Quasi-monomial Valuations in Birational Geometry

• Quasi-monomial valuations are real valuations that become monomial on log-smooth models using weight vectors over simple normal crossing divisors.
• They structure valuation spaces as polyhedral complexes, unifying perspectives in birational geometry, asymptotic invariants, and K-stability.
• Their finite generation properties enable flat degenerations and test configurations key to understanding Fano varieties and klt singularities.

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 (Jonsson et al., 2010, Xu, 2019, Amini et al., 2022).

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 $X$ (Kim, 20 Apr 2026).

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 $K(X)$0 cutting out the components through $K(X)$1, and a weight vector $K(X)$2. If

$K(X)$3

then one sets

$K(X)$4

This gives a valuation that is literally monomial in the chosen coordinates (Xu, 2019).

In the rank-one framework, quasi-monomial valuations are equivalently characterized by the Zariski–Abhyankar condition

$K(X)$5

and, in the regular excellent setting, an Abhyankar valuation is quasi-monomial (Xu, 2019, Jonsson et al., 2010). The divisor case is the special case $K(X)$6; higher rational rank corresponds to monomiality along several components simultaneously.

For projective klt pairs $K(X)$7, log discrepancy is defined for a prime divisor $K(X)$8 over $K(X)$9 by

$\nu:K(X)^*\to\mathbb R$0

and the theory extends from divisorial valuations to arbitrary real valuations through the valuative log discrepancy function $\nu:K(X)^*\to\mathbb R$1 (Kim, 20 Apr 2026, Xu, 2019).

2. Quasi-monomial valuations inside valuation spaces

Quasi-monomial valuations coming from a fixed log-smooth pair $\nu:K(X)^*\to\mathbb R$2 form a simplicial cone complex

$\nu:K(X)^*\to\mathbb R$3

with cones indexed by strata of $\nu:K(X)^*\to\mathbb R$4 and coordinates given by the values on irreducible components of the boundary. On each cone, the map

$\nu:K(X)^*\to\mathbb R$5

is a homeomorphism, and the cone complex carries a natural $\nu:K(X)^*\to\mathbb R$6-affine structure (Jonsson et al., 2010).

A basic construction is the retraction

$\nu:K(X)^*\to\mathbb R$7

which sends an arbitrary valuation to the unique monomial valuation agreeing on the boundary divisors. One has

$\nu:K(X)^*\to\mathbb R$8

with equality if and only if $\nu:K(X)^*\to\mathbb R$9. As $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$0 ranges over log-smooth models, the full valuation space is recovered as a projective limit: $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$1 Moreover, each $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$2 is dense in $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$3, so quasi-monomial valuations are dense in the ambient valuation space (Jonsson et al., 2010).

This polyhedral picture is also implicit in more recent existence arguments. In the proof of the quasi-monomiality of the $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$4- and $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$5-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 (Kim, 20 Apr 2026).

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 $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$6, Xu proved a version of the Jonsson–Mustaţă conjecture: if

$$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$7

then there exists a quasi-monomial valuation $$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$8 computing the infimum

$$\nu(fg)=\nu(f)+\nu(g), \qquad \nu(f+g)\ge \min\{\nu(f),\nu(g)\}, \qquad \nu(c)=0 \text{ for } c\in k.$$9

This establishes quasi-monomiality for minimizers of log canonical threshold type invariants for graded sequences (Xu, 2019).

For a projective klt pair $X$0 and an ample $X$1-Cartier $X$2-divisor $X$3, the $X$4-invariant and $X$5-invariant are defined by

$X$6

and

$X$7

Kim proved that there exist quasi-monomial valuations $X$8 and $X$9 in $X$0 such that

$X$1

independently of whether the base field is countable. Equivalently, the infima defining $X$2 and $X$3 are minima achieved by valuations of toroidal type (Kim, 20 Apr 2026).

The proof strategy in the $X$4-case proceeds by approximating $X$5 with basis-type divisors $X$6, stabilizing multiplier ideals

$X$7

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 $X$8 and $X$9, hence of the ratios $c\cdot \operatorname{ord}_E$0 and $c\cdot \operatorname{ord}_E$1 (Kim, 20 Apr 2026).

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 (Jonsson et al., 2010). 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 $c\cdot \operatorname{ord}_E$2 (Bao et al., 26 May 2025).

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

For a klt singularity $c\cdot \operatorname{ord}_E$3, the valuation ideals

$c\cdot \operatorname{ord}_E$4

define the valuation volume

$c\cdot \operatorname{ord}_E$5

and the normalized volume

$c\cdot \operatorname{ord}_E$6

Xu proved that any minimizer of $c\cdot \operatorname{ord}_E$7 is quasi-monomial, confirming Chi Li’s conjecture in this form (Xu, 2019).

Li and Xu analyzed the converse direction under finite generation of the associated graded ring. If $c\cdot \operatorname{ord}_E$8 is a quasi-monomial valuation with finitely generated $c\cdot \operatorname{ord}_E$9, then $E$0 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

$E$1

produces a flat degeneration to

$E$2

and the minimizer is unique among quasi-monomial valuations up to rescaling (Li et al., 2017).

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 (Xu, 2019).

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 (Li et al., 2017).

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 $E$3, show that $E$4 is finitely generated. This step is described as crucial in K-stability, because it produces a flat degeneration of $E$5 and identifies $E$6 with the limiting Fano or cone (Chen, 12 Oct 2025).

Chen gave a higher-rank finite-generation theorem using an extended Rees algebra. Under dlt Fano-type hypotheses, with

$E$7

and the extended Rees algebra

$E$8

the paper proves that $E$9 is finitely generated, flat over $(Y,E=\sum E_i)$0, and that for $(Y,E=\sum E_i)$1 with $(Y,E=\sum E_i)$2 one has a canonical graded-algebra isomorphism

$(Y,E=\sum E_i)$3

In particular $(Y,E=\sum E_i)$4 is finitely generated over the base ring, so $(Y,E=\sum E_i)$5 is a projective flat degeneration of $(Y,E=\sum E_i)$6 (Chen, 12 Oct 2025).

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,E=\sum E_i)$7 (Chen, 12 Oct 2025). In the context of the $(Y,E=\sum E_i)$8-invariant, Kim records that if $(Y,E=\sum E_i)$9 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 $\eta$0, 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 (Kim, 20 Apr 2026).

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 $\eta$1. For a smooth irreducible variety $\eta$2 with an SNC divisor $\eta$3, and a cone $\eta$4 of the dual cone complex $\eta$5, a weight datum $\eta$6 defines a valuation

$\eta$7

for admissible expansions of $\eta$8 (Amini et al., 2022).

Amini and Iriarte proved a duality theorem identifying the space of such valuations with the order-$\eta$9 tangent-cone bundle of the dual cone complex: $K(X)$00 They further showed that for a rational function $K(X)$01, the value of a higher-rank quasi-monomial valuation can be read off from successive directional derivatives of the tropicalization $K(X)$02: $K(X)$03 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 (Amini et al., 2022).

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 $K(X)$04 is quasi-monomial (Jonsson et al., 2010). By contrast, in the two-variable rational function field $K(X)$05, rank-two monomial valuations with value group $K(X)$06 can exhibit unexpectedly bad behavior on the polynomial subring: Mosteig constructs a valuation that is nonpositive on $K(X)$07 but whose value semigroup

$K(X)$08

is not reversely well-ordered (Mosteig, 2017).

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 (Chen, 12 Oct 2025, Li et al., 2017).