memo

# Stability theory over toroidal or Novikov-type base and canonical
modifications

Yuji Odaka

###### Abstract.

We set up a generalization of ubiquitous one parameter families in algebraic
geometry and its use for stability theories ([Mum65, HL14, AHLH23]) to
families over toric varieties, Novikov type rings, or its complex analytic
analogues. The language allows us to reformulate degenerations of “irrational”
direction in various literatures as canonical objects in a unified manner.

Compatibly, we generalize the stable reduction type theorem for
$\Theta$-stratification in [AHLH23] of Langton type to our higher rank setup.

##  1\. Introduction

In the classical geometric invariant theory [Mum65], the so-called Hilbert-
Mumford criterion for stability of an algebro-geometric object, say ${\bf X}$,
with respect to an action of a reductive algebraic group $G$, shows that it is
enough to analyze degenerations of ${\bf X}$ along actions of $\mathbb{G}_{m}$
which is inside $G$ ($1$-parameter subgroups). The resulted families are
isotrivial degenerating families of ${\bf X}$ over $\mathbb{A}^{1}$. In the
case when ${\bf X}$ is a polarized projective variety, they are characterized
as test configurations by Donaldson [Don02] (cf., also [Don05]).

More precisely speaking, for the Hilbert-Mumford criterion, one associates GIT
weight to each $1$-parameter subgroup and their positivity is equivalent to
the stability. On the other hand, it is also well-known that we can
alternatively think of maximal tori $T$ of $G$, instead of $1$-parameter
subgroups, and encode all the data of GIT weights inside $T$ in the weight
polytope. The corresponding natural extension of test configurations or the
above type isotrivial degenerations, replacing $\mathbb{G}_{m}$ by a maximal
torus, is isotrivial family over $T$ and its partial equivariant
compactification i.e., toric varieties. At the risk of being misunderstood, a
primitive toy model of the perspective in this paper, is such isotrivial
degeneration over toric variety of higher dimension.

In this paper, we introduce more extensive algebraic (partly stack-theoretic)
generalities with such perspective, which leads to introduce e.g., isotrivial
degenerating families over toroidal bases, Novikov type valuation rings of
high rank, and their analytic analogues. In particular, focusing on the
context of stability and moduli theories of varieties, we introduce the notion
of generalized test configurations and their limit versions, and show that it
is useful to treat the following types of degenerations in “irrational
direction” in literatures, a more canonical and unified manners:

  * •

klt singularities to irregular Ricci-flat cones ([DS17, Li18]),

  * •

Kähler-Ricci flow on Fano manifolds converging to Kähler-Ricci solitons
([CSW18, HL20]),

  * •

negative weight deformations to asymptotically conical Calabi-Yau manifolds
([CH22]).

In our other papers we plan to discuss and explore these one by one, but this
paper still focuses on the general framework. In particular, the latter half
of this paper gives some generalization of the $\Theta$-stability theory and
stable reduction theorem ([HL14] and [AHLH23] respectively). Here, recall
$\Theta:=[\mathbb{A}^{1}/\mathbb{G}_{m}]$ over a field $k$ is the quotient
stack whose coarse moduli is just one point which is used as a source of map
to express equivariant isotrivial families over $\mathbb{A}^{1}$. [HL14,
AHLH23] introduces the notion of $\Theta$-strata and $\Theta$-stratification
on (highly non-separated) algebraic stacks, as a stacky reminicsents of the
Hesselink-Kempf-Kirwan-Ness stratification [Kir84, Nes84] and the Harder-
Narasimhan filtration of coherent sheaves [HN74]. We call a natural higher
rank version of it as higher $\Theta$-strata. Also, as an irrational analogue
of DVR, we introduce Novikov type spectra $\overline{\Delta_{\xi}}(R)$
(Definition 2.3). This gives an extension of the valuation ring $R$.

We give a brief outlook of our main theorem but we refer to later Theorem 3.8,
3.15, Corollary 3.10 for the precise statements and generalizations.

######  Theorem 1.1 (Higher $\Theta$-stable reduction).

Take any quotient stack $\mathcal{M}=[M/G]$ where $M$ is an algebraic
$\mathbbm{k}$-scheme and $G$ is a linear algebraic $\mathbbm{k}$-group and let
us set $\Delta=\mathop{\mathrm{Spec}}\nolimits\mathbbm{k}[[t]]$ (resp., a
complex unit disk $\Delta$, if $\mathbbm{k}=\mathbb{C}$).

Consider any $\mathbbm{k}$-morphism $f\colon\Delta\to\mathcal{M}$ from to
$\mathcal{M}$ whose closed point (resp., origin) maps into a closed substack
$\mathcal{Z}^{+}$ corresponding higher $\Theta$-strata for the cone
$\tau\ni\xi$ (see Definition 3.3) while the generic point (resp., general
points) maps outside $\mathcal{Z}^{+}$. We denote the restriction of $f$ to
$\mathop{\mathrm{Spec}}\nolimits(K)$ (resp., $\Delta\setminus\\{0\\}$) as
$f^{o}$.

Then, after a finite extension of $R$ (resp., $\Delta$), there is a canonical
(for each $\xi$) modification of $f$ as follows:

$f_{{\rm new},\xi}\colon\overline{\Delta_{\xi}}(R)\quad(\text{resp., the
universal cover of }\Delta\setminus 0)\to\mathcal{M},$

which still “extends” $f^{o}$ and sends the closed point to a point outside
$Z^{+}$, which still isotrivially degenerates to a point in $Z^{+}$.

We call this replacement of $f$ by $f_{{\rm new},\xi}$ as higher
$\Theta$-stable reduction and call its limit as higher $\Theta$-stable limit.
When $\xi$ is a rational vector, it is nothing but the result of [AHLH23, §6]
(“$\Theta$-stable reduction, “$\Theta$-stable limit”). In general case,
relations between higher $\Theta$-stable reduction (resp., higher
$\Theta$-strata) and the original stable reduction (resp., $\Theta$-stara) are
unclear at the moment. Nevertheless, under the assumption of their close
connection, we could also apply our method to study certain deformation of
$\Theta$-stratification along an algebraic torus and how it affects the
$\Theta$-stable limits and show its essentially construtible behaviour
(Corollary 3.13).

The framework and theorems in this paper are applied in our companion papers
([Od24a, Od24c]).

##  2\. Generalities on multi-parameter families

In this section, we generalize some part of the theories of one parameter
families and degenerations, which appear in the theory of K-stability or its
local analogues for klt singularities (cf., [Li18] and subsequent
developments), and the stacky stability theories in [HL14, AHLH23], to those
over toroidal base of often higher dimension. Some germs of our theory can be
seen in [Spi90, Tei03, DS17, LX18, HL20], for instance. We observe that our
language allows re-explanation in a unified manner of those existing
K-stability theories.

Firstly, we introduce a notion of generalized test configurations as certain
type of multi-parameter isotrivial family over toroidal bases and their
analogues, which unifies various families in the field of K-stability and
Kähler geometry, such as

  1. (1)

test configurations and their affine versions ([Don02, CS18, CS19]) cf.,
Example 2.7,

  2. (2)

negative weight deformation in the sense of [CH22, Definition 1.7] cf.,
Example 2.18 and

  3. (3)

each step of the recent $2$-step degeneration theory (originated in [DS17])

     * •

e.g., the canonical degeneration of klt singularities to its possibly
irregular K-semistable Fano cone via the volume-minimizing valuation of klt
singularities ([Li18]) in Example 2.9,

     * •

$\mathbb{R}$-test configurations in [CSW18, HL20] cf., Example 2.14,

     * •

(the $2$nd steps to the polystable replacements ([DS17, LWX21]) as Examples
2.13),

  4. (4)

(or more generally, degeneration of GIT semistable objects to GIT polystable
objects ([Mum65]) cf., Remark 2.12).

Further, as we will later explain, the notion allows us to treat the above
examples of degenerations (2) and (3) in a (more) canonical manner, to remove
the broad ambiguity of choices of $1$-parameter subgroups, except for those
with parentheses.

###  2.1. Setup of the bases

Since our bases will be often toric geometric, first we set the usual
notations for toric geometry.

######  Notations 1.

We fix a lattice $N\simeq\mathbb{Z}^{r}$ and its rational polyhedral cone
$\tau\subset N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$. The dual
lattice of $N$ is denoted as $M:={\rm Hom}(N,\mathbb{Z})$ and we set
$M_{\mathbb{R}}:=M\otimes\mathbb{R}$. We denote the algebraic torus
$T:=\mathbb{G}_{m}\otimes_{\mathbb{Z}}N$. If we regard $M$ as the character
lattice of $T$, we denote by $\chi(\vec{m})$ the character of $T$ which
corresponds to each $\vec{m}\in M$. An affine toric partial compactification
of $T$ for $\tau$ is denoted by $U_{\tau}(\supset T)$. We write
$\tau^{\vee}:=\\{x\in M\otimes\mathbb{R}\mid\langle
x,\tau\rangle\subset\mathbb{R}_{\geq 0},$ its interior as $\tau^{\vee,o}$,
$\mathcal{S}_{\tau}:=\tau^{\vee}\cap M$, and
$\mathcal{S}_{\tau}^{o}:=\mathcal{S}_{\tau}\cap\tau^{\vee,o}$.

We take $(0\neq)\xi\in\tau$ which is not contained in any proper
$\mathbb{Q}$-linear subspace of
$N_{\mathbb{Q}}:=N\otimes_{\mathbb{Z}}\mathbb{Q}$ i.e., of rational rank $r$.
Henceforth, we simply call such $\xi$ as being irrational. The latter is not
finitely generated if $r>1$. We introduce the monoids

$\displaystyle M_{\xi\geq 0}$ $\displaystyle:=\\{m\in M\mid\langle
m,\xi\rangle\geq 0\\},$ $\displaystyle\Gamma_{\geq 0}$ $\displaystyle:=\langle
M_{\geq 0},\xi\rangle\subset\mathbb{R}.$

Here, $\langle,\rangle$ means the natural pairing which comes from the
definition of $M$. For our later applications to relate with differential
geometry, $\xi$ is often rather a given data, rather than $\tau$ (or even
$N$).

Also, for any algebraic scheme $V$ over $\mathbb{C}$, $V^{\rm an}$ (or
$V(\mathbb{C})^{\rm an}$) refers to the (complex) analytification i.e.,
$V(\mathbb{C})$ with the complex analytic (euclidean) topology.

We prepare the bases of “irrational deformations” in an algebraic manner,
following [Nov81, Nov82, FOOO09, AJ10, AI21] for instance.

######  Definition 2.1 (Novikov-type rings and spectra).

Under the Notations 1, the global N-ring 111N stands for Novikov as these
rings are variants of so-called Novikov ring, in the contexts of Morse theory,
Gromov-Witten theory, symplectic geometry, which seem to have different
versions in the literatures cf., e.g., [Nov81, Nov82, FOOO09, AJ10, AI21].
over a ring $A$ for $\xi\in N_{\mathbb{R}}$ is the semiring $A[\Gamma_{\geq
0}]$ and we call its spectra $\mathop{\mathrm{Spec}}\nolimits A[\Gamma_{\geq
0}]$ as global N-spectra, which we denote as $\Delta_{\xi}^{\rm gl}(A)$ or
simply $\Delta_{\xi}^{\rm gl}$ if the base ring $A$ is obvious from the
context. Then, we define the following.

  1. (1)

Let $I$ be the maximal ideal generated by the monomials corresponding to
$\Gamma_{\geq 0}$ and the formal completion of $A[\Gamma_{\geq 0}]$ with
respect to $I$ is denoted as $A[[\Gamma_{\geq 0}]]$. With its formal variable
$t$, the N-ring over a ring $A$ for $\xi\in N_{\mathbb{R}}$ is its subring
defined as

$A_{\xi}:=\\{\sum_{i=1,2,\cdots}a_{i}t^{r_{i}}\mid a_{i}\in
A,r_{i}\in\Gamma_{\geq 0},r_{1}<r_{2}<\cdots,r_{i}\to\infty(i\to\infty)\\}.$

We call ${\rm Spec}(A_{\xi})$ the N-spectrum and denote it by
$\Delta_{\xi}(A)$. Note that $T$ over ${\rm Spec}(A)$ acts on
$\Delta_{\xi}(A)$. Later, we often consider the case when the base $A$ is a
field $\mathbbm{k}$ and in that case we denote $A_{\xi}$ as
$\mathbbm{k}_{\xi}$. Note that it is clearly a valuation ring of rational rank
$r$. When the base field is obvious, we often abbreviate $\Delta_{\xi}(A)$ as
$\Delta_{\xi}$.

  2. (2)

If $A\subset\mathbb{C}$ (or more generally, if $A$ is associated a valuation)
and a semigroup $\Gamma_{\geq 0}$ is a submonoid of $\mathbb{R}_{\geq 0}$, we
define the convergent N-ring $A[[\Gamma_{\geq 0}]]^{\rm conv}$ as

$\\{f=\sum_{i=1,2,\cdots}a_{i}t^{r_{i}}\in A_{\xi}\mid\exists\epsilon>0\text{
s.t. }f(t)\text{ absolutely converge for any }t\in(0,\epsilon)\\}.$

We write its fraction field as $A((\Gamma_{\geq 0}))^{\rm conv},$ and denote
their spectrum $\mathop{\mathrm{Spec}}\nolimits A[[\Gamma_{\geq 0}]]^{\rm
conv}$ (resp., $\mathop{\mathrm{Spec}}\nolimits A((\Gamma_{\geq 0}))^{\rm
conv}$) as $\Delta_{\Gamma_{\geq 0}}^{\rm conv}(A)$ (resp.,
$\Delta_{\Gamma_{\geq 0}}^{*,{\rm conv}}(A)$) and call them convergent
N-spectrum. We often consider the case when $\Gamma_{\geq 0}$ is of the form
of Notation 1 with some $\xi\in N_{\mathbb{R}}$, and denote them by
$\Delta_{\xi}^{\rm conv}(A)$ (resp., $\Delta_{\xi}^{*,{\rm conv}}(A)$).

We can interpret the above notion in terms of the framework of limit toric
variety in [Od22b].

######  Proposition 2.2.

We fix a base integral domain $A$ and irrational $\xi\in N_{\mathbb{R}}$ as
Notation 1. Consider the set of all rational polyhedral cones $\tau$ of
$N_{\mathbb{R}}$ which includes $\xi$, and associate the obvious partial order
with respect to the inclusion.

Then, the global N-spectra $\mathop{\mathrm{Spec}}\nolimits A[\Gamma_{\geq
0}]$ is canonically isomorphic, as locally ringed spaces, to a limit toric
variety over $A$ in the sense of [Od22b, §2.4, Definition 2.16]222which
(mainly) discussed the case when $A$ is a (valuation-) field $\mathbbm{k}$ but
Definition 2.16 in loc.cit works over any base ring $A$, which is the
projective limit $\varprojlim_{\tau\ni\xi}U_{\tau}$ of the above projective
system (or its sub-system which is still cofinal to it). Here, each $U_{\tau}$
is the affine toric variety over $A$ which corresponds to $\tau$. The
N-spectra $\Delta_{\xi}(A)$ is the spectrum of the completion of the stalk at
its natural prime ideal $I_{\xi}$. If $A=\mathbb{C}$, similarly, $A_{\xi}^{\rm
conv}(\mathbb{C})$ is the stalk of the corresponding the limit toric space, in
the sense of [Od22b, Definition 2.16].

Note that each $U_{\tau}(\mathbbm{k})$ for a field $\mathbbm{k}$ has Krull
dimension $r$ while the limit $\Delta_{\xi}(\mathbbm{k})$ has dimension $1$,
interestingly.

###### Proof.

Since the projective system $\\{U_{\tau}\\}_{\tau}$ is that of affine schemes,
it is enough to check at the level of rings. Then, it is a straightforward
confirmation so we leave the details to the readers. ∎

From the perspective of the above proposition, it is naturally to consider the
quotient stack $[\Delta_{\xi}^{\rm gl}/T]$ over the category of (non-finite
type) schemes, and denote it as $\Delta_{\xi}$ henceforth. For later purpose
in the next section, we also introduce a $R$-algebra variant:

######  Definition 2.3 (N-spectra over $R$).

For $N,\xi$ as notation 1, we fix a local ring $R$ with its non-unit non-zero
element $\pi$. We also take and fix $c\in\mathbb{R}_{>0}$, set
$\xi^{\prime}:=(\xi,c)\in N_{\mathbb{R}}\times\mathbb{R}_{>0}.$

Consider the minimum lattice $N^{\prime}\subset N\oplus\mathbb{Z}$ whose
spanning $\mathbb{R}$-linear subspace contains $\xi^{\prime}$, which we
suppose to have rank $r$. We take the projective system of rational polyhedral
cones $\tau^{\prime}$ of $N^{\prime}$ which contains $\xi^{\prime}$. Then
consider the natural projection

$p\colon M\oplus\mathbb{Z}\twoheadrightarrow M^{\prime},$

which is dual to $N^{\prime}\hookrightarrow N\oplus\mathbb{Z}$. Then, consider
the affine scheme $U_{\tau^{\prime}}(R)$ over $R$, defined as
$\mathop{\mathrm{Spec}}\nolimits(R[M^{\prime}\cap\tau^{\prime\vee}]/(\pi-\chi^{p(\vec{0},1)})),$
and its projective limit
$\varprojlim_{\tau^{\prime}}\mathop{\mathrm{Spec}}\nolimits(R[M^{\prime}\cap\tau^{\prime\vee}]/(\pi-\chi^{p(\vec{0},1)}))$
is denoted as $\overline{\Delta_{\xi}}(c,\pi,R)$. Here, $\tau^{\prime\vee}$
means the dual cone of $\tau^{\prime}$. As $p(\vec{0},1)\in
M^{\prime}\cap\tau^{\prime\vee},$ $\chi^{p(\vec{0},1)}$ means the
corresponding monomial (character) of $R[M^{\prime}\cap\tau^{\prime\vee}]$.

We also note that the above ring
$R[M^{\prime}\cap\tau^{\prime\vee}]/(\pi-\chi^{p(\vec{0},1)})$ is a quotient
ring of $(R[X])[\tau^{\vee}\cap M]/(X\chi^{p(\vec{0},1)}-\pi)$, where $X$ is
an extra variable and the quotient is by the ideal $(X-1)$.
$\varprojlim_{\tau^{\prime}}\mathop{\mathrm{Spec}}\nolimits((R[X])[\tau^{\vee}\cap
M]/(X\chi^{p(\vec{0},1)}-\pi))$ is denoted as
$\widetilde{\Delta_{\xi}}(c,\pi,R)$. (These and their further extensions will
appear in the proofs of stable reduction type theorems in §3.2 and later. See
Definition 3.5, Remark 3.7 Theorem 3.15, and Corollary 3.10.)

Note that for a fixed $\xi$ and $\tau$, there is a canonical choice of $c$
such that the lift of $\tau$ as $\tau^{\prime}\subset
N_{\mathbb{R}}\oplus\mathbb{R}$ contains $(v_{i},1)$ for each of the basis
$\\{v_{i}\\}_{i}$ of the regular generator of the cone $\tau\cap N$ thanks to
the irrationality of $\xi$. $\pi$ is also naturally decided in some context
later. In such cases, we abbreviate and denote as
$\overline{\Delta_{\xi}}(R,\pi)$ (resp., $\widetilde{\Delta_{\xi}}(R,\pi)$),
$\overline{\Delta_{\xi}}(R)$ (resp., $\widetilde{\Delta_{\xi}}(R)$), or even
$\overline{\Delta_{\xi}}$ (resp., $\widetilde{\Delta_{\xi}}$) for brevity.
These are also regarded as versions of N-spectra.

A real ($\mathbb{R}$) analogue of $\Delta_{\xi}$ and $\overline{\Delta}_{\xi}$
is $\tilde{R}_{\xi}$ below.

######  Definition 2.4.

If the base ring is $A=\mathbbm{k}=\mathbb{C}$, then we consider
$r_{\xi}\colon\mathbb{R}\to N_{\mathbb{R}}$ which simply maps $x$ to $x\xi$.
$\exp\colon\mathbb{R}\mapsto\mathbb{R}_{>0}\subset\mathbb{C}^{*}=\mathbb{G}_{m}(\mathbb{C})$
means the exponential map $r\mapsto e^{r}$. We also set the notation
$(\cdot)^{\xi}:=R_{\xi}\circ\log\colon\mathbb{R}_{>0}\to\mathbb{R}_{>0}\otimes
N$, which informally means the $\xi$-th power map. Consider the map

$R_{\xi}:=({\rm id}_{N}\otimes\exp)\circ r_{\xi}\colon\mathbb{R}\to
T(\mathbb{C}),$

where ${\rm id}_{N}\colon N\simeq N$ is the identity map. Note that because of
the irrationality of $\xi$, this can not be extended to $\mathbb{C}$. We
denote the universal cover of $\mathbb{C}^{*}$ as
$a\colon\widetilde{\mathbb{C}^{*}}\to\mathbb{C}^{*}$, which is isomorphic to
$\mathbb{R}_{>0}\times(\widetilde{U(1)}\simeq\mathbb{R})$, and for each
$\epsilon\in\mathbb{R}_{>0}$. $\widetilde{\mathbb{C}}_{\epsilon}^{*}$ denotes
the preimage of $\\{t\in\mathbb{C}^{*}\mid|t|<\epsilon\\}$.

Then, $R_{\xi}\circ a$ nevertheless extends to $\widetilde{\mathbb{C}^{*}}\to
T(\mathbb{C})$ as a holomorphic (analytic) homomorphism between complex Lie
groups. We denote it as

$\tilde{R}_{\xi}\colon\widetilde{\mathbb{C}^{*}}\to T(\mathbb{C}).$

Note that the image is not inside the real part $N\otimes\mathbb{R}_{>0}$ as
it is also nontrivial along the argument direction $\widetilde{U(1)}$, hence
$R_{\xi}\circ|\cdot|\circ a\neq\tilde{R}_{\xi}$. Nevertheless, they coincide
at the level of absolute values i.e.,

$(\cdot)^{\xi}\circ|\cdot|\circ a=|\cdot|^{r}\circ\tilde{R}_{\xi}=:|R_{\xi}|.$

The analogy with N-spectrum is partially explained by the following easy
lemma.

######  Lemma 2.5.

Under Notationss 1 and Definition 2.1 (ii), take any algebraic
$\mathbb{C}$-scheme $X$. Then, any $\mathbb{C}$-morphism (as schemes)
$\varphi\colon\Delta_{\xi}^{*,{\rm conv}}(\mathbb{C})\to X$ induces a
holomorphic map $\varphi_{\epsilon}^{\rm
an}\colon\widetilde{\mathbb{C}}_{\epsilon}^{*}\to X(\mathbb{C})$ for
$0<\epsilon\ll 1$.

We leave the proof to readers as it is straightforward. Now, we generalize the
notion of test configuration as follows. Henceforth, we stick to non-log setup
i.e., of (often polarized) varieties without boundary divisors, as we leave
the natural extensions to log pairs, which is very often almost verbatim now,
to readers if necessary.

###  2.2. Generalized test configurations and examples

Henceforth, we always fix a base field $\mathbbm{k}$ throughout the paper
(which is sometimes assumed to be $\mathbb{C}$ if mentioned). The following
concepts implicitly or explicitly (compare [HL20, §2.2], [BJ21, Appendix A3],
[Ino22, §2.2]) appeared in the literature. We introduce the terms just for
slight extensions and convenience for our later usage, without any
originality.

######  Definition 2.6.

  1. (1)

A generalized (unpolarized affine) test configuration (resp., generalized
(polarized, projective) test configuration) is the following set of data:

    1. (a)

free abelian groups $N$ of rank $r$.

    2. (b)

algebraic tori $T$ defined as $T=N\otimes_{\mathbb{Z}}\mathbb{G}_{m}$.

    3. (c)

a $r$-dimensional rational polyhedral cone $\tau(\supset\\{0\\})$ of
$N_{\mathbb{R}}:=N\otimes\mathbb{R}$. and its corresponding affine toric
variety $U_{\tau}$ over $\mathbbm{k}$. We denote the $0$-dimensional strata of
$U_{\tau}$ as $p_{\tau}$ or $0=p_{\tau}$ for simplicity.

    4. (d)

a variety $\mathcal{Y}$ (resp., a polarized variety
$(\mathcal{Y},\mathcal{L})$) and a $T$-action on it.

    5. (e)

faithfully flat $T$-equivariant affine morphism
$p\colon\mathcal{Y}\twoheadrightarrow U_{\tau}$ (resp., $T$-equivariant
projective flat morphism $p\colon(\mathcal{Y},\mathcal{L})\twoheadrightarrow
U_{\tau}.$)

    6. (f)

In the context when general fibers are affine cones (in the sense of [Od24a]),
with respect to an additional algebraic $\mathbbm{k}$-torus $T^{\prime}$, we
require more on the generalized (affine) test configuration as follows.
$\mathcal{Y}$ admits $p$-fiberwise $T^{\prime}$-action so that it can be
written as $\mathop{\mathrm{Spec}}\nolimits\mathcal{R}$ with a
$\Gamma(U_{\tau})$-algebra $\mathcal{R}$ which we decompose as
$\mathcal{R}=\oplus_{\vec{m}\in M}\mathcal{R}_{\vec{m}}$ via the
$T^{\prime}$-action to the $T^{\prime}$-eigensubmodules over
$\Gamma(U_{\tau})$, then each $R_{\vec{m}}$ is locally free over
$\Gamma(U_{\tau})$.

  2. (2)

For an affine integral reduced scheme $Y$ of finite type over a field
$\mathbbm{k}$ (resp., polarized variety $(Y,L)$ over $\mathbbm{k}$), an
(affine) algebraic test deformation (resp., (polarized) algebraic test
deformation of $(T\curvearrowright Y,\xi)$), where $T$ is an algebraic
$\mathbbm{k}$-torus $N\otimes\mathbb{G}_{m}$ and $\xi$ is an element of
$N\otimes\mathbb{R}$, is a generalized (unpolarized affine) test configuration
$p\colon\mathcal{Y}\to U_{\tau}$ with $p^{-1}(0)=Y$.

Note that if $\mathbbm{k}$ contains $\mathbb{R}$ and $T$ is
$\mathbbm{k}$-split, $\xi$ can be identified with its holomorphic333we take
this choice rather than real vector field, as in usual choice of Reeb vector
field, as we want to work also over $\mathbbm{k}\neq\mathbb{C}$ vector field
$\xi$ over the smooth locus $Y^{\rm sm}$ of $Y$. Indeed, if we consider the
induced action of $T$ on $p^{-1}(0)$ and the natural map $a\colon
N_{\mathbb{R}}\to H^{0}(p^{-1}(0),T_{p^{-1}(0)^{\rm sm}})$ induced by it
(here, $T_{p^{-1}(0)^{\rm sm}}$ denotes the Zariski tangent sheaf of the
smooth locus of $p^{-1}(0)$), the image ${\rm Im}(a)$ contains $a(\xi)$ and
$a$ is often injective, so that we identify $\xi$ with a holomorphic vector
field.

  3. (3)

An affine algebraic test deformation of $(Y,\xi)$ over $U_{\tau}$ is algebraic
positive (resp., negative) weight deformation (or $Y$ is algebraic positive
(resp., negative) $\xi$-weight deformation) over $U_{\overline{\tau}}$ if

    1. (a)

$T\curvearrowright Y$ is a good action in the sense of [LS13] (e.g., Fano cone
as in [CS18, CS19]) and

    2. (b)

$\tau\ni\xi$ (resp., $-\tau\ni\xi$).

We also propose alternative names of algebraic positive (resp., negative)
weight deformation: scale up (resp., scale down) degeneration, motivated by
the differential geometric context.

######  Example 2.7.

Underlying family of affine test configuration (cf., [CS18]) is a generalized
(unpolarized) test configuration with $r=1,\tau=\mathbb{R}_{\geq 0}$.
Similarly, (polarized projective) test configuration ([Don02]) is a
generalized (polarized projective) test configuration with
$r=1,\tau=\mathbb{R}_{\geq 0}$.

######  Example 2.8.

If we consider an embedded affine variety
$Y^{\prime}\subset\mathbb{A}_{\mathbbm{k}}^{l}$ and the natural action of
$T=\mathbb{G}_{m}^{l}$ on the ambient space $\mathbb{A}_{\mathbbm{k}}^{l}$
(with all weights $1$). Then we consider
$Y^{\prime}\times\\{(1,\cdots,1))\\}\subset\mathbb{A}_{\mathbbm{k}}^{l}\times
T$ and its orbit $T\cdot(Y^{\prime}\times\\{(1,\cdots,1))\\})$. If the orbit
closure $\overline{T\cdot(Y^{\prime}\times\\{(1,\cdots,1))\\})}$ inside
$\mathbb{A}_{\mathbbm{k}}^{l}\times U_{\tau}$ is faithfully flat over
$U_{\tau}$ for some (not necessarily strictly) convex rational polyhedral cone
$\tau$ of $N_{\mathbb{R}}$, then

$\overline{T\cdot(Y^{\prime}\times\\{(1,\cdots,1))\\})}\twoheadrightarrow
U_{\tau}$

gives a generalized (unpolarized) test configuration. This generalizes the
classical construction of test configuration à la Mumford ([Mum65]) and
Donaldson ([Don02]) for the $r=1$ case. However, the required (faithful)
flatness becomes nontrivial in general for $r>1$.

######  Example 2.9 (The degenerations of klt singularities to its
K-semistable Fano cone).

We discuss an algebraic version of the local version of the Donaldson-Sun
theory [DS17]. Suppose the base field $\mathbbm{k}$ is algebraically closed of
characteristic $0$. For any affine klt variety $X$ with its closed point $x$,
there is a canonical quasi-monomial valuation $v_{X}$ with the center $x$
([Li18, Blu18]) and
$W=\mathop{\mathrm{Spec}}\nolimits\mathop{\mathrm{gr}}\nolimits_{v_{X}}\mathcal{O}_{X,x}$
is a K-semistable Fano cone, in particular
$\mathop{\mathrm{gr}}\nolimits_{v_{X}}\mathcal{O}_{X,x}$ is finitely generated
([LX18, XZ22]). Here, recall that
$\mathop{\mathrm{gr}}\nolimits_{v_{X}}\mathcal{O}_{X,x}:=\oplus_{a\in\mathbb{R}_{\geq
0}}\\{f\in\mathcal{O}_{X,x}\mid v_{X}(f)\geq a\\}/\\{f\in\mathcal{O}_{X,x}\mid
v_{X}(f)>a\\}$.

Choose its homogenous generators $\overline{z}_{1},\cdots,\overline{z}_{l}$,
and we write their weights as $w_{1},\cdots,w_{l}\in\mathbb{R}_{>0}$ in the
holomorphic spectrum (cf., [DS17]) and denote $\xi:=(w_{1},\cdots,w_{l})$.
Naturally, we have the corresponding embedding
$W\hookrightarrow\mathbb{A}_{\mathbbm{k}}^{l}$. We then take lifts of
$\overline{z}_{1},\cdots,\overline{z}_{l}$ to the element of
$\Gamma(\mathcal{O}_{X})$, say $z_{1},\cdots,z_{l}\in\Gamma(\mathcal{O}_{X})$.
By taking additional $z_{i}$s (and $\overline{z}_{i}$s) if necessary, we can
assume that the corresponding map
$X\hookrightarrow\mathbb{A}_{\mathbbm{k}}^{l}$, which maps $x$ to the origin
$\vec{0}$, is an embedding.

Recall that by a slight perturbation of $v_{X}$ to a divisorial valuation
$v^{\prime}$, we can easily show there is a natural isomorphism of the graded
ring as $gr_{v_{X}}(\mathcal{O}_{X,x})=gr_{v^{\prime}}(\mathcal{O}_{X,x})$
(cf., e.g., [LX18, 2.10]) and we take associated test configuration as the
spectrum of graded Rees algebra ([Mat89, Chapter 5, section 4]), which
degenerates $x\in X$ to $W$ with the vertex. Note that this test configuration
depends on the choice of the perturbation $v^{\prime}$. We integrate these
degenerations for various approximation $v^{\prime}$ as follows.

For that, we first observe the following. We leave the proof to the readers as
it is easy.

######  Lemma 2.10.

For any $\xi=(w_{1},\cdots,w_{r})\in\mathbb{R}^{r}$, we take a lattice $N$
which is the minimum sublattice of $\mathbb{Z}^{r}$ such that
$N\otimes\mathbb{R}(\subset\mathbb{R}^{r})$ which contains $\mathbb{R}\xi$.
Then the natural pairing map

$\langle\xi,-\rangle\colon M\to\mathbb{R},$

which sends $\vec{m}=(m_{1},\cdots,m_{l})\in M:={\rm Hom}(N,\mathbb{Z})$ to
$\langle\vec{m},\xi\rangle=\sum_{i}m_{i}w_{i}$, is a group isomorphism onto
its image $\Gamma:=\sum_{i}\mathbb{Z}w_{i}\subset\mathbb{R}$.

We apply the above for our weight vector
$\xi=(w_{1},\cdots,w_{r})\in\mathbb{R}^{r}$, and also take a rational
polyhedral cone $\tau$ of $N$ which contains $\xi$, to later discuss its
perturbations inside $\tau$.

We define the algebraic subtorus $T\subset\mathbb{G}_{m}^{r}$ as
$T:=\mathbb{G}_{m}\otimes N$. We take generators of the defining ideal of
$X\subset\mathbb{A}_{\mathbbm{k}}^{l}$, say $f_{1},\cdots f_{a}$, which are
even universal Gröbner basis of the generating ideal of $X$ ([BCRV22, §5.1]).

We then take small enough $\tau$ which includes $\xi$ and applies Example 2.8
which results in a flat $T$-equivariant family
$\pi_{\tau}\colon\mathcal{X}_{\tau}\twoheadrightarrow U_{\tau}$ whose fiber is
$X$ over $(1,\cdots,1)$ and the fiber over the $T$-invariant point $p_{\tau}$
is $W$ as the proof [LX18, 2.10] shows as $\tau$ is small enough.

By using the fact that $f_{i}$s form a universal Gröbner basis, it easily
follows that for any rational ray $\mathbb{R}_{>0}\xi^{\prime}$ in $\tau$, the
inducing valuation on $X$ is represented by $\mathbb{R}_{>0}\xi^{\prime}$,
which we denote as $v_{\xi^{\prime}}$, ts extended Rees construction ([Mat89,
Chapter 5, section 4]) is nothing but the restriction of $\pi_{\tau}$ to the
orbit closure
$\overline{(\mathbb{G}_{m}\otimes(N\cap\mathbb{R}\xi^{\prime}))\cdot[X]}$
along $\xi^{\prime}$-direction. Further, from the Zariski openness of the
irreducibility condition, by replacing $\tau$ by a smaller enough one if
necessary, we can and do assume that all the fibers of $\pi_{\tau}$ are
irreducible. (Even more is true, as [LX18, 2.10] again shows that the fibers
over the toric boundary are all isomorphic.) Thus its inducing valuation on
$X$ is represented by $\mathbb{R}_{>0}\xi^{\prime}$. Further, from the
construction, the above generalized test configuration $\pi_{\tau}$ is a
positive weight deformation (scale up degeneration) in the sense of Definition
2.6.

To make the construction even more canonical, to remove the ambiguity of the
choice of $\tau$, if we take a projective limit with respect to the injective
system of $\tau$ via inclusions, we obtain the canonical deformation family
over $\varprojlim_{\tau}U_{\tau}$.

We end Example 2.9 by the following conclusion of our discussion.

######  Theorem 2.11.

For any affine klt variety $X$ with its closed point $x$, over an
algebraically closed field $\mathbbm{k}$ of characteristic $0$, consider the
volume-minimizing valuation $v_{X}$ of rational rank $r$ ([Li18, Blu18, Xu20])
and $W:=\mathop{\mathrm{Spec}}\nolimits({\rm gr}_{v_{X}}(\mathcal{O}_{X,x}))$.
Take the groupificiation $N$ of ${\rm Im}(v_{X})$ and its small enough
rational polyhedral cone $\tau$ which includes $\xi=(w_{1},\cdots,w_{l})$
where $w_{i}$ are the values of $v_{X}$ for homogeneous generators of ${\rm
gr}_{v_{X}}(\mathcal{O}_{X,x})$.

There is a canonical construction of algebraic positive weight deformation
$\pi_{\tau}\colon\mathcal{X}_{\tau}\twoheadrightarrow U_{\tau}$ of $W$ to $X$
for each $\tau$. The (degenerated) fibers over the toric boundary
$U_{\tau}\setminus T$ are all isomorphic to $W$ if $\tau$ is small enough. To
remove the ambiguity of $\tau$, we can also take affine morphisms
$\varprojlim_{\tau}\mathcal{X}_{\tau}\to\varprojlim_{\tau}U_{\tau}$, which is
completely canonical whose general fiber is (the base change of) $X$ while the
closed special fiber is $W$ i.e., the canonical degeneration of $X$ to $W$. In
particular, if we pass to the completion of a stalk (see Proposition 2.2), we
obtain a canonical affine family $\mathcal{X}_{\xi}\to\Delta_{\xi}$ over the
N-spectrum $\Delta_{\xi}$.

See closely related studies in [Tei03, §2] and [LX18, the proof of 2.10] which
follows loc.cit.

######  Remark 2.12.

As the general context of [Mum65], consider a reductive algebraic group $G$
and its action on a polarized variety $(M,L)$ over an algebraically closed
field $\mathbbm{k}$. Take a semistable closed point $x$ of $X$.

Then, the orbit closure $\overline{G\cdot x}\subset M$ contains a unique
$G$-orbit which is closed inside the semistable locus. We denote it by $G\cdot
y$ with $y\in X$. By [IM65] (cf., also [Mum65, Chapter 2, §1]), there is a one
parameter subgroup $\lambda\colon\mathbb{G}_{m}\to G$ with $\lim_{t\to
0}\lambda(t)\cdot x=y$ though it is far from unique. If we take a maximal
torus of $G$ which contains the image of $\lambda$, then clearly we have
$\overline{T\cdot x}\ni y$, with a little less isomorphism classes. If $M$ is
a moduli of certain varieties, this will give us a generalized test
configuration but we do not know its canonicity in general (since $T$ is
unique only up to the conjugation).

######  Remark 2.13.

Let us consider isotrivial degenerations of K-semistable Fano cone $W$ to
K-polystable Fano cone $C$, which first appeared in [DS17] where the $C$ is
the metric tangent cone of the Gromov-Hausdorff limits of Kähler-Einstein
manifolds, and later [LWX21] gave an algebraic proof to the fact that such
isotrivial degeneration exists for any $W$ as an affine test configuration.

Recall that $C$ is given an action of algebraic torus $T$ from the definition,
and by the reductivity of $G:={\rm Aut}_{T}(C)$ ([DS17, Appendix], cf., also
[Od24a]), which means the group of automorphisms of $C$ which commutes with
the $T$-action, and the action of its centralizer to the multi-Hilbert scheme
[HM04] $H$ as in [DS17]. From these facts, it is easy to see similar remark
apply as above Remark 2.12.

######  Example 2.14 ($\mathbb{R}$-test configurations, Kähler-Ricci flow).

For a general polarized variety $(X,L)$ over $\mathbbm{k}$, there is a notion
of $\mathbb{R}$-test configuration (cf., [HL20, BLXZ23] as originated in
[CSW18, §3]) which we recall now.

######  Definition 2.15.

If we set $R_{m}:=H^{0}(X,L^{\otimes m})$ for $m\in\mathbb{Z}_{>0}$, and
$R_{(X,L)}:=\oplus_{m\in\mathbb{Z}_{\geq 0}}R_{m}$, the filtration
$\mathcal{F}^{\lambda}R_{m}$ of $R_{m}$ for $\lambda\in\mathbb{R}$ which
satisfies the following is called filtration of $(X,L)$ (cf., [Sze15]).

  1. (1)

$\mathcal{F}^{\lambda}R_{m}$ is a $\mathbbm{k}$-vector subspace of $R_{m}$ for
each $\lambda\in\mathbb{R},m\in\mathbb{Z}$.

  2. (2)

(decreasingness)
$\mathcal{F}^{\lambda^{\prime}}R_{m}\supset\mathcal{F}^{\lambda}R_{m}$ for any
$\lambda^{\prime}\leq\lambda$.

  3. (3)

(linearly boundedness) there is $e_{-}$ and $e_{+}$ such that
$\mathcal{F}^{e_{-}m}R_{m}=R_{m}$ and $\mathcal{F}^{e_{+}m}R_{m}=0$ for any
$m$.

  4. (4)

(multiplicativity)
$\mathcal{F}^{\lambda_{1}}R_{m_{1}}\cdot\mathcal{F}^{\lambda_{2}}R_{m_{2}}\subset\mathcal{F}^{\lambda_{1}+\lambda_{2}}R_{m_{1}+m_{2}}$
for any $\lambda_{1},\lambda_{2}\in\mathbb{R},m_{1},m_{2}\in\mathbb{Z}$.

  5. (5)

(as a convention: left-continuous)
$\cap_{\lambda^{\prime}\leq\lambda}\mathcal{F}^{\lambda^{\prime}}R_{m}=\mathcal{F}^{\lambda}R_{m}$.

For a filtration $\mathcal{F}^{\lambda}R_{m}$, we can define the graded ring

$\displaystyle{\rm gr}_{\mathcal{F}}(R_{(X,L)})$
$\displaystyle:=\oplus_{m\in\mathbb{Z}_{\geq 0}}{\rm gr}_{\mathcal{F}}R_{m}$
$\displaystyle=\oplus_{m\in\mathbb{Z}_{\geq
0},\lambda\in\mathbb{R}}(\mathcal{F}^{\lambda}R_{m}/\cup_{\lambda^{\prime}>\lambda}\mathcal{F}^{\lambda^{\prime}}R_{m}).$

A filtration is called a $\mathbb{R}$-test configuration if ${\rm
gr}_{\mathcal{F}}(R_{(X,L)})$ is finitely generated algebra over
$\mathbbm{k}$.

Now we briefly recall the following geometric realization of $\mathbb{R}$-test
configuration as a generalized test configuration, as essentially proved in
[Tei03, HL20, BlJ20, Ino22] under the name of “geometric $\mathbb{R}$-test
configuration II” ([HL20, p.10]), “rank $r$ test configuration” ([BlJ20,
Appendix A.3]), and “polyhedral configuration” ([Ino22, §2.2.2, Prop 2.20])
respectively. See also loc.cit for more information on this correspondence.

######  Proposition 2.16 ([HL20, BlJ20, Ino22]).

For a $\mathbb{R}$-test configuration $\mathcal{F}$ of $(X,L)$, with its
central fiber $(Y:={\rm Proj}({\rm gr}_{\mathcal{F}}(R(X,L))),\mathcal{O}(1))$
and its natural holomorphic vector field $\xi$, we can associate a canonical
(polarized) algebraic test deformation of $(Y,\mathcal{O}(1),\xi)$ over a
certain affine toric variety $U_{\tau}$.

###### Proof.

We proceed similarly as the proof of Theorem 2.11 in Example 2.9. Take
homogeneous generators of ${\rm gr}_{\mathcal{F}}(R_{(X,L)})$ and their
$\mathbb{Z}$-homogeneous lifts to $R(X,L)$ as $f_{1},\cdots,f_{l}$ and denote
their gradings as $(m_{i},w_{i})(i=1,\cdots,l)$ with $m_{i}\in\mathbb{Z}_{\geq
0},w_{i}\in\mathbb{R}$. We set $\xi:=(w_{1},\cdots,w_{l})$ and take
$\tau\ni\xi$ similarly. Then we apply Example 2.9 to the cone
$Y^{\prime}:={\rm Spec}R(X,L)$ with the natural $\mathbb{G}_{m}$-action. By
dividing the resulting family by $\mathbb{G}_{m}$, we obtain the desired
family. We omit the details as the proof is almost verbatim to Example 2.9. ∎

Similarly to the remark after Theorem 2.11, if we use the language of limit
toric variety $\varprojlim_{\tau}U_{\tau}$ (or its germ at the $0$-dimensional
strata) in [Od22b, §2.4] or use its spectra of the completion (Proposition
2.2) as the N-spectra $\mathbbm{k}_{\xi}$, we can eliminate the ambiguity of
$\tau$ by the set of all its subdivisions.

Recall that [CW20, Theorem 1.6] and then [Bam18, Corollary 1.3], [WZ21,
Theorem 1.1] proved the Hamilton-Tian conjecture which says that the Gromov-
Hausdorff limit $W$ of the normalized Kähler-Ricci flow on (possibly
K-unstable) Fano manifold $X$ is $\mathbb{Q}$-Fano variety with weak Kähler-
Ricci soliton, defined as

$\frac{\partial}{\partial t}\omega_{t}=\omega_{t}-{\rm Ric}(\omega_{t}),$

with $[\omega_{t}]\in 2\pi c_{1}(X).$ Then [CSW18] proved even more i.e., it
is obtained as the central fiber of $\mathbb{R}$-test configuration of
$(X,\mathcal{O}(-K_{X}))$, which minimizes the H-invariant as proved in [DS20,
HL20] (cf., also [BLXZ23]).

######  Corollary 2.17 (cf., [CSW18, HL20, BLXZ23]).

For any K-unstable $\mathbb{Q}$-Fano variety $X$, consider the corresponding a
(klt) $\mathbb{Q}$-Fano variety $W$ with action of an algraic torus $T$ and
$\xi\in({\rm Hom}(\mathbb{G}_{m},T)\otimes\mathbb{R})\setminus({\rm
Hom}(\mathbb{G}_{m},T)\otimes\mathbb{Q})$, which satisfies the (modified)
K-semistability in the sense of [BN14, §3.5], [HL20, §2]. This is the $1$-st
process of the degeneration to the limit of the normalized Kähler-Ricci flow
on $X$ as [CSW18, HL20], also characterized as the linear bounded filtration
whose $H^{\rm NA}$-invariant is minimized ([HL20, BLXZ23]). Thus, there is a
canonical algebraic test deformation of $W$ to $X$.

######  Example 2.18 (Negative weight deformations).

Conlon-Hein [CH22] introduces a notion of negative weight deformation of
Calabi-Yau cones in terms of sequence of affine test configurations, for their
classification of asymptotically conical Calabi-Yau manifolds. We
alternatively introduce the notion of algebraic negative deformation in
Definition 2.6 without using sequence (to answer Open Questions 1.4(2) of
[CH22]). For our later purpose, we also clarify the relation as follows. Here,
we assume $\mathbbm{k}=\mathbb{C}$ as [CH22]. We take a affine
$\mathbb{C}$-variety $Y$ and an algebraic torus $T=N\otimes\mathbb{G}_{m}$
acting on $Y$, with $\xi\in N_{\mathbb{R}}$.

######  Lemma 2.19 (Rational approximation - relation with [CH22]).

  1. (1)

Suppose $p\colon\mathcal{Y}\twoheadrightarrow U_{\tau}$ is an algebraic
negative weight deformation (Definition 2.6) of $(Y,\xi)$. In particular,
$\xi\in-\tau$. Then, its general fiber is a negative $\xi$-weight deformation
in the sense of [CH22, Definition 1.7].

  2. (2)

([Od24c]) Conversely, if $Y$ has only isolated singularities and $T$ acts as a
good action in the sense of [LS13], any negative $\xi$-weight deformation of
normal affine variety with $Y$ and $\xi$ arise in this way.

###### Proof.

For the item (1), we can assume that $\tau=\mathbb{Z}_{\geq 0}^{r}$,
$N=\mathbb{Z}^{r}$. Then we approximate $\xi$ by
$\xi_{i}=(\frac{r_{i,1}}{m_{1}},\frac{r_{i,2}}{m_{1}},\cdots,\frac{r_{i,r}}{m_{1}})\in-\tau\cap
N_{\mathbb{Q}}$, with nonnegative integers $r_{i,j}(j=1,\cdots,r)$ and
$m_{i}$s so that a convergence $\xi_{i}\to\xi$ holds for $i\to+\infty$. Then
take a morphism $\varphi_{\vec{r}}\colon\mathbb{A}^{1}\to U_{\tau}$ defined by
$t\mapsto(t^{r_{i,1}},t^{r_{i,2}},\cdots,t^{r_{i,r}})$ and consider the base
change $\mathcal{Y}\times_{p,U_{\tau},\varphi_{\vec{r}}}\mathbb{A}^{1}$, which
we denote as $p^{\prime}_{i}\colon W^{\prime}_{i}\to\mathbb{A}_{1}$. It can be
naturally seen as an affine test configuration but we modify the
$\mathbb{G}_{m}$-action $\sigma^{\prime}_{i}$ on it by compositing with the
$m_{i}$-th power map $\mathbb{G}_{m}\to\mathbb{G}_{m}$. In particular,
$p^{\prime}_{i}(\sigma^{\prime}_{i}(z,x))=z^{m_{i}}p^{\prime}_{i}(x)$ for
$z\in\mathbb{G}_{m}(\mathbbm{k}),x\in W^{\prime}_{i}$. Then, we take the base
change of $p^{\prime}_{i}\colon W^{\prime}_{i}\to\mathbb{A}^{1}$ with
$\sigma^{\prime}_{i}$ by the $m_{i}$-th power map
$q_{i}\colon\mathbb{A}^{1}\to\mathbb{A}^{1}$ whose target is regarded as the
base of $p^{\prime}_{i}$. It is straightforward confirmation that the
sequence, with respect to $i=1,2,\cdots$, of the obtained fiber product
$p_{i}\colon
W_{i}(:=W^{\prime}_{i}\times_{p^{\prime}_{i},\mathbb{A}^{1},q_{i}}\mathbb{A}^{1})\to\mathbb{A}^{1}$
with the $\mathbb{G}_{m}$-action induced by $\sigma^{\prime}_{i}$ satisfies
the conditions of the [CH22, Definition 1.7], with $c_{i}$ of loc.cit as
$m_{i}$, $\mu_{i}$ of loc.cit as $1$, hence the latter being auxiliary
parameters. Suppose $Z$ is the maximum Artin scheme supported at the
$T$-invariant point $p_{\tau}\in U_{\tau}$, such that $\mathcal{Y}|_{Z}$ is
trivial family. Such $Z$ exists due to the Zorn’s lemma combined with the
formal smoothness ([Gro64, Part I, Part IV]) of ${\rm Aut}_{T}(Y)$ which is an
algebraic group as the stabilizer of $[Y]$ in the multi-Hilbert scheme. By
using this $Z$, $k_{i}$ of loc.cit can be written as $m_{i}{\rm
length}\mathcal{O}_{\varphi_{\vec{r}}^{*}Z}$, which explains the nature of the
$\xi$-weight in loc.cit. ∎

The item (2) is proved in [Od24c] in which some more related materials are
also explored.

##  3\. Higher rank semistable reduction theorem

In this section, we generalize the theorem of [AHLH23, §6] to the families
over higher dimensional toroidal base or Novikov type rings.

###  3.1. Toroidal stack-theoretic setup

Here are some preparations on stack-theoretic framework, following [HL14,
Hei18, AHLH23].

######  Definition 3.1.

Under the notation 1 in the previous subsection, we set the following Artin
stacks.

  1. (1)

Consider the quotient stack of the affine toric variety $U_{\tau}$:

$\Theta_{\tau}:=[U_{\tau}/T],$

generalizing the case $r=1,\tau=\mathbb{R}_{\geq 0}$ treated in [HL14,
AHLH23], which is denoted simply as $\Theta$. We denote the $T$-invariant
point of $U_{\tau}$ as $p_{\tau}$. In [HL14], (the isomorphic object)
$\Theta_{\tau}$ for general $r$ and regular simplicial $\tau$ is also
introduced under the disguise of $\Theta^{r}$, the $r$-times self-product of
$\Theta$. For our purpose, since we will encounter a setup with various $\tau$
for a fixed $N$ later, we prefer the above setup and notation.

  2. (2)

We fix a valuation ring $(R,v)$ of height $1$ with $\Gamma_{\geq
0}:=v(R\setminus\\{0\\})\subset\mathbb{R}$. Further, take a polyhedral cone
$\tau^{\prime}$ of $N\oplus\Gamma_{\geq 0}$ which projects as PL isomorphism
(in particular, bijectively) to $\tau\subset N_{\mathbb{R}}$ and the normal
vector of $\tau$ can be taken as $(\vec{m},1)\in M\oplus(\Gamma_{\geq
0})^{*}$, i.e., $\langle\tau,(\vec{m},1)\rangle=0$. We denote the character on
$T$ which corresponds to $\vec{m}$ as $\chi_{\vec{m}}$. Then we define

(1) $\displaystyle\widetilde{\rm ST}_{R}(\tau^{\prime})$
$\displaystyle:=\mathop{\mathrm{Spec}}\nolimits R[\tau^{\vee}\cap
M]\biggl{(}\frac{\pi}{\chi_{\vec{m}}}\biggr{)},$ (2)
$\displaystyle{\overline{\rm ST}}_{R}(\tau^{\prime})$
$\displaystyle:=[\widetilde{\rm ST}_{R}(\tau^{\prime})/T],$

again generalizing the case $r=1,\tau=\mathbb{R}_{\geq 0}$ introduced in
[Hei18] (cf., also [AHLH23]). Note that the adjunction of
$\frac{\pi}{\chi_{\vec{m}}}$ in the right hand side of (1) is a localization
corresponding to a “toroidal” blow up of $R\times U_{\tau}$, with respect to
$\hat{\tau}^{\prime}:=\tau^{\prime}+\mathbb{R}_{\geq 0}(\vec{0},1)$. In
particular, we have a natural (birational) map $b\colon{\overline{\rm
ST}}_{R}(\tau^{\prime})\to{\rm Spec}(R)$ and a canonical open embedding ${\rm
Spec}(K)\hookrightarrow{\overline{\rm ST}}_{R}(\tau^{\prime}),$ where $K$
denotes the fraction field of the valuation ring $R$.

######  Definition 3.2 (Complex analytic analogue).

If $\mathbbm{k}=\mathbb{C}$, then we can literally make sense of the
toroidality of the blow up $b^{\rm an}\colon
U_{\hat{\tau}^{\prime}}\to\mathbb{C}\times U_{\tau}^{\rm an}$. We denote the
complex analytic space $(b^{\rm an})^{-1}(\Delta\times U_{\tau}^{\rm an})$ as
$\widetilde{\rm ST}_{\Delta}^{\rm an}(\tau^{\prime})$ its analytic quotient
stack ${\overline{\rm ST}}_{\Delta}^{\rm an}(\tau^{\prime}):=[\widetilde{\rm
ST}_{\Delta}^{\rm an}(\tau^{\prime})/T(\mathbb{C})^{\rm an}]$.

Now we extend the notion of $\Theta$-strata ([HL14]) and elementary
modifications ([AHLH23]) accordingly, which morally corresponding to the
“optimal destabilization”. The point of our generalization is that they are
not necessarily one parameter algebraic families but rather higher dimensional
families, which contains “irrational” directions.

Below, we fix an algebraic stack $\mathcal{M}$ over $\mathbbm{k}$.

######  Definition 3.3 (Higher $\Theta$-strata cf., [HL14, §2.1]).

Consider the mapping stack ${\rm Map}(\Theta^{r},\mathcal{M})$ of $\Theta^{r}$
to $\mathcal{M}$, as an algebraic stack, after [AHR20, 5.10, 5.11], [AHR19,
6.23]. Extending [HL14, Definition 2.1.1], a higher $\Theta$-stratum of rank
$r$ in $\mathcal{M}$ consists of a union of connected components
$\mathcal{Z}^{+}\subset{\rm Map}(\Theta^{r},\mathcal{M})$ such that the
natural evaluation morphism ${\rm
ev}_{(1,\cdots,1)}\colon\mathcal{Z}^{+}\to\mathcal{M}$ is a closed embedding.
If we fix an isomorphism $\Theta^{r}\simeq\Theta_{\tau}$ for a rational
polyhedral cone $\tau$, then we also call higher $\Theta$-stratum as higher
$\Theta$-stratum for the cone $\tau$.

Further, if $\mathcal{M}$ is quotient stack of the form $[M/T]$ where $M$ is
an algebraic $\mathbbm{k}$-scheme $M$ and $T$ is an algebraic torus again, we
often write ${\rm ev}_{(1,\cdots,1)}(\mathcal{Z}^{+})=[Z^{+}/T]$ for some
$T$-invariant closed subscheme $Z^{+}\subset M$.

######  Definition 3.4 (Elementary ($\tau$-)modification cf., [AHLH23, §3.5,
Definition 3.36]).

An elementary ($\tau$-)modification of $f\colon{\rm Spec}(R)\to\mathcal{M}$ is
the data of a morphism $\overline{f}\colon\overline{\rm
ST}_{R}(\tau)\to\mathcal{M}$ along with an isomorphism
$f\simeq\overline{f}|_{\rm Spec(R)}$. Here, to make sense of the restriction
symbol, $\rm Spec(R)$ is regarded as a localization of $\overline{{\rm
ST}}_{R}(\tau^{\prime})$ since we have
$\mathop{\mathrm{Spec}}\nolimits(R)=[\mathop{\mathrm{Spec}}\nolimits R[M]/T]$.

The following lemma lies the ring-theoretic aspect of Definition 2.3, as both
appear in Theorem 3.8 later.

######  Lemma 3.5 (Extension of valuation ring).

Suppose $(R,v=v_{R})$ is a valuation local ring, such that the associated
graded ring $gr_{v}(R):=\oplus_{w\in{\rm Im}(v)}\\{a\in R\mid v(a)\geq
w\\}/\\{a\in R\mid v(a)>w\\}$ is isomorphic to a semigroup ring over a field
for a submonoid of $\mathbb{R}^{r}$ with some $r\in\mathbb{Z}_{>0}$ as a
graded ring (which holds e.g. if $R$ is of the form $k[[\Gamma_{\geq 0}]]$ in
Definition 2.1), and $\xi,N,M_{\xi\geq 0}$ are as Notation 1. Now, we set

$R[[M_{\xi\geq 0}]]:=\\{\sum_{i=1,2,\cdots}a_{i}t^{m_{i}}\mid a_{i}\in
R,m_{i}\in M_{\xi\geq 0},m_{1}<m_{2}<\cdots,m_{i}\to\infty(i\to\infty)\\},$

as Definition 2.1 (i), and consider

$R_{\rm new}(\xi)^{\prime}:=R[[M_{\xi\geq 0}]]/(\pi-\chi^{m}),$

for $(0\neq)\pi\in R\setminus R^{*}$ and $m\in M_{\xi\geq 0}$ which we suppose
to be primitive i.e., there is no positive integer $a>1$ with $m^{\prime}\in
M_{\xi\geq 0}$ such that $am^{\prime}=m$.

Then, $R_{\rm new}(\xi)^{\prime}$ has a natural valuation $\overline{v}_{\rm
new}$ induced from $v_{R}$, with $\overline{v}_{\rm
new}(\overline{\pi})=v(\pi)$. We take its associated valuation ring $R_{\rm
new}(\xi)$ in the fraction field of $R_{\rm new}(\xi)^{\prime}$. Then, the
associated graded ring is again isomorphic to a semigroup ring over a field,
for a submonoid of $\mathbb{R}^{r}$ with some $r\in\mathbb{Z}_{>0}$ as a
graded ring.

###### Proof.

Set $b:=\frac{v(\pi)}{\langle\xi,m\rangle}>0$ and put $\tilde{v}(\sum
a_{i}t^{m_{i}}):=\min_{i}\\{v(a_{i})+b\langle m_{i},\xi\rangle\\}$ for
$a_{i}\in R,m_{i}\in M_{\xi\geq 0}$. This $\tilde{v}$ defines a valuation on
$R[[M_{\xi\geq 0}]]$ and consider its corresponding filtration
$\mathcal{F}_{\tilde{v}}$ as $\mathcal{F}_{\tilde{v}}(x)=\\{f\in R[[M_{\xi\geq
0}]]\mid\tilde{v}(f)\geq x\\}$ for each $x\in\mathbb{R}$. It induces a
filtration $\overline{\mathcal{F}}$ on $R_{\rm new}(\xi)$ as a quotient of
$R[[M_{\xi\geq 0}]]$, and the corresponding graded ring is clearly $({\rm
gr}_{v}(R))[M_{\xi}/\mathbb{Z}_{\geq 0}m]$ which is still a semigroup ring for
a submonoid of $\mathbb{R}^{r}$ for a bigger $r$. Since it is an integral
domain, $\overline{\mathcal{F}}$ also induces a valuation on $R_{\rm
new}(\xi)$. ∎

######  Remark 3.6 (Novikov-type case).

Even if $R$ is of the form $k[[\Gamma_{\geq 0}]]$ in Definition 2.1, hence a
valuation ring, $R_{\rm new}(\xi)^{\prime}$ is not necessarily of the form
$k[[\Gamma_{\geq 0}]]$ for a different $\Gamma_{\geq 0}$ as in Definition 2.1,
and it is not a valuation ring in general. For instance, such a pathology
often happens if we continue the above process $R\rightsquigarrow R_{\rm
new}(\xi)$ for the same irrational $\xi\in M_{\mathbb{R}}$ twice.
Nevertheless, if $R$ is of such form $k[[\Gamma_{\geq 0}]]$, $R_{\rm
new}(\xi)$ is again of such form $k[[\Gamma_{\geq 0}+bM_{\xi\geq 0}]]$ for
$b\in\mathbb{R}_{>0}$ as in the proof of Lemma 3.5.

######  Remark 3.7 (Relation with $\overline{\Delta_{\xi}}(R)$).

In the setting of Definition 2.3, a formal completion of the inductive limit
of $R[M^{\prime}\cap\tau^{\prime\vee}]/(\pi-\chi^{p(\vec{0},1)})$ is nothing
but $R_{\rm new}(\xi^{\prime})$ defined in the above Lemma 3.5. Hence, $R_{\rm
new}(\xi^{\prime}(=(\xi,c))$ can be understood as the completion of a stalk of
$\overline{\Delta_{\xi}}(c,\pi,R)$ of Definition 2.3.

###  3.2. Higher $\Theta$-stable reduction theorem

Below, we generalize the semistable reduction type theorem in [AHLH23, §6.1],
which was for $r=1$ with particular emphasis on canonicity and uniqueness.
Even in that $r=1$ case, the following proof (for the quotient stack case)
seems to be different from loc.cit and more toric geometric.

Morally speaking, it says that a one parameter family which strictly
degenerates into a higher $\Theta$-strata, there is a canonical modification
of the family to avoid the higher $\Theta$-strata i.e., with “less
degeneration”. This is a version of stable reduction type theorem, which we
later refine in Theorem 3.18.

######  Theorem 3.8.

Let $(R,\mathfrak{m},v)$ be a valuation $\mathbbm{k}$-algebra and
$\mathcal{M}$ be an algebraic stack over $\mathbbm{k}$ with affine diagonal.

As in Notation 1, we set $T=\mathbb{G}_{m}\otimes_{\mathbb{Z}}N$, where $N$ is
a free abelian group of rank $r$, and fix a rational regular polyhedral cone
$\tau\subset N_{\mathbb{R}}$ and $(0\neq)\xi\in\tau$ which is irrational if
$r>1$.

Consider an arbitrary morphism
$f\colon\mathop{\mathrm{Spec}}\nolimits(R)\to\mathcal{M}$. Suppose that
$f(\mathfrak{m})$ sits in a higher $\Theta$-stratum $\mathcal{Z}^{+}$ in
$\mathcal{M}$ while $f(\eta_{R})$ is not in $\mathcal{Z}^{+}$, where
$\eta_{R}$ denotes the generic point of $R$. Now we fix an isomorphism
$\Theta^{r}\simeq\Theta_{\tau}$ for some rational polyhedral cone $\tau$ of a
lattice $N$ of rank $r$, and fix some $\xi\in\tau\setminus
N\otimes\mathbb{Q}$. Then, after

  * •

replacing $\tau$ by a smaller rational poyhedral cone of $N_{\mathbb{R}}$
which still contains $\xi$ (if necessary)

  * •

replacing $R$ by another valuation $\mathbbm{k}$-algebra with local
$\mathbbm{k}$-morphism $R\to R^{\prime}$, so that the valuation of
$R^{\prime}$ restricts to that of $R$.

  * •

taking a rational polyhedral cone $\tau^{\prime}$ of $N\oplus\mathbb{Z}$ which
projects as PL isomorphism (in particular, bijectively) to $\tau\subset
N_{\mathbb{R}}$, as in Definition 3.1,

we have the following canonical extension:

  1. (1)

$f$ extends to a unique (for fixed $\tau$) extends to its elementary
($\tau$-)modification $\overline{f}\colon{\overline{\rm
ST}}_{R}(\tau^{\prime})\to\mathcal{M}$ which satisfy the generic point
$\eta_{V(\tau^{\prime})}$ of $V(\tau^{\prime})$ maps to outside $Z^{+}$ and
there is a $T$-equivariant morphism from the N-spectrum
$\iota\colon\Delta_{-\xi}\to\mathcal{M}$ which satisfies that

     * •

$\overline{f}(\eta_{V(\tau^{\prime})})\notin Z^{+},$

     * •

$\iota(\eta_{-\xi})=\overline{f}(\eta_{V(\tau^{\prime})}),$ where
$\eta_{-\xi}$ denotes the generic point of the N-spectrum $\Delta_{-\xi}$,

     * •

$\iota(c_{-\xi})=\overline{f}(\eta_{V(\hat{\tau}^{\prime})})\in Z^{+}$, where
$c_{-\xi}$ denotes the closed point of the N-spectrum $\Delta_{-\xi}$.

  2. (2)

(Canonicity of degenerations) Such $\iota$ is unique, up to translation by
$T(\mathbbm{k})\simeq(\mathbbm{k}^{*})^{r}$.

  3. (3)

For any $\xi^{\prime\prime}\in(N\otimes\mathbb{Q})\cap\tau$, if we consider
the one parameter subgroup
$\lambda_{\xi^{\prime\prime}}\colon\mathbb{G}_{m}\to T$, then we have
$\lim_{t\to
0}\lambda_{\xi^{\prime\prime}}(t)\cdot\iota(\eta_{-\xi})=\iota(c_{-\xi}).$

######  Remark 3.9.

If $\mathcal{M}$ is a moduli stack of varieties and $\overline{f}$ lifts to a
morphism from $\tilde{\rm ST}_{R}(\tau^{\prime})$ to a chart $M$ of
$\mathcal{M}$, on which $T$ acts, its restrictions to each of the two strata
corresponding to $\tau^{\prime}$ and $\tau$ gives a generalized test
configurations in the sense of Definition 2.6.

######  Corollary 3.10.

In the setup of Theorem 3.8, we obtain a canonical modification
$f|_{\xi^{\prime}}\colon\overline{\Delta_{\xi}}(R^{\prime})\to\mathcal{M}$ for
$\xi^{\prime}=(\xi,c)\in\tau^{\prime}$ (cf., Definition 2.3) of
$f\colon\mathop{\mathrm{Spec}}\nolimits(R)\to\mathcal{M}$ such that

  * •

it extends $f|_{\rm
Spec(K)}\colon\mathop{\mathrm{Spec}}\nolimits(K)\to\mathcal{M}$,

  * •

the closed point $c_{\xi}$ maps to outside $\mathcal{Z}^{+}$,

  * •

$\lim_{t\to\infty}\lambda_{\xi^{\prime\prime}}(t)\cdot
f|_{\xi^{\prime}}(c_{\xi})\in\mathcal{Z}^{+}$ for any $\xi^{\prime\prime}\in
N\otimes\mathbb{Q}$ which approximates $\xi$ enough (as (3)).

In particular, we obtain a canonical modification $f_{{\rm
new},\xi}\colon\mathop{\mathrm{Spec}}\nolimits R_{\rm new}(\xi)\to\mathcal{M}$
which maps the closed point to outside $\mathcal{Z}^{+}$.

###### proof of Corollary 3.10 (assuming Theorem 3.8).

If we consider the direction along $\tau^{\prime}$, note that $\overline{f}$
restricts to $[(\mathop{\mathrm{Spec}}\nolimits R[(\tau^{\prime})^{\vee}\cap
M^{\prime}]/(\pi-p(\vec{0},1))/T]\to\mathcal{M}$ which maps the closed point
to outside $\mathcal{Z}^{+}$, where $M^{\prime}$ is the dual lattice of the
lattice $N^{\prime}$ which is the minimal sublattice of $N$ such that
$N^{\prime}_{\mathbb{R}}$ contains $\xi^{\prime}$, and
$(\tau^{\prime})^{\vee}$ denotes the dual cone. We denote the morphism as
$\overline{f}|_{\tau^{\prime}}$, for simplicity.

Then, if we take the projective limit with respect to $\tau^{\prime}$, and
take composite with the completion map (see Proposition 2.2)
$f|_{\xi^{\prime}}$ of Corollary 3.10 is obtained. ∎

###### proof of Theorem 3.8.

As [HL14, Lemma 4.4.6] (cf., also [AHHLR22]) shows, there is an affine
$\mathbbm{k}$-scheme $M$, algebraic $\mathbbm{k}$-split torus
$T=N\otimes\mathbb{G}_{m}$ of rank $r$ and a smooth surjective representable
morphism $[M/T]\to\mathcal{M}$ which induces ${\rm Filt}^{r}([M/T])\to{\rm
Filt}^{r}(\mathcal{M})$, at the level of $\mathbb{Z}^{r}$-filtration stack (as
algebraic stacks by [AHR20, 5.10, 5.11], [AHR19, 6.23]) for $r={\rm rank}(N)$
in particular. Hence we reduce the proof to the case when $\mathcal{M}$ is a
quotient stack of the form $M$ an algebraic scheme over $\mathbbm{k}$ acted by
an algebraic $\mathbbm{k}$-split torus $T$ of dimension $r$. We assume it
henceforth.

From the structure of higher $\Theta$-strata on $Z^{+}$, as we assume, there
is a $T$-equivariant morphism $Z^{+}\times U_{\tau}\to M$, following Notation
1, for some polyhedral cone $\tau$. Since there is also the (compatible)
$T$-action on $M$, they induce a $T$-equivariant rational map
$U_{\tau}\times\mathop{\mathrm{Spec}}\nolimits(R)\dashrightarrow M$. From the
condition that $Z^{+}$ gives a union of connected components in the mapping
stack, it follows that
$U_{\tau}\times\mathop{\mathrm{Spec}}\nolimits(R)\dashrightarrow M$ is indeed
not defined in the neighborhood of $(p_{\tau},\mathfrak{m})$, hence we want to
resolve the indeterminancy as follows.

######  Lemma 3.11.

If necessary, by replacing $\tau$ by a smaller rational polyhedral cone which
still contains $\xi$, there is a regular simplicial subdivision fan of finite
rational polyhedral subcones of $\tau\times\mathbb{R}_{\geq 0}$, which we
denote as $\Sigma^{\prime}$ such that the following holds:

  * •

the corresponding toroidal blow up of
$U_{\tau}\times\mathop{\mathrm{Spec}}\nolimits(R)$, which we denote as
$\mathcal{U}_{\Sigma^{\prime}}$, there is a $T$-equivariant morphism
$\tilde{f}\colon\mathcal{U}_{\Sigma^{\prime}}\to M$ which extends $\tilde{f}$.

  * •

There is some $c^{\prime}\in\mathbb{R}_{>0}$ such that $\mathbb{R}_{\geq
0}\xi^{\prime}\subset N_{\mathbb{R}}\oplus\mathbb{R}$ with
$\xi^{\prime}=(\xi,c^{\prime})$ lies inside a $r$-dimensional rational
polyhedral cone $\tau^{\prime}$ of $\Sigma^{\prime}$ and the convex hull
$\hat{\tau}^{\prime}:=\tau^{\prime}+\mathbb{R}_{\geq 0}(\vec{0},1)$ is a cone
inside $\Sigma^{\prime}$. Further, $\tilde{f}(V({\tau^{\prime}}),0)\notin
Z^{+}$, where $V(\tau^{\prime})$ denotes the strata of
$\mathcal{U}_{\Sigma^{\prime}}$ which corresponds to $\tau^{\prime}$ as in
usual toric geometry.

###### proof of Lemma 3.11.

We consider $T$-equivariant local coordinates $z_{1},\cdots,z_{N}$ around the
image of the $T$-invariant point $p_{\tau}\in U_{\sigma}$ and consider its
pullback (as rational functions) to
$U_{\tau}\times\mathop{\mathrm{Spec}}\nolimits(R)$ and denote them as
$\frac{f_{i}}{g_{i}}$ with holomorphic (algebraic) $f_{i}$s and $g_{i}$s with
$T$-semiinvariant $g_{i}$s. Consider the ideal
$\mathcal{J}:=\langle\\{f_{i},g_{i}\mid 1\leq i\leq N\\}\rangle$ which can be
decomposed as

$\mathcal{J}=\sum I_{\vec{m}}z^{\vec{m}},$

where $\vec{m}$ runs over elements of $\tau^{\vee}\cap M$, $z^{\vec{m}}$ means
$\prod_{i}z_{i}^{m_{i}}$, and $I_{\vec{m}}$ are ideals of $R$. With respect to
the natural partial order on $\tau^{\vee}\cap M$ as
$\vec{m}\leq\vec{m^{\prime}}:\Leftrightarrow\vec{m^{\prime}}-\vec{m}\in\tau^{\vee}\cap
M$, we have $I_{\vec{m}}\subset I_{\vec{m}}$ if $\vec{m}\leq\vec{m^{\prime}}$.
If we take a uniformizer of $R$ as $\pi$ and we can write $I_{\vec{m}}$ as
$(\pi^{l_{m}})$ since $R$ is a discrete valuation ring, with
$l_{m}\in\mathbb{Z}_{\geq 0}$. We have $l_{\vec{m}}\geq
l_{\vec{m^{\prime}}}\geq 0$ if $\vec{m}\leq\vec{m^{\prime}}$.

The blow up of $\mathcal{J}$ resolves the indeterminancy of the rational map
$U_{\tau}\times\mathop{\mathrm{Spec}}\nolimits R\to M$ from the definition,
and it is “toroidal” as there is some subdivision of the rational polyhedral
cone $\tau\oplus\mathbb{R}_{\geq 0}\subset N_{\mathbb{R}}\oplus\mathbb{R}$
which realizes it. We can and do assume henceforth that $\Sigma^{\prime}$ is a
regular simplicial fan, by taking subdivision each time if necessary.

Suppose that the rays of $\Sigma^{\prime}|_{\mathbb{R}_{\geq
0}\xi+\mathbb{R}_{\geq 0}(\vec{0},1)}$ are spanned by $(\xi,0)$ and
$\\{(\xi,c_{i})\mid i=1,2,\cdots,l\\}$ with $c_{1}>c_{2}>\cdots>c_{l}$ and
denote the $r$-dimensional cone which includes $(\xi,c_{i})$ as
$\tau^{\prime}_{i}$ and which includes $(\xi,0)$ as
$\tau_{l+1}=\tau^{\prime}_{l+1}$. By further subdividing $\Sigma^{\prime}$ if
necessary, we can and do assume that all $\tau^{\prime}_{i}$ projects down to
the same cone $\tau$ in $N_{\mathbb{R}}$. By shrinking $\tau$ and subdividing
$\Sigma^{\prime}$ further if necessary, we can and do assume that
$\\{\tau^{\prime}_{i}\\}_{i}\cup\\{\tau\times\\{0\\}\\}$ are the only
$r$-dimensional cones (walls) in $\Sigma^{\prime}$.

Now, we take a cone of $\Sigma^{\prime}$ which includes
$(\xi,[c_{i}-\epsilon,c_{i}])$ for $0<\epsilon\ll 1$ as $\tilde{\tau_{i}}$ for
each $i=1,\cdots,l$. If $\tilde{f}(V(\tilde{\tau_{l}}))\in Z^{+}$, then from
the assumption that $\mathcal{Z}^{+}$ gives a union of connected components in
${\rm Map}(\Theta_{\tau},\mathcal{M})$, it follows that
$\tilde{f}(V(\tau_{l+1})\times\mathop{\mathrm{Spec}}\nolimits(R))\in Z^{+}$ so
that $f(\eta_{R})\in Z^{+}$ which gives a contradiction.

Therefore, there is some $i\in\\{1,\cdots,l\\}$ such that

  * •

$\tilde{f}(V(\tau_{i}))\notin Z^{+}$,

  * •

For any cone $\sigma$ of $\Sigma^{\prime}$ which intersects with
$(\xi,c_{i}+\mathbb{R}_{>0})$, the corresponding strata $V(\sigma)$ of
$\mathcal{U}_{\Sigma^{\prime}}$ maps into $Z^{+}$ i.e.,
$\tilde{f}(V(\sigma))\subset Z^{+}.$

Suppose $i>1$, we do some modification as follows. We take any
$j\in\\{1,\cdots,i-1\\}$. If
$\tilde{f}(V(\tau^{\prime}_{j}))\neq\tilde{f}(V(\tau^{\prime}_{j}))$, then
take the cone $\tau^{\prime}_{j,j+1}$ of $\Sigma^{\prime}$ which contains
$\tau^{\prime}_{j}+\tau^{\prime}_{j+1}$ (the Minkowski sum) consider a point
$z\in\tilde{f}(V(\tau^{\prime}_{j,j+1}))\subset Z^{+}$. Then the existence of
two degenerating families from $z$ via $T$-action along $U_{\tau}$ and
$U_{-\tau}$ respectively, to each $T$-invariant point, contradicts again to
the higher $\Theta$-strata condition of $\mathcal{Z}^{+}$. Indeed, as both
families degenerate to the isotrivial families of $\tilde{f}(V(\tau_{j-1}))$
so that it contradicts with the condition of higher $\Theta$-strata.
Therefore, $\tilde{f}(V(\tau^{\prime}_{j}))=\tilde{f}(V(\tau^{\prime}_{j}))$
for any $j\in\\{1,\cdots,i-1\\}$. Therefore, if we do coarsening of
$\Sigma^{\prime}$ to remove the walls
$\tau^{\prime}_{1},\cdots,\tau^{\prime}_{i-1}$, the morphism
$\mathcal{U}_{\Sigma^{\prime}}\to M$ still descend as a morphism. Therefore,
we can and do assume $i=1$ henceforth and simply write $c_{1}$ as $c^{\prime}$
and $\tau^{\prime}_{1}$ as $\tau^{\prime}$. We complete the proof of Lemma
3.11. ∎

We take a regular simplicial subcone $\sum_{i=1}^{r}\mathbb{R}_{\geq
0}v^{\prime}_{i}$ of $\tau^{\prime}$, where
$\\{v^{\prime}_{i}\\}_{i}\cup(\vec{0},1)$ is a basis of of $N$, which still
contains $\xi^{\prime}$. We also take the cone which contains
$(\xi,[c^{\prime}-\epsilon,c^{\prime}])$ for $0<\epsilon\ll 1$ and denote it
as $\widetilde{\tau}^{\prime}$. Write $v^{\prime}_{i}=(v_{i},\frac{1}{m_{i}})$
with $m_{i}\in\mathbb{Z}_{>0}$ and set $m:={\rm lcm}\\{m_{i}\\}$.

Take an extension of discrete valuation ring $R\to R^{\prime}$ with
ramification degree $m$, which is obtained as a localization of
$R[x]/(x^{m}-\pi)$ at $(x)$. Note that
$\\{\frac{m_{i}v^{\prime}_{i}}{m}\\}_{i}$s together with
$(\vec{0},\frac{1}{m})$ gives a basis of a regular cone of
$N\oplus\frac{1}{m}\mathbb{Z}$. From the construction, (base change of)
$\tilde{f}$ is extendable to $\widetilde{\rm
ST}_{R^{\prime}}(\tau^{\prime})\to M$ and ${\overline{\rm
ST}}_{R^{\prime}}(\tau^{\prime})\to\mathcal{M}$ and the desired properties are
satisfied from the construction and finish the proof of (1). Note that the
extension $R\to R^{\prime}$ does not change when we shrink $\tau^{\prime}$, as
it follows from the construction.

Now we show the uniqueness assertion (2). First, note that for each positive
real number $c^{\prime\prime}$, if $(\xi,c^{\prime\prime})$ is contained in a
rational hypersurface, then it is defined only by $c^{\prime\prime}$ because
$\xi$ is assumed to be irrational if $r>1$. We recall that from the proof of
Lemma 3.11, we have $\tilde{f}(\eta_{V(\tau^{\prime})})\notin Z^{+}$ and
$\lim_{t\in T,t\to p_{-\tau}}\tilde{f}(\eta_{V(\tau^{\prime}+\mathbb{R}_{\geq
0}(\vec{0},1))})\in Z^{+}$, where $p_{-\tau}$ denotes the $T$-invariant point
of affine toric variety $U_{-\tau}$ over $R/\mathfrak{m}$. Here the meaning of
$\lim$ is naturally justified by the extension of the $T$-action to the
N-spectrum $\Delta_{-\xi}$ and leave the detailed definition to the readers.
From these conditions, $c^{\prime}$ is determined canonically. Indeed, in case
$c^{\prime\prime}>c^{\prime}$ and a rational polyhedral cone
$\tau^{\prime\prime}$ which contains $(\xi,c^{\prime\prime})$, we would have
$\tilde{f}(\eta_{V(\tau^{\prime\prime})})\in Z^{+}$ and in case
$c^{\prime\prime}<c^{\prime}$ then $\lim_{t\in T,t\to
p_{-\tau}}\tilde{f}(\eta_{V(\tau^{\prime\prime}+\mathbb{R}_{\geq
0}(\vec{0},1))})\notin Z^{+}$. Indeed, the latter limit is $\lim_{t\in T,t\to
p_{-\tau}}\tilde{f}(\eta_{V(\tau^{\prime\prime}+\mathbb{R}_{\geq
0}(\vec{0},1))})=\tilde{f}(V(\hat{\tau}^{\prime\prime}))$, where
$\hat{\tau}^{\prime\prime}$ is the cone of $\Sigma^{\prime}$ which contains
$(\xi,(c^{\prime\prime},c^{\prime\prime}+\epsilon))$. However
$\tilde{f}(V(\hat{\tau}^{\prime\prime}))$ is not contained in $Z^{+}$ from the
higher $\Theta$-strata condition. Hence we complete the proof of the
uniqueness assertion. ∎

The following is a corollary to the above proof of Theorem 3.8. For that, we
prepare the following, to connect to the original $\Theta$-strata ([AHLH23]).

######  Definition 3.12.

Consider $\xi\in\tau\subset N_{\mathbb{R}}$, following Notation 1, but we take
$\xi$ to be integral i.e., $\xi\in N\cap\tau$ here. If we set
$\iota\colon\mathbb{Z}\xi\hookrightarrow N$ the natural inclusion, note that
there is a $({\rm exp}\otimes\iota)\colon\mathbb{C}\xi\to T(\mathbb{C})$
(compare Definition 2.4).

For a higher $\Theta$-stratum of the cone $\tau$ on an algebraic stack
$\mathcal{M}$, which we denote as $\mathcal{Z}^{+}\subset{\rm
Map}(\Theta_{\tau},\mathcal{M})$, consider restriction map $({\rm
exp}\otimes\iota)^{*}\colon\mathcal{Z}\to{\rm Map}(\Theta_{\tau},\mathcal{M})$
and suppose it gives the $\Theta$-strata structure for each $\xi$ i.e., the
images are connected. 444at the moment, we are not sure how weak or strong
this condition is Then we call the obtained $\Theta$-strata as restriction
along $\xi$ of the $\Theta$-strata $\mathcal{Z}$ and often denote as
$\mathcal{Z}_{\xi}$.

######  Corollary 3.13 (of proof of Theorem 3.8).

For an algebraic stack $\mathcal{M}$ with affine diagonal. We take a higher
$\Theta$-stratum of the cone $\tau$ on an algebraic stack $\mathcal{M}$, which
we denote as $\mathcal{Z}^{+}$ which has restrictions $\mathcal{Z}^{+}$ along
each $\xi\in N\cap\tau$. As in Theorem 3.8 or [AHLH23, §6], take an arbitrary
morphism $f\colon\mathop{\mathrm{Spec}}\nolimits(R)\to\mathcal{M}$ where $R$
is a valuation $\mathbbm{k}$-algebra. Suppose that $f(\mathfrak{m})$ sits in a
higher $\Theta$-stratum $\mathcal{Z}^{+}$ in $\mathcal{M}$ while $f(\eta_{R})$
is not in $\mathcal{Z}^{+}$.

Then, the limit of the obtained $\Theta$-stable reduction $f|_{{\rm
new},\xi}(0)$ by [AHLH23, Theorem 6.3] or Corollary 3.10 (and the one
parameter family obtained $\mathcal{T}^{\prime\prime}$ in the proof of Theorem
3.8) depends only on the strata of the cone $\tau$ where $\xi$ sits in.

It often occurs in various moduli problems that a priori we only have a
uniform degeneration along $\xi$-direction but not along a higher dimensional
toric variety $U_{\tau}$. Nevertheless, the following lemma builts the
connection, following Proposition 2.2.

######  Lemma 3.14 (Limit mapping stack).

For an algebraic stack $\mathcal{M}$ over $\mathbbm{k}$ with affine diagonal,
following Notation 1, take irrational $\xi\in N_{\mathbb{R}}$, set
$\Gamma_{\geq 0}:=\langle M_{\geq 0},\xi\rangle\subset\mathbb{R}$, and
consider the global N-spectrum $\Delta_{\xi}^{\rm gl}=\Delta_{\xi}^{\rm
gl}(\mathbbm{k})$ as in Definition 2.1. Note that the algebraic torus
$T:=\mathbb{G}_{m}\otimes N$ naturally acts on $\Delta_{\xi}$.

Consider various rational polyhedral cone $\tau$ which contains $\xi$. The
mapping stack from the quotient stack (a priori, fibered only over the étale
site of $\mathbbm{k}$-schemes which are not necessarily finite type)
$[\Delta_{\xi}^{\rm gl}/T]$ is a locally finite type algebraic stack over
$\mathbbm{k}$ and has a description as follows.

(3) $\displaystyle{\rm Filt}_{\xi}(\mathcal{X}):$ $\displaystyle={\rm
Map}(\Theta_{\xi}=[\Delta_{\xi}^{\rm gl}/T],\mathcal{X})$ (4)
$\displaystyle=\varinjlim_{\tau}{\rm Map}([\Theta_{\tau}/T],\mathcal{X}).$

Here, the injective system is given by the inclusion of the cones
$\tau^{\prime}\supset\tau$ which induces the pull back morphism
$\iota_{\tau,\tau^{\prime}}^{*}\colon{\rm
Map}([\Theta_{\tau}^{\prime}/T],\mathcal{X})\to{\rm
Map}([\Theta_{\tau}/T],\mathcal{X})$ through the morphism
$\iota_{\tau,\tau^{\prime}}\colon U_{\tau}\to U_{\tau^{\prime}}$. The obtained
restriction morphism $\iota_{\tau,\tau^{\prime}}^{*}$ is an open immersion of
algebraic stacks so that we have the injective limit and the identification
(4) makes sense.

What we use later in the sequel papers [Od24a, Od24c] more efficiently is the
following proof, rather than only the statements themselves.

###### Proof.

Note that ${\rm Filt}_{\xi}(\mathcal{X})$ is a priori only defined as a
$2$-functor. Consider any base $\mathbbm{k}$-scheme $B^{\prime}$, on which $T$
acts, and take an object of the groupoid ${\rm Filt}_{\xi}(\mathcal{X})(B)$,
and represent it as a $T$-equivariant morphism $\varphi\colon B\to\mathcal{X}$
after passing to an étale covering $B\to B^{\prime}$. We can and do assume
that $B$ is an affine scheme $\mathop{\mathrm{Spec}}\nolimits(R)$ for a
$\mathbbm{k}$-algebra $R$ and the map is represented by that of $B$ to a chart
of $\mathcal{X}$, which we write $\mathop{\mathrm{Spec}}\nolimits(S)$, for
simplicity. It corresponds to a ring homomorphism
$S\to\mathbbm{k}[\Gamma_{\geq 0}]$. Now we exploit the similar discussions to
Proposition 2.2. Since $S$ is finitely generated $\mathbbm{k}$-algebra, there
is a finitely generated saturated submonoid $\Gamma^{\prime}_{\geq
0}\subset\Gamma_{\geq 0}$ so that the image of $S$ lies inside
$\mathbbm{k}[\Gamma^{\prime}_{\geq 0}]$. This shows that

$\varinjlim_{\tau}{\rm Map}([\Theta_{\tau}/T],\mathcal{X})(B)\to{\rm
Map}([\Delta_{\xi}/T],\mathcal{X})(B)$

is surjective for each $k$-scheme $B$.

What remains is to show that $\iota_{\tau,\tau^{\prime}}^{*}\colon{\rm
Map}([\Theta_{\tau}^{\prime}/T],\mathcal{X})\to{\rm
Map}([\Theta_{\tau}/T],\mathcal{X})$ is an open immersion. For that take
$T$-equivariant $B\times U_{\tau}\to\mathop{\mathrm{Spec}}\nolimits(S)$ for
$B=\mathop{\mathrm{Spec}}\nolimits(R)$ and consider for which
$\mathop{\mathrm{Spec}}\nolimits(\mathbbm{k}^{\prime})=b\in B$, it extends to
$U_{\tau}^{\prime}\times b\to\mathop{\mathrm{Spec}}\nolimits(S)$. We consider
the corresponding $\mathbbm{k}$-algebra homomorphism $f\colon S\to
B[\tau^{\vee}\cap M]$. Take a finite generator $s_{1},\cdots,s_{m}$ of $S$ as
$\mathbbm{k}$-algebra and consider all the coefficients $r_{i,\vec{m}}$ of
$f(s_{i})$ of the monomial terms which correspond to
$\vec{m}\in\Gamma^{\prime}\setminus\Gamma_{\geq 0}$. Then it is easy to see
that $U_{\tau}\times b\to\mathop{\mathrm{Spec}}\nolimits(S)$ extends to
$U_{\tau}^{\prime}\times b\to\mathop{\mathrm{Spec}}\nolimits(S)$ if and only
if $r_{i,\vec{m}}(b)=0$ for all $i,\vec{m}$. Hence we complete the proof. ∎

An analytic variant of Theorem 3.8 is as follows.

######  Theorem 3.15 (Analytic analogue of Theorem 3.8).

Let $\Delta:=\\{t\in\mathbb{C}\mid|t|<1\\}$ and $M$ a complex variety acted
algebraically by $T(\mathbb{C})^{\rm an}\simeq(\mathbb{C}^{*})^{r}$ i.e., it
is the complex analytification of an algebraic $T$-variety.

Consider an arbitrary holomorphic map $\tilde{f}\colon\Delta\to M$ induces
$f\colon\Delta\to\mathcal{M}:=[M/T(\mathbb{C})^{\rm an}]$ whose target is
regarded as a complex analytic stack. Suppose that for a (algebraic) higher
$\Theta$-stratum $\mathcal{Z}^{+}$ with ${\rm
ev}_{(1,\cdots,1)}(\mathcal{Z}^{+})=Z^{+}\subset M$,
$\tilde{f}^{-1}(Z^{+})\subset\Delta$ is set theoritically
$\\{0\\}\subset\Delta$.

Then, the following claims hold.

  1. (1)

We can modify $f$ to avoid the higher $\Theta$-stratum in the following sense.
If we choose an isomorphism $\Theta^{r}\simeq\Theta_{\tau}$ for some rational
regular polyhedral cone $\tau$ of a lattice $N$ of rank $r$, and fix an
irrational $\xi\in\tau\setminus N_{\mathbb{Q}}$, then the map $\overline{f}$
can be taken canonically.

     * •

For some $m\in\mathbb{Z}_{>0}$, we replace $\Delta$ by
$\Delta^{\prime}:=\\{s\in\mathbb{C}\mid|s|<1\\}$ with
$\Delta^{\prime}\twoheadrightarrow\Delta$ which maps $s$ to $t=s^{m}$,

     * •

replace $\tau$ by a smaller rational poyhedral cone of $N_{\mathbb{R}}$ (if
necessary)

     * •

take a rational polyhedral cone $\tau^{\prime}$ of $N\oplus\mathbb{Z}$ which
projects as PL isomorphism (in particular, bijectively) to $\tau\subset
N_{\mathbb{R}}$, as in Definition 3.1,

so that the following hold:

> $\tilde{f}$ extends to a $T(\mathbb{C})^{\rm an}$-equivariant morphism
> ${\widetilde{\rm ST}}^{\rm an}_{\Delta^{\prime}}(\tau^{\prime})\to M$, hence
> extends $f$ to $\overline{f}\colon{\overline{\rm ST}}^{\rm
> an}_{\Delta^{\prime}}(\tau^{\prime})\to\mathcal{M}$, and maps
> $V(\tau^{\prime})$ outside $Z^{+}$ and there is a holomorphic map
> $\iota\colon\tilde{\mathbb{C}}^{*}\to M(\mathbb{C})$ which is equivariant
> with respect to the complex Lie group homomorphism
> $\tilde{R}_{-\xi}\colon\tilde{\mathbb{C}}^{*}\to T(\mathbb{C})$ (Definition
> 2.4) and satisfies that
>
>     1. (A)
>
> $T(\mathbb{C})\cdot{\rm Im}(\iota)\supset\tilde{f}(V(\tau^{\prime})),$ which
> is disjoint from $Z^{+}$,
>
>     2. (B)
>
> there is a sequence $t_{i}(i=1,2,\cdots)$ in $\mathbb{R}$ with $t_{i}\to
> 0(i\to\infty)$ such that $\lim_{i\to\infty}\iota(t_{i})\in Z^{+}.$ (Actually
> the above limit is $\tilde{f}(p_{\hat{\tau}^{\prime}})$ automatically.)
>
>     3. (C)
>
> Each point of ${\rm Im}(\iota)$ is obtained as
> $\lim_{i\to\infty}t(i)\tilde{f}(s_{i})$ where $t(i)\in T(\mathbb{C})$,
> $s_{i}(\in\tilde{\mathbb{C}}^{*})$, $s_{i}\to 0$ for $i\to\infty$.

  2. (2)

Such $\iota$ which satisfies (1A), (1B), (1C) is unique, up to a translation
by the action of $T(\mathbb{C})$,

  3. (3)

In particular, as the projective limit of $\overline{f}|_{V(\tau^{\prime})}$,
we obtain a canonical map

(5) $\displaystyle f_{\rm new}\colon\Delta_{\xi^{\prime}}^{\rm
conv}(\mathbb{C})\to\mathcal{M}$

for some $\xi^{\prime}=(\xi,c)$ with $c\in\mathbb{R}_{>0}$, which we denote by
$f_{\rm new}$. Note $f_{\rm new}$ maps the closed point (origin) outside
$Z^{+}$. This is the desired canonical modification of $f$.

###### Proof.

Since the proof is along the same line as that of Theorem 3.8, we only note
the technical differences and the way to overcome them. The assumption that
$T(\mathbb{C})\curvearrowright M$ is algebraic implies that
$U_{\tau}\otimes\Delta\dashrightarrow M$ is meromorphic hence the
$T$-invariant toroidal resolution of indeterminancy exists similarly as Lemma
3.11. If we apply the same arguments as Theorem 3.8 for
$R:=\mathbb{C}[[t]]^{\rm conv}=\mathcal{O}^{\rm an}_{\mathbb{C},0}$ (the
convergent power series ring), the rest of the arguments is almost the same
except for following points:

  1. (a)

$\iota$ is obtained as a morphism from $\tilde{\mathbb{C}^{*}}$, rather than
the N-spectrum algebraically.

  2. (b)

the second item (1A) of Theorem 3.15 is slightly weaker than the algebraic
version in Theorem 3.8 in that it only concerns a sequence of real numbers
$\\{t_{i}\\}_{i}$. Thus the uniqueness statement (2) is a priori stronger.

Now we make comments to overcome these differences to complete the proof.
Firstly, the difference (a) does not become a problem as we can simply take a
(closed) point in $\tilde{f}(V(\hat{\tau}))$, consider the
$T(\mathbb{C})$-action on it, take limit along $U_{-\tau}(\mathbb{C})$ to the
$T(\mathbb{C})$-invariant point $p_{-\tau}$ and combine with
$\tilde{R}_{-\xi}$.

For the difference (b) can be overcome as follows. The main points are that
$\mathbb{C}^{*}$ acts continuously on $M(\mathbb{C})$ with respect to their
complex analytic topolgies, so that any orbit of $U(1)\otimes N$ in
$M(\mathbb{C})^{\rm an}$ is compact. First we take $c^{\prime}$ and
$\tau^{\prime}$ as Lemma 3.11. From the first condition of item (1B), there is
some $c_{j}(j\in\\{1,\cdots,l\\})$ in the proof of Theorem 3.8 (before Lemma
3.11) so that the limit $\lim_{i\to\infty}\iota(t_{i})$ in the second item of
(1B) is $\tilde{f}(V(\sigma_{j}))$ for the $r+1$-dimensional cone $\sigma_{j}$
of $\Sigma^{\prime}$ which intersects with $(\xi,(c_{j},c_{j}+\epsilon))$ for
$0<\epsilon\ll 1$.

In case $c_{j}>c^{\prime}$ and a rational polyhedral cone $\tau^{\prime}_{j}$
which contains $(\xi,c_{j})$, we would have
$\tilde{f}(\eta_{V(\tau^{\prime}_{j})})\in Z^{+}$ and in case
$c_{j}<c^{\prime}$ then $\lim_{t\in T(\mathbb{C}),t\to
p_{-\tau}}t\cdot\tilde{f}(x)=\tilde{f}(V(\sigma_{j}))\notin Z^{+}$, for any
$x\in V(\tau^{\prime}+\mathbb{R}_{\geq 0}(\vec{0},1))$. Since any orbit of
$U(1)\otimes N$ in $M(\mathbb{C})^{\rm an}$ is compact, we have
$\lim_{i\to\infty}\iota(t_{i})=\lim_{t\in T(\mathbb{C}),t\to
p_{-\tau}}t\cdot\tilde{f}(x)=\tilde{f}(V(\sigma_{j}))$ for any $x\in
V(\tau^{\prime}+\mathbb{R}_{\geq 0}(\vec{0},1))$. Therefore, our $c^{\prime}$
coincides with that of Lemma 3.11. Hence the rest of the proof works similarly
as Theorem 3.8. ∎

######  Remark 3.16.

If we apply Theorem 3.8 with convergent N-ring $R=\mathbb{C}[[\Gamma_{\geq
0}]]^{\rm conv}$ of Definition 2.1 (ii), and use Lemma 2.5, we reprove Theorem
3.10 and even extends to general submonoid $\Gamma_{\geq 0}$. We wrote Theorem
3.8 as a special (simpler) case for simplicity.

###  3.3. Stable reduction along higher $\Theta$-stratification

We use the following setup, which is a straightforward extension of the notion
of $\Theta$-stratification of [HL14].

######  Definition 3.17 (cf., [HL14, §2.1]).

For an algebraic $\mathbbm{k}$-stack $\mathcal{M}$ with affine diagonal,
consider the mapping stack ${\rm Map}(\Theta_{\tau},\mathcal{M})$ of
$\Theta_{\tau}$ to $\mathcal{M}$ with various $N$ and $\tau$, as algebraic
$\mathbbm{k}$-stacks, after [AHR20, 5.10, 5.11], [AHR19, 6.23]. Extending
[HL14, Definition 2.1.2], a higher $\Theta$-stratification in $\mathcal{M}$
consists of

  1. (1)

a totally ordered 555our order is converted from the notation in [HL14] for
convenience to study problems related to K-stability, as in [BHLLX21] set
$\Gamma$, which has a maximal element $1$,

  2. (2)

closed666it is a priori not necessarily reduced, although in our stable
reduction type theorem, the scheme structure is not really important substacks
$\mathcal{M}_{<c}$ of $\mathcal{M}$ for $c\in\Gamma$ which satisfies that

     * •

they are monotonely increasing in the sense that we have
$(\mathcal{M}_{<c})_{\rm red}\supset(\mathcal{M}_{<c^{\prime}})_{\rm red}$ if
$c>c^{\prime}$ in $\Gamma$ (here, $(\mathcal{M}_{<c})_{\rm red}$ denotes the
natural reduced subscheme structure on the same support),

     * •

$\bigcap_{c\in\Gamma}\mathcal{M}_{\geq c^{\prime}}=\emptyset$.

  3. (3)

for each $c\in\Gamma$, higher $\Theta$-strata structure of type $\tau$
(Definition 3.3) for some $N,\tau$, on $\mathcal{M}_{=c}$ (we do not require
to identify $N,\tau$ for different $c$)

  4. (4)

for any point $p\in|\mathcal{M}|$, $\\{c\in\Gamma\mid
p\notin|\mathcal{M}_{<c}|\\}$ has a maximal element (the corresponding
condition, for $r=1$ case, is assumed in [HL14] while it is called well-
ordered in [BHLLX21].)

Following [AHLH23, §6.2], we have the corresponding generalization as below.

######  Theorem 3.18.

Following the notation of Definition 3.17, we take an algebraic stack
$\mathcal{M}$ of finite type with affine diagonal over $\mathbbm{k}$, and its
higher $\Theta$-stratification $\\{\mathcal{M}_{<c}\\}_{c}$ as Definition
3.17.

Consider any $\mathbbm{k}$-morphism
$f\colon\mathop{\mathrm{Spec}}\nolimits(R)\to\mathcal{M}$ for a valuation
$\mathbbm{k}$-ring $(R,\mathfrak{m})$ with its fraction field $K$, which
satisfies that $f(\mathop{\mathrm{Spec}}\nolimits(K))\in\mathcal{M}_{=1}$
while $f(\mathfrak{m})\in\mathcal{M}_{<1}$. Then, there is a finite sequence
of modifications of $f$ and $R$ which we denote as
$f_{i}\colon\mathop{\mathrm{Spec}}\nolimits(R_{i})\to\mathcal{M}(i=0,1,\cdots,l)$
with $R\to R_{i}$ such that

  * •

$R_{i}$ is a valuation ring with injective local morphisms $\mathbbm{k}\to
R\to R_{i}$ (we denote the fraction field of $R_{i}$ as $K_{i}$ and the
maximal ideal as $\mathfrak{m}_{i}$),

  * •

$f=f_{0}$,

  * •

$f|_{\mathop{\mathrm{Spec}}\nolimits(K_{i})}=f_{i}|_{\mathop{\mathrm{Spec}}\nolimits(K_{i})}$
for each $i$,

  * •

$f_{i}(\mathfrak{m}_{i})\in\mathcal{M}_{=c_{i}}$ with
$c_{0}<c_{1}<\cdots<c_{l}=1$,

  * •

in particular, $f_{l}((\mathfrak{m}_{l})\in\mathcal{M}_{=1}$.

For each $f_{i}$, if we consider the source stack $\Theta_{i}\simeq\Theta^{r}$
and fix an isomorphism to $\Theta_{\tau_{i}}$ for some $N_{i},\tau_{i}$,
together with choices $\xi_{i}\in\tau_{i}\setminus N_{\mathbb{Q}}$. Then the
above sequence of modifications $f_{i}$ can be taken canonically.

###### Proof.

We apply Theorem 3.8 to $f$ with the higher $\Theta$-stratification on
$\mathcal{M}_{={c_{0}}}$ and take the restriction of the obtained extension
$\overline{f}$ to the closed substack corresponding to a ray $\mathbb{R}_{\geq
0}v$ in $v\in N_{\mathbb{Q}}\cap\tau^{\prime}$ in the proof of Theorem 3.8,
and denote it by $f_{1}$. Then, we continue the process. Then, from the
condition (4), it stops after finite times so that we are done. The last
statement about the canonical choices of $f_{i}$ follows also from Theorem
3.8. We can characterize such $f_{i}$ by the natural condition in terms of
$\xi_{i}$ by Theorem 3.8 but we avoid the lengthy repetition. ∎

Note that from the proof and canonicity statement of Theorem 3.8, $f_{i}$ are
taken almost canonically in the sense that $f_{i}(\mathfrak{m})$ are unique,
and have certain characterizations.

#### Acknowledgement

The author thanks R.Conlon, E.Inoue, M.Jonsson for the comments on the
manuscript. During the research, the author was partially supported by Grant-
in-Aid for Scientific Research (B) 23H01069, 21H00973, Grant-in-Aid for
Scientific Research (A) 21H04429, 20H00112, and Fund for the Promotion of
Joint International Research (Fostering Joint International Research)
23KK0249. Part of this work is done during the author’s stay at Institut de
mathématiques de Jussieu - Paris Rive Gauche. We appreciate the kind
hospitality.

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