mu({\mathcal{F}})$ is invariant under $\xi$ -twist, and linear under rescaling. Hence $\mu({\mathcal{F}}_{t})=t\mu({\mathcal{F}}^{\prime})$ . We also have $G_{{\mathcal{F}}}(y)=(1-t)\langle\alpha,\xi\rangle+tG_{{\mathcal{F}}^{\prime}}(y)$ where $y=(\alpha,y^{\prime})$ . Hence

$\displaystyle\mathbf{H}^{g}({\mathcal{F}}_{t})$	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g(\mu({\mathcal{F}}_{t})-G_{{\mathcal{F}}_{t}}(y)){\rm d}y\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\big{(}-\langle\alpha,\xi\rangle+t(\mu({\mathcal{F}}^{\prime})-G_{{\mathcal{F}}^{\prime}}(y)+\langle\alpha,\xi\rangle)\big{)}{\rm d}y\Big{)}$
	$\displaystyle=$	$\displaystyle{\rm log}\Big{(}\int_{\mathbf{O}}g\big{(}-\langle\alpha,\xi\rangle+t(\mu({\mathcal{F}}^{\prime}_{\xi})-G_{{\mathcal{F}}^{\prime}_{\xi}}(y))\big{)}{\rm d}y\Big{)},$
$\displaystyle\frac{{\rm d}}{{\rm d}t}\|_{t=0}\,\mathbf{H}^{g}({\mathcal{F}}_{t})$	$\displaystyle=$	$\displaystyle\frac{\int_{\mathbf{O}}g^{\prime}\big{(}-\langle\alpha,\xi\rangle\big{)}\cdot\big{(}\mu({\mathcal{F}}^{\prime}_{\xi})-G_{{\mathcal{F}}^{\prime}_{\xi}}(y)\big{)}{\rm d}y}{\int_{\mathbf{O}}g(-\langle\alpha,\xi\rangle){\rm d}y}$
	$\displaystyle=$	$\displaystyle\frac{\mathbf{v}^{g^{\prime}}}{\mathbf{v}^{g}}\mathbf{D}^{g^{\prime},\xi}_{W,\Delta_{W}}({\mathcal{F}}^{\prime}_{\xi})\,\,\,\geq\,\,\,0,$

where $y=(\alpha,y^{\prime})$ . Hence the second inequality in (19) holds and the proof is finished. ∎

Remark 4.15.

If $(X,\Delta)$ admits a connected reductive group ${\mathbb{G}}$ -action, then by Theorem 4.10, the $\mathbf{H}^{g}$ -minimizer $v_{0}$ is ${\mathbb{G}}$ -invariant, hence ${\rm gr}_{v_{0}}R$ admitting the ${\mathbb{G}}$ -action and inducing a ${\mathbb{G}}$ -equivariant multistep special degeneration. In other word, the $g$ -optimal degeneration of $(X,\Delta)$ is ${\mathbb{G}}$ -equivariant.

As a corollary, we have the following characterization of $g$ -optimal degeneration.

Corollary 4.16.

Let $(X,\Delta)$ be a log Fano pair admitting a torus ${\mathbb{G}}_{m}^{r}$ -action, and $\xi_{0}\in N_{\mathbb{R}}$ . Then the filtration ${\mathcal{F}}_{{\rm triv},\xi_{0}}$ minimizes $\mathbf{H}^{g}$ if and only if $(X,\Delta,\xi_{0})$ is $g^{\prime}$ -weighted K-semistable.

Now we can finish the proof of the main theorem in this paper.

Proof of Theorem 1.1.

The existence and uniqueness of the minimizer $v_{0}$ of $\mathbf{H}^{g}$ follows from Theorem 4.9 and 3.3 respectively. The valuation is special by Theorem 4.11. Moreover, the central fiber $({\mathcal{X}}_{0},\Delta_{{\mathcal{X}},0},\xi_{0})$ of the multistep special degeneration induced by $v_{0}$ is $g^{\prime}$ -weighted K-semistable by Theorem 4.14. Finally, $({\mathcal{X}}_{0},\Delta_{{\mathcal{X}},0},\xi_{0})$ has a unique $g^{\prime}$ -weighted K-polystable degeneration $(Y,\Delta_{Y},\xi_{0})$ by [HL24, Theorem 1.3], and $(Y,\Delta_{Y},\xi_{0})$ admits a $g^{\prime}$ -soliton by [BLXZ23, Theorem 1.3] and [HL23, Theorem 1.7]. ∎

5. Examples

In this section, we give some examples that Question 1.4 has positive answer.

5.1. Weighted K-stable Fano varieties for any weight function

Let $(X,\Delta)$ be a log Fano pair with a ${\mathbb{T}}={\mathbb{G}}_{m}^{r}$ -action, $M={\rm Hom}({\mathbb{T}},{\mathbb{G}}_{m}),N=M^{\vee}$ be the weight, coweight lattices respectively. Let $\mathbf{P}\subseteq M_{\mathbb{R}}$ be the moment polytope of the ${\mathbb{T}}$ -action and ${\rm DH}_{\mathbf{P}}$ be the DH measure of the ${\mathbb{T}}$ -action on $\mathbf{P}$ (see for example [MW23, Section 2.5 and 3.3]). A continuous function $g_{0}:\mathbf{P}\to{\mathbb{R}}_{>0}$ is called a weight function if

\displaystyle\int_{\mathbf{P}}\alpha_{i}\cdot g_{0}(\alpha){\rm DH}_{\mathbf{P}}({\rm d}\alpha)=0,

for any $1\leq i\leq r$ . Similar to Definition 4.13, one can define the $g_{0}$ -weighted K-stability and Ding-stability of the log Fano ${\mathbb{T}}$ -pair $(X,\Delta)$ . In the setting of $g$ -optimal degenerations, we will choose

\displaystyle g_{0}(\alpha)=g^{\prime}(-\langle\alpha,\xi_{0}\rangle),

where $\xi_{0}$ is the minimizer of $\mathbf{H}^{g}$ on $N_{\mathbb{R}}$ . We have the following easy consequence of Corollary 4.16, which gives some trivial examples answering Question 1.4 positively.

Corollary 5.1.

Assume that $(X,\Delta)$ is $g_{0}$ -weighted K-polystable for any weight function $g_{0}$ . Then $(X,\Delta)$ is the $g$ -optimal degeneration of itself for any function $g$ satisfying (1).

Let $(X,\Delta)$ be a toric log Fano pair. Then $(X,\Delta)$ is $g_{0}$ -weighted K-polystable for any weight function $g_{0}$ . Indeed, any ${\mathbb{T}}$ -invariant filtration ${\mathcal{F}}$ is equivalent to ${\mathcal{F}}_{{\rm triv},\xi}$ for some $\xi\in N_{\mathbb{R}}$ . Hence

\displaystyle\mathbf{D}^{g_{0}}({\mathcal{F}})=\frac{1}{\mathbf{v}^{g_{0}}}\int_{\mathbf{P}}(-\langle\alpha,\xi\rangle)\cdot g_{0}(\alpha){\rm DH}_{\mathbf{P}}({\rm d}\alpha)=0.

In particular, the $g$ -optimal degenerations of $(X,\Delta)$ are always itself.

The following non-trivial examples follow from [Wan24, Example 5.5].

Theorem 5.2.

Any Fano threefold $X$ in the families №2.28 and №3.14 of Mori-Mukai’s list is $g_{0}$ -weighted K-polystable for any weight function $g_{0}$ . In particular, the $g$ -optimal degenerations of $X$ are always $X$ itself for any function $g$ satisfying (1).

5.2. Non-trivial $g$ -optimal degenerations

The Fano threefolds in the family №2.23 of Mori-Mukai’s list are K-unstable and admit discrete automorphism group [MT22]. Hence they could not be weighted K-semistable and admit no $g_{0}$ -soliton [HL23, (1.3)] for any weight function $g_{0}$ . Their optimal degenerations were determined by [MW24]. It’s natural to ask what are their $g$ -optimal degenerations for other functions $g$ satisfying (1).

Recall that any Fano threefold $X$ in №2.23 is obtained by blowing up the quadric threefold $Q$ along the complete intersection $C$ of a hyperplane section $H\in|{\mathcal{O}}_{Q}(1)|$ and a quadric section $Q^{\prime}\in|{\mathcal{O}}_{Q}(2)|$ . The family №2.23 is divided into two subfamilies by the smoothness of $H$ ,

•

$X\in$ №2.23(a), if $H\cong{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ ,
•

$X\in$ №2.23(b), if $H\cong{\mathbb{P}}(1,1,2)$ .

The optimal degeneration $X_{0}$ of $X$ in №2.23(a) is induced by the divisorial valuation ${\rm ord}_{H}$ by [MW24, Corollary 1.4]. Hence $X_{0}={\rm Bl}_{C}Q_{0}$ where $Q_{0}\subseteq{\mathbb{P}}^{4}$ is the cone over a smooth quadric surface $H\subseteq{\mathbb{P}}^{3}$ , and $C\subseteq H\cong{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ is a biconic curve (i.e. $C\in|{\mathcal{O}}_{{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}}(2,2)|$ ).

Theorem 5.3.

For any Fano threefold $X$ in family №2.23(a), the $g$ -optimal degenerations are always $X_{0}$ for any function $g$ satisfying (1).

Proof.

We need to prove that $X_{0}$ is the $g$ -optimal degeneration of $X$ for any function $g$ satisfying (1). This is equivalent to $\mathbf{H}^{g}_{X}$ being minimized by $a\cdot{\rm ord}_{H}$ for some $a\in{\mathbb{R}}_{>0}$ , hence is equivalent to $(X_{0},a\cdot\xi)$ being $g^{\prime}$ -weighted K-polystable for some $a\in{\mathbb{R}}_{>0}$ , where $\xi\in N\cong{\mathbb{Z}}$ whose filtration is a shift of ${\mathcal{F}}_{{\rm ord}_{H}}$ . We conclude by [Wan24, Example 5.7], which says that $X_{0}$ is $g_{0}$ -weighted K-polystable for any weight function $g_{0}$ . ∎

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Generalized optimal degenerations of Fano varieties

Abstract.

1. Introduction

Theorem 1.1 (Generalized Tian conjecture).

Remark 1.2.

Theorem 1.3 (Theorem 4.14).

Question 1.4.

Theorem 1.5.

2. Preliminaries

2.1. Filtrations, concave transforms and DH measures

Definition 2.1.

Definition 2.2.

Lemma 2.3.

Definition 2.4.

2.2. Log canonical slopes and 𝐋\mathbf{L}-functionals

Definition 2.5.

Lemma 2.6.

Lemma 2.7.

Lemma 2.8.

Proof.

Remark 2.9.

Definition 2.10.

Lemma 2.11.

2.3. Multistep special degenerations and higher rank finite generation

Theorem 2.12.

Theorem 2.13.

Proof.

3. Generalized 𝐇\mathbf{H}-invariants

Definition 3.1 (𝐇g\mathbf{H}^{g}-invariants).

Remark 3.2.

3.1. Convexity

Theorem 3.3.

Proof.

Corollary 3.4.

Proof.

Theorem 3.5.

Proof.

3.2. Approximation by valuations

Definition 3.6 (β~g\tilde{\beta}^{g}-invariants).

Remark 3.7.

Theorem 3.8.

Proof.

Remark 3.9.

3.3. Weighted delta invariants

Definition 3.10.

Lemma 3.11.

Proof.

4. Existence of 𝐇g\mathbf{H}^{g}-minimizers and finite generation

4.1. Approximation by test configurations

Lemma 4.1.

Corollary 4.2.

Theorem 4.3.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

4.2. Approximation by special test configurations

Theorem 4.6.

Proof.

Remark 4.7.

Corollary 4.8.

4.3. Existence of 𝐇g\mathbf{H}^{g}-minimizer

Theorem 4.9.

Proof.

Theorem 4.10.

Proof.

4.4. Finite generation and weighted K-stability

Theorem 4.11.

Proof.

Lemma 4.12.

Definition 4.13.

Theorem 4.14.

Proof.

Remark 4.15.

Corollary 4.16.

Proof of Theorem 1.1.

5. Examples

5.1. Weighted K-stable Fano varieties for any weight function

Corollary 5.1.

Theorem 5.2.

2.2. Log canonical slopes and $\mathbf{L}$ -functionals

3. Generalized $\mathbf{H}$ -invariants

Definition 3.1 ( $\mathbf{H}^{g}$ -invariants).

Definition 3.6 ( $\tilde{\beta}^{g}$ -invariants).

4. Existence of $\mathbf{H}^{g}$ -minimizers and finite generation

4.3. Existence of $\mathbf{H}^{g}$ -minimizer

5.2. Non-trivial $g$ -optimal degenerations