Log Canonical Threshold (LCT)

• Log canonical threshold (LCT) is a fundamental invariant that measures the severity of singularities in algebraic, analytic, and birational geometry.
• LCT computation involves log resolutions, Newton polygon methods, and valuation techniques, providing explicit formulas in monomial and binomial settings.
• LCT plays a crucial role in applications such as multiplier ideals, vanishing theorems, the minimal model program, and statistical learning theory.

The log canonical threshold (LCT) is a fundamental invariant in algebraic, analytic, and birational geometry, capturing the severity of singularities of ideal sheaves, analytic functions, or divisors. It plays a central role in the minimal model program, the theory of multiplier ideals, the study of vanishing theorems, and applications in both complex and real algebraic geometry, as well as in singular learning theory and representation-theoretic motivic integration.

1. Definition and Fundamental Properties

Given a smooth variety $X$ over a field $k$ of characteristic zero and a nonzero ideal sheaf $\mathfrak{a} \subset \mathcal{O}_X$, the log canonical threshold of $\mathfrak{a}$ at a point $x \in X$ is defined as

$$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$

When $\mathfrak{a} = (f)$ for a regular function $f$, this is written $\operatorname{lct}_x(f)$.

For a log resolution $\mu: Y \to X$ so that $k$0 and $k$1, the threshold is

$k$2

For analytic or plurisubharmonic functions $k$3 with isolated singularity at $k$4, the LCT is the supremum $k$5 such that $k$6 is locally integrable:

$k$7

This analytic definition coincides with the algebro-geometric one for functions defining singular hypersurfaces (Demailly et al., 2012).

Key properties include:

• $k$8 is always a positive rational number.
• It remains unchanged under smooth base change.
• The set of all LCTs in dimension $k$9 satisfies the ascending chain condition (ACC) (Libgober et al., 2010).
• For real analytic functions, the real LCT (rlct) has analogous definitions and appears as the "learning coefficient" in statistical applications (Kosta et al., 2024).

2. Sharp Lower Bounds and Monotonicity

Classical lower bounds for the LCT include:

• Skoda's bound: $\mathfrak{a} \subset \mathcal{O}_X$0, where $\mathfrak{a} \subset \mathcal{O}_X$1 is the Lelong number (Demailly et al., 2012).
• Geometric mean bound: $\mathfrak{a} \subset \mathcal{O}_X$2, where $\mathfrak{a} \subset \mathcal{O}_X$3 is the top mixed Lelong number.

Demailly–Hiệp established a sharp, optimal lower bound:

$\mathfrak{a} \subset \mathcal{O}_X$4

with $\mathfrak{a} \subset \mathcal{O}_X$5, generalizing Skoda and achieving equality in toric and diagonal cases. This bound is strictly stronger than either classical estimate, is implied by log-convexity properties among the $\mathfrak{a} \subset \mathcal{O}_X$6, and is optimal in monomial (toric) examples (Demailly et al., 2012).

The "additive" and "geometric" bounds for a psh function $\mathfrak{a} \subset \mathcal{O}_X$7 are:

• $\mathfrak{a} \subset \mathcal{O}_X$8,
• $\mathfrak{a} \subset \mathcal{O}_X$9, and rigidity at equality characterizes the precise locus and asymptotic structure of extremal psh singularities (Rashkovskii, 2015).

3. Computational Methods and Special Cases

For monomial, binomial, and $\mathfrak{a}$0-binomial ideals, explicit reduction procedures exist:

• The LCT of a binomial ideal can be computed by minimizing a piecewise-linear function $\mathfrak{a}$1 over the rays of an associated polyhedral fan, with the exponent data derived from the monomial/binomial structure (Blanco et al., 2014).
• For plane curve singularities, valuation-theoretic and Newton polygon approaches yield explicit formulas. For an irreducible plane curve $\mathfrak{a}$2,

$\mathfrak{a}$3

where $\mathfrak{a}$4 denotes the curve semivaluation (Lee, 31 Jan 2026). For arbitrary reduced plane curves, the LCT is given in terms of the first two maximal contact values of each branch and intersection multiplicities (Galindo et al., 2012).

In two variables, construction of the Newton tree allows for computation of the LCT using only Newton polygon data and associated combinatorics (Cassou-Noguès et al., 2013). For du Val singularities (ADE types), explicit values are provided in terms of the type and defining equations (Watanabe, 2023).

For hyperplane arrangements, the (real) LCT and its multiplicity are computable via an intersection lattice and flat weights, with formulas

$\mathfrak{a}$5

where $\mathfrak{a}$6 is the weight and $\mathfrak{a}$7 the intersection lattice (Kosta et al., 2024).

4. Real and $\mathfrak{a}$8-adic Analogs

In statistical learning theory, the real log canonical threshold (RLCT) appears as the learning coefficient controlling the asymptotics of Bayesian generalization error and marginal likelihood:

$\mathfrak{a}$9

with $x \in X$0 (the RLCT) controlling subleading error terms (Yoshida et al., 2023, Kurumadani, 2024). RLCT can be calculated at non-singular parameter points using explicit formulas involving the codimension, the Fisher information rank, and the order of vanishing:

$x \in X$1

with $x \in X$2 the codimension of the realizable set, $x \in X$3 the rank, and $x \in X$4 the smallest order with nontrivial Taylor term (Kurumadani, 2024).

For non-Archimedean local fields $x \in X$5 (including positive characteristic), the $x \in X$6-analytic log canonical threshold $x \in X$7 is the supremum $x \in X$8 such that $x \in X$9 is locally integrable. Positivity and effective lower bounds are proven via Weierstrass preparation and sublevel set estimates:

$$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$0

where $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$1 is the degree of the Weierstrass polynomial. Uniform bounds in the algebraic category reflect multiplicity and degree data (Glazer et al., 3 Nov 2025).

5. Birational and Topological Applications

LCTs govern the formation and behavior of multiplier ideals $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$2, dictate the thresholds for vanishing theorems (e.g., Kawamata–Viehweg), and serve as obstructions in birational rigidity questions (Demailly et al., 2012, Rashkovskii, 2015, Ambro, 2014). The LCT detects rational singularities: for instance, $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$3 if and only if the hypersurface $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$4 has rational singularities; otherwise $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$5 (Cluckers et al., 2019, Cluckers et al., 2022).

In Floer-theoretic and topological contexts, the LCT and multiplicity of a hypersurface are encoded as invariants of the link of the singularity via fixed-point Floer cohomology, linking symplectic geometry and singularity theory (McLean, 2016).

6. Variation, Families, and Polytope Structures

The set of all LCTs in fixed dimension forms an ACC set, and more generally, the set of LCT polytopes associated to $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$6-tuples of ideals is closed under Hausdorff limits and satisfies the strong form of the ACC property (Libgober et al., 2010). In linear systems, the function $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$7 is Zariski lower semi-continuous and takes only finitely many values (Ambro, 2014).

Refinements such as the potential log canonical threshold (plct) account for both singularities and the positivity of $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$8 in the context of the minimal model program and confirm ACC properties in broader moduli-theoretic settings (Choi et al., 2022).

The G-stable rank of the defining ideal provides a bound $$\operatorname{lct}_x(\mathfrak{a}) = \sup \left\{ c > 0 \mid \text{the pair } (X, c \cdot \mathfrak{a}) \text{ is log canonical at } x \right\}.$$9, with equality in the monomial case, tying invariant-theoretic measures of instability directly to singularity invariants (Jiang, 2022).

7. Explicit Examples and Sharpness

• For the diagonal monomial $\mathfrak{a} = (f)$0, $\mathfrak{a} = (f)$1, and $\mathfrak{a} = (f)$2, achieving the lower bound (Demailly et al., 2012).
• For plane curve singularities $\mathfrak{a} = (f)$3, $\mathfrak{a} = (f)$4 over any characteristic (Lee, 31 Jan 2026).
• For du Val singularities $\mathfrak{a} = (f)$5:

$\mathfrak{a} = (f)$6

and explicit values for $\mathfrak{a} = (f)$7 (Watanabe, 2023).

The optimality of the sharp lower bounds is evidenced by explicit construction in the toric and monomial settings, and tightness in real and $\mathfrak{a} = (f)$8-adic cases is demonstrated in model-theoretic and analytic computations (Demailly et al., 2012, Glazer et al., 3 Nov 2025).

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References:

• (Demailly et al., 2012): "A sharp lower bound for the log canonical threshold"
• (Rashkovskii, 2015): "A log canonical threshold test"
• (Glazer et al., 3 Nov 2025): "A lower bound on the analytic log-canonical threshold over local fields of positive characteristic"
• (Kosta et al., 2024): "Classification of real hyperplane singularities by real log canonical thresholds"
• (Blanco et al., 2014): "A procedure for computing the log canonical threshold of a binomial ideal"
• (Yoshida et al., 2023): "Upper Bound of Real Log Canonical Threshold of Tensor Decomposition and its Application to Bayesian Inference"
• (Cassou-Noguès et al., 2013, Galindo et al., 2012, Lee, 31 Jan 2026): Plane curve and Newton polygon LCT computation
• (Watanabe, 2023): "Log canonical threshold of du Val singularities"
• (McLean, 2016): "Floer Cohomology, Multiplicity and the Log Canonical Threshold"
• (Jiang, 2022): "G-stable rank of symmetric tensors and log canonical threshold"
• (Libgober et al., 2010): "Sequences of LCT-polytopes"
• (Ambro, 2014): "Variation of log canonical thresholds in linear systems"
• (Choi et al., 2022): "ACC of plc thresholds"
• (Cluckers et al., 2019, Cluckers et al., 2022): LCT and rational singularities

For further computational methodologies, applications in singular learning theory, and connections with other birational invariants, see the cited works.