---
title: Quasi-monomial Valuations in Birational Geometry
url: https://www.emergentmind.com/topics/quasi-monomial-valuations
type: topic
---

# Quasi-monomial Valuations in Birational Geometry

Quasi-monomial valuations are real valuations on function fields that become monomial after passing to a suitable log-smooth birational model. In local coordinates adapted to a simple normal crossings divisor, they are determined by a weight vector and evaluate a function by taking the minimum weighted exponent occurring in its admissible expansion. This class of valuations occupies a central position in birational geometry, asymptotic invariants of linear series, normalized volume theory, and the algebraic theory of K-stability; it also admits a polyhedral and tropical interpretation through dual complexes and skeleta of valuation spaces [1011.3699] [1907.01114] [2208.06237].

## 1. Definition and local structure

Let $X$ be a normal variety with function field $K(X)$. A real valuation is a map
\[
\nu:K(X)^*\to\mathbb R
\]
satisfying
\[
\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.
\]
It is centered on $X$ if its valuation ring contains an affine chart of $X$ [2604.18465].

A divisorial valuation is a rank-one valuation of the form $c\cdot \operatorname{ord}_E$ for a prime divisor $E$ on some birational model. A quasi-monomial valuation is defined on a log-smooth model $(Y,E=\sum E_i)$ by choosing the generic point $\eta$ of a stratum, local parameters $y_1,\dots,y_r$ cutting out the components through $\eta$, and a weight vector $\alpha=(\alpha_1,\dots,\alpha_r)\in \mathbb R_{\ge 0}^r\setminus\{0\}$. If
\[
f=\sum_{\beta\in \mathbb Z_{\ge 0}^r} c_\beta\, y^\beta,
\]
then one sets
\[
v_\alpha(f)=\min\bigl\{\langle \alpha,\beta\rangle \mid c_\beta\neq 0\bigr\}.
\]
This gives a valuation that is literally monomial in the chosen coordinates [1907.01114].

In the rank-one framework, quasi-monomial valuations are equivalently characterized by the Zariski–Abhyankar condition
\[
\operatorname{trdeg}(v)+\operatorname{rank}_{\mathbb Q}(v)=\dim X,
\]
and, in the regular excellent setting, an Abhyankar valuation is quasi-monomial [1907.01114] [1011.3699]. The divisor case is the special case $r=1$; higher rational rank corresponds to monomiality along several components simultaneously.

For projective klt pairs $(X,\Delta)$, log discrepancy is defined for a prime divisor $E$ over $X$ by
\[
A_{X,\Delta}(E):=\operatorname{mult}_E(K_{X'}-f^*(K_X+\Delta))+1>0,
\]
and the theory extends from divisorial valuations to arbitrary real valuations through the valuative log discrepancy function $A_{X,\Delta}(-)$ [2604.18465] [1907.01114].

## 2. Quasi-monomial valuations inside valuation spaces

Quasi-monomial valuations coming from a fixed log-smooth pair $(Y,D)$ form a simplicial cone complex
\[
\operatorname{QM}(Y,D),
\]
with cones indexed by strata of $D$ and coordinates given by the values on irreducible components of the boundary. On each cone, the map
\[
v\mapsto (v(D_1),\dots,v(D_r))
\]
is a homeomorphism, and the cone complex carries a natural $\mathbb Z$-affine structure [1011.3699].

A basic construction is the retraction
\[
r_{Y,D}:\operatorname{Val}_X\to \operatorname{QM}(Y,D),
\]
which sends an arbitrary valuation to the unique monomial valuation agreeing on the boundary divisors. One has
\[
r_{Y,D}(v)\le v, \qquad A(r_{Y,D}(v))\le A(v),
\]
with equality if and only if $v\in \operatorname{QM}(Y,D)$. As $(Y,D)$ ranges over log-smooth models, the full valuation space is recovered as a projective limit:
\[
\operatorname{Val}_X \cong \varprojlim_{(Y,D)} \operatorname{QM}(Y,D).
\]
Moreover, each $\operatorname{QM}(Y,D)$ is dense in $\operatorname{Val}_X$, so quasi-monomial valuations are dense in the ambient valuation space [1011.3699].

This polyhedral picture is also implicit in more recent existence arguments. In the proof of the quasi-monomiality of the $\alpha$- and $\delta$-invariants, Xu’s retraction onto the dual complex is used after passing to a simultaneous fiber-wise log resolution, and this retraction is the step that converts a sequence of divisorial approximants into quasi-monomial valuations on a fixed simplicial cone [2604.18465].

A plausible implication is that quasi-monomial valuations are not merely a convenient subclass but the natural skeleta of the valuation space: they are the points on which birational, polyhedral, and asymptotic structures can be organized simultaneously.

## 3. Asymptotic thresholds and valuation-theoretic minimizers

The modern role of quasi-monomial valuations is anchored in asymptotic invariants. For a graded sequence of ideals $a_\bullet$, Xu proved a version of the Jonsson–Mustaţă conjecture: if
\[
\operatorname{lct}(X,\Delta;a_\bullet)<+\infty,
\]
then there exists a quasi-monomial valuation $v$ computing the infimum
\[
\frac{A_{X,\Delta}(v)}{v(a_\bullet)}
=
\inf_{w\in \operatorname{Val}_X}\frac{A_{X,\Delta}(w)}{w(a_\bullet)}.
\]
This establishes quasi-monomiality for minimizers of log canonical threshold type invariants for graded sequences [1907.01114].

For a projective klt pair $(X,\Delta)$ and an ample $\mathbb Q$-Cartier $\mathbb Q$-divisor $L$, the $\alpha$-invariant and $\delta$-invariant are defined by
\[
\alpha(X,\Delta,L)=\inf_{0\le D\sim_{\mathbb Q}L}\operatorname{lct}(X,\Delta;D),
\]
and
\[
\delta_m(X,\Delta,L)=\inf\{\operatorname{lct}(X,\Delta;D_m)\},\qquad
\delta(X,\Delta,L)=\inf_{m\gg1}\delta_m(X,\Delta,L)=\lim_{m\to\infty}\delta_m.
\]
Kim proved that there exist quasi-monomial valuations $v_\alpha$ and $v_\delta$ in $\operatorname{Val}_X^*$ such that
\[
v_\alpha(L)=\alpha(X,\Delta,L), \qquad v_\delta(L)=\delta(X,\Delta,L),
\]
independently of whether the base field is countable. Equivalently, the infima defining $\alpha$ and $\delta$ are minima achieved by valuations of toroidal type [2604.18465].

The proof strategy in the $\delta$-case proceeds by approximating $\delta$ with basis-type divisors $D_{m_i}$, stabilizing multiplier ideals
\[
\mathcal J\bigl(X,\Delta+(1+\varepsilon)\delta_{m_i}D_{m_i}\bigr)
\]
via Lehmann’s perturbation trick, using Nadel vanishing to obtain bounded Hilbert polynomials, extracting Kollár components through local bounded complements, and then passing to a simultaneous log resolution where Xu’s retraction yields quasi-monomiality. The final passage from an approximating sequence to an actual minimizer uses semicontinuity of the normalized quantities $S_m$ and $T_m$, hence of the ratios $A/S$ and $A/T$ [2604.18465].

The older conjectural landscape is broader. Jonsson–Mustaţă showed that asymptotic jumping numbers are always computed by some real valuation and conjectured that every valuation computing such an invariant is quasi-monomial; they proved this in dimension two [1011.3699]. More recently, the weak algebraic Jonsson–Mustaţă conjecture was shown to be equivalent to the statement that every algebraic Zhou valuation is quasi-monomial, and Xu’s theorem implies this for the case $\mathfrak q=\mathcal O_X$ [2505.19451].

## 4. Normalized volume, uniqueness, and K-semistable cones

For a klt singularity $x\in (X,\Delta)$, the valuation ideals
\[
\mathfrak a^v_k=\{\,f\in \mathcal O_{x,X}\mid v(f)\ge k\,\}
\]
define the valuation volume
\[
\operatorname{vol}_{X,x}(v)=\lim_{k\to\infty}\frac{\ell\bigl(\mathcal O_{x,X}/\mathfrak a^v_k\bigr)}{k^n/n!},
\]
and the normalized volume
\[
\widehat{\operatorname{vol}}_{(X,\Delta),x}(v)=A_{X,\Delta}(v)^n\,\operatorname{vol}_{X,x}(v).
\]
Xu proved that any minimizer of $\widehat{\operatorname{vol}}$ is quasi-monomial, confirming Chi Li’s conjecture in this form [1907.01114].

Li and Xu analyzed the converse direction under finite generation of the associated graded ring. If $v$ is a quasi-monomial valuation with finitely generated $\operatorname{gr}_v R$, then $v$ minimizes normalized volume if and only if it induces a degeneration to a K-semistable log Fano cone singularity. In this framework, the Rees algebra
\[
\mathcal R:= \bigoplus_{m\in \Gamma_v}\mathfrak a_m(v)\,t^{-m}\subset R[t,t^{-1}]
\]
produces a flat degeneration to
\[
X_0:=\operatorname{Spec}\operatorname{gr}_v R,
\]
and the minimizer is unique among quasi-monomial valuations up to rescaling [1707.05561].

This circle of ideas has strong consequences in families. Xu showed that the volume of klt singularities is a constructible function in a family, that in a family of klt log Fano pairs the K-semistable fibers form a Zariski open set, and, together with earlier work, that K-semistable klt Fano varieties with fixed dimension and volume are parametrized by an Artin stack of finite type admitting a separated good moduli space whose geometric points parametrize K-polystable klt Fano varieties [1907.01114].

In the metric direction, Li–Xu also showed that for a point on a Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated to the metric tangent cone is uniquely determined by the algebraic structure of the singularity, thereby confirming the Donaldson–Sun conjecture described in their paper [1707.05561].

## 5. Finite generation and algebraic degenerations

A recurrent difficulty in applications is the finite-generation problem for valuation-graded algebras:
given a quasi-monomial valuation $v$, show that $\operatorname{gr}_v(R)$ is finitely generated. This step is described as crucial in K-stability, because it produces a flat degeneration of $X=\operatorname{Proj}R$ and identifies $\operatorname{Proj}(\operatorname{gr}_v R)$ with the limiting Fano or cone [2510.10737].

Chen gave a higher-rank finite-generation theorem using an extended Rees algebra. Under dlt Fano-type hypotheses, with
\[
R:=R(L_1,\dots,L_s)=\bigoplus_{n\in \mathbb N^s} H^0\bigl(Y,\mathcal O_Y(n_1L_1+\cdots+n_sL_s)\bigr),
\]
and the extended Rees algebra
\[
\mathcal R:=\bigoplus_{n\in\mathbb N^s}\bigoplus_{m\in\mathbb Z^r}
H^0\bigl(Y,\mathcal O_Y(n\!\cdot\! L-E(m))\bigr)t^{-m}
\subset R[t_1^\pm,\dots,t_r^\pm],
\]
the paper proves that $\mathcal R$ is finitely generated, flat over $\Bbbk[t_1,\dots,t_r]$, and that for $v=v_\alpha\in \operatorname{QM}_\zeta^\circ(Y,E)$ with $\alpha\in \mathbb R_{>0}^r$ one has a canonical graded-algebra isomorphism
\[
\operatorname{gr}_v R \simeq \mathcal R/(t_1,\dots,t_r).
\]
In particular $\operatorname{gr}_v R$ is finitely generated over the base ring, so $\operatorname{Proj}(\operatorname{gr}_v R)$ is a projective flat degeneration of $\operatorname{Proj}R$ [2510.10737].

The same source states that this recovers earlier global Fano and local klt-singularity finite-generation results and extends to Fano-type fibrations and multi-section rings of arbitrary divisors on $Y$ [2510.10737]. In the context of the $\delta$-invariant, Kim records that if $\delta(X,\Delta,L)$ is computed by a quasi-monomial valuation of finite rational rank, then the induced filtration on the section ring is finitely generated; this yields a special test-configuration whose Donaldson–Futaki invariant vanishes at $\delta$, and it leads to structure theorems in the study of K-stability and moduli of Fano varieties, including the existence of K-polystable degenerations and the openness of uniform K-stability in families [2604.18465].

This suggests that quasi-monomiality is the bridge from valuative minimization problems to algebraic degeneration theory: the valuation identifies the optimal asymptotic direction, while finite generation turns that direction into an actual test configuration or cone degeneration.

## 6. Higher-rank geometry, tropical models, and special cases

Higher-rank quasi-monomial valuations take values in the lexicographically ordered group $\mathbb R^k$. For a smooth irreducible variety $X$ with an SNC divisor $D=\sum D_i$, and a cone $\sigma$ of the dual cone complex $\Sigma(D)$, a weight datum $\underline\alpha=(\alpha_i)_{i\in I_\sigma}\in (\mathbb R_{\succeq 0}^k)^{I_\sigma}$ defines a valuation
\[
\nu_{\sigma,\underline\alpha}(f)
=
\min\nolimits_{\preceq}
\left\{
\sum_{i\in I_\sigma}\beta_i\alpha_i
\ \middle|\
c_\beta\neq 0
\right\}
\]
for admissible expansions of $f$ [2208.06237].

Amini and Iriarte proved a duality theorem identifying the space of such valuations with the order-$(k-1)$ tangent-cone bundle of the dual cone complex:
\[
\operatorname{Val}^k(D)\cong C^{k-1}\Sigma(D).
\]
They further showed that for a rational function $f$, the value of a higher-rank quasi-monomial valuation can be read off from successive directional derivatives of the tropicalization $trop(f)$:
\[
\nu_{x;\underline w}(f)
=
\Bigl(
trop(f)(x),\,
D_{w_1}trop(f)(x),\,
D_{(w_1,w_2)}trop(f)(x),\dots
\Bigr).
\]
The same paper introduces a refined tropicalization that remembers initial terms on each cone, proves a tropical weak approximation theorem asserting that any coherent family of antichains arises from a single rational function, defines the tropical topology on spaces of higher-rank valuations, and recovers the full higher-rank valuation space as a projective limit of higher-rank skeleta [2208.06237].

At the opposite end of the spectrum, low-dimensional cases show both rigidity and pathology. Jonsson–Mustaţă proved that in dimension two any valuation computing the asymptotic jumping number $\operatorname{Arn}^q(\mathfrak a_\bullet)$ is quasi-monomial [1011.3699]. By contrast, in the two-variable rational function field $K(x,y)$, rank-two monomial valuations with value group $\mathbb Z\oplus \mathbb Z$ can exhibit unexpectedly bad behavior on the polynomial subring: Mosteig constructs a valuation that is nonpositive on $K[x,y]$ but whose value semigroup
\[
v(K[x,y]^*)=\{(0,0)\}\cup \mathbb Z_{\ge 0}(-1,-1)+\mathbb Z_{\ge 0}(0,1)
\]
is not reversely well-ordered [1712.08325].

These examples clarify a common source of confusion. Quasi-monomiality is a birational local property of the valuation itself; it does not automatically imply that every induced semigroup or graded structure on a chosen coordinate ring has the simplest possible order-theoretic behavior. The stronger finiteness and moduli consequences arise only after additional hypotheses, such as klt or Fano-type geometry, bounded complement constructions, or finite generation of the associated graded ring [2510.10737] [1707.05561].

Source: https://www.emergentmind.com/topics/quasi-monomial-valuations