Log Canonical Threshold in Algebraic Geometry

• Log Canonical Threshold is a fundamental invariant that quantifies divisor singularities by measuring integrability conditions and discrepancies in resolution models.
• It is computed using methods like log resolutions, jet schemes, and Newton polyhedra, providing practical formulas such as min((kₙ+1)/aₙ) for effective divisors.
• Applications include determining Kähler–Einstein metric existence, assessing Fano variety stability, and influencing singular learning theory through statistical zeta functions.

The log canonical threshold (lct) is a fundamental birational invariant quantifying the singularities of divisors or ideals on algebraic varieties. It is central to higher-dimensional algebraic geometry, singularity theory, complex analytic geometry, and the theory of Kähler–Einstein metrics. The threshold governs vanishing theorems, measures the “severity” of singularities, appears in stability conditions for Fano varieties, and unifies algebraic and analytic notions of singularity.

1. Definition and Fundamental Properties

Let $X$ be a smooth variety (or complex manifold) and $D$ an effective $\mathbb{Q}$-divisor (or ideal sheaf). The pair $(X,D)$ is log canonical at $x\in X$ if, for some log resolution $\pi:Y\to X$,

$K_Y + D_Y = \pi^*(K_X + D)$

has all coefficients in $D_Y$ (proper transform and exceptional divisors with their multiplicities) at most $1$ over $\pi^{-1}(x)$. The log canonical threshold at $x$ is: $\lct_x(D) = \sup \left\{ c > 0\,\big|\, (X, cD) \text{ is log canonical at } x \right\}$ Equivalently, if $D$ is given by a holomorphic function or analytic ideal $(f)$ in local coordinates, $\lct_x(f)$ is the supremum of $c>0$ such that $|f|^{-2c}$ is integrable in a punctured neighborhood of $x$ (Hiep, 2014).

For an effective $\mathbb{Q}$-divisor $D$, upon taking a log resolution $\pi$ such that $\pi^*D = \sum a_i E_i$, $K_{Y/X} = \sum k_i E_i$, the lct is

$\lct_x(D) = \min_i \frac{k_i+1}{a_i}$

where the minimum is taken over all $E_i$ mapping to $x$ (Bhatt et al., 2011, Watanabe, 2023).

For ideals, analogous definitions apply: for $I\subset\mathcal{O}_{X,x}$,

$\lct_x(I) = \min_E \frac{a(E)+1}{\nu_E(I)}$

where $a(E)$ is the discrepancy and $\nu_E(I)$ the vanishing order along the divisor $E$ over $x$ (Bivià-Ausina, 21 Sep 2024).

2. Combinatorial and Jet-Theoretic Characterizations

Beyond resolution-of-singularity approaches, alternative computations are critical:

• Jet schemes/arcs: For smooth $X$ in arbitrary characteristic and $Y\subset X$, Mustaţă's formula expresses the lct via jet scheme codimensions: $\lct(X,Y) = \inf_{m\ge0} \frac{\codim_{X_m}(Y_m)}{m+1}$ Similarly, there is a correspondence between closed cylinders in the arc space $X_\infty$ and divisorial valuations, allowing a description

$\lct(X,Y) = \inf_{C} \frac{\codim(C)}{\operatorname{ord}_C(Y)}$

(Zhu, 2013).

• Newton polyhedron/toric techniques: For ideals in two variables, the lct can be extracted directly from the Newton polygon. For each compact face $S$ of the polygon given by $p\alpha + q\beta = N$, the lct is the minimum of $(p+q)/N$ over all faces $S$: $\lct(I) = \min_{S} \frac{p_S + q_S}{N_S}$ (Cassou-Noguès et al., 2013, Paemurru, 28 Apr 2024).
• Binomial and monomial ideals: For binomial ideals, explicit formulae in terms of exponents and a finite set of rays in a fan are known, enabling practical computation: $\lct(\mathfrak{a}) = \min_{\rho \text{ ray in }\Sigma} \varphi(v_\rho)$ where $\varphi$ is piecewise linear in exponent data and $v_\rho$ is the primitive ray generator (Blanco et al., 2014).

3. Analytic and Weighted Formulations

In pluripotential theory and complex analysis, the lct of a plurisubharmonic (psh) function $\varphi$ at 0 is

$$\mathrm{lct}(\varphi; 0) = \sup\left\{ c > 0 : e^{-2c\varphi} \text{ is } L^1 \text{ near } 0\right\}$$

This analytic lct coincides with the algebraic lct when $\varphi = \log|f|$ for holomorphic $f$ (Hiep, 2014).

Weighted variants, $\mathrm{lct}_f(\varphi; 0)$, account for additional holomorphic weights, and reflect how multiplicities of zero sets interact with singularities of $\varphi$. Effective semicontinuity and strong openness properties for (weighted) lct play a structural role in the theory of multiplier ideals and their stability under parameter variation (Hiep, 2014).

Demailly–Pham (Demailly et al., 2012) established a sharp lower bound for lct in terms of the sequence of intermediate multiplicities (Lelong numbers) $e_j(\varphi)$: $$\mathrm{lct}(\varphi; 0) \ge \sum_{j=0}^{n-1} \frac{e_j}{e_{j+1}}$$ which unifies previous bounds and is sharp for monomial psh functions.

4. Birational and Geometric Applications

Birational Rigidity and Fano Varieties

The global lct for a Fano variety $X$,

$\glct(X) = \inf_{D \sim_{\mathbb{Q}} -K_X} \lct(X, D)$

is a critical invariant. For generic smooth Fano complete intersections of index 1 and codimension $k$ in $\mathbb{P}^{M+k}$ with $d_{\max} \geq 8$ and $M \geq 3k+4$, $\glct(X) = 1$ (Eckl et al., 2014). This exact threshold is “optimal” for analytic existence of Kähler–Einstein metrics (via Nadel–Demailly–Kollár criteria): $\glct(X) = 1 \quad \Leftrightarrow \quad X \text{ admits a Kähler–Einstein metric}$ The computation of $\glct$ is also used in K-stability for Fano varieties and plays a role in the minimal model program.

Sharp Inequalities and Bounds

The lct is tightly connected to various algebraic invariants. For a $\mathfrak{m}$-primary ideal $I$ in $\mathcal{O}_{\mathbb{C}^n,0}$, the Demailly–Pham number

$\DP(I) = \sum_{i=1}^n \frac{1}{e_i(I)}$

satisfies $\DP(I) \le \lct(I)$, and further inequalities relate $\lct(IJ)$, the mixed Łojasiewicz exponents $\mathcal{L}_k$, and multiplicities for products of ideals (Bivià-Ausina, 21 Sep 2024).

For ideals $I$, $J$: $\sum_{k=1}^n \frac{1}{\mathcal{L}_k(I) + \mathcal{L}_k(J)} \leq \DP(IJ) \leq \lct(IJ)$ with equality conditions tied to integral closure and maximal multiplicity.

In projective geometry, lct bounds the regularity of an ideal via the inequality

$1 < \lct(\mathcal{I}) \cdot \operatorname{reg}(\mathcal{I})$

both for Castelnuovo–Mumford regularity on $\mathbb{P}^n$ and $\Theta$-regularity on principally polarized abelian varieties (Oygarden et al., 2018).

5. Interactions with Positive Characteristic and Singular Learning Theory

F-pure Thresholds and Reduction Mod $p$

In positive characteristic, the F-pure threshold (fpt) is the analogous invariant defined via Frobenius action and test ideals. Bhatt–Hernández–Miller–Mustaţă (Bhatt et al., 2011) proved:

• Any lct in characteristic $0$ arises as a limit of F-pure thresholds for reductions mod $p$.
• Conversely, every limit of F-pure thresholds is an lct in characteristic $0$.

This connection is achieved via comparison of test ideals and multiplier ideals, and non-standard ultraproduct methods.

Statistical/Real LCT

In singular learning theory, the real log canonical threshold (RLCT) controls the learning-theoretic generalization error. The RLCT is defined as the rightmost pole (largest $\lambda$) of the statistical zeta function

$\zeta(z) = \int K(w)^z \varphi(w) dw$

where $K$ is the Kullback–Leibler divergence. Asymptotically, the generalization error has leading term $\lambda/n$, with $\lambda$ measurable via resolution of singularities: $G_n \sim S + \frac{\lambda}{n} + o(1/n)$ (Yoshida et al., 2023). For du Val singularities, there is a distinction between complex and real LCTs, with the latter depending intricately on the number of real branches and real blowups (Watanabe, 2023).

6. Special Cases: Plane Curves, Surface Singularities, and Explicit Formulae

Plane Curves

For a reduced plane curve germ $C = C_1 \cup \dots \cup C_r$, there are explicit formulas (Galindo–Hernando–Monserrat) for $\lct(C)$ in terms of the first two maximal contact values $m_{1,i}, m_{2,i}$ of each branch $C_i$ and intersection multiplicities. In particular,

$\lct(C) = \min \left\{ a_{t_i}, a_k \right\}$

where $a_{t_i}$, $a_k$ are given by explicit rational expressions in maximal contact values and intersection numbers (Galindo et al., 2012).

Weighted Blowups and Newton Polyhedra

Paemurru (Paemurru, 28 Apr 2024) extends Varchenko’s formula for non-isolated plane curve singularities: the lct at the origin of $f(x, y)$ is bounded above by $1/c$, where $(c, c)$ lies on a facet of the Newton polyhedron of $f$. Equality is achieved under weak normalization conditions, leading to a direct recipe for any convergent power series in two variables.

ADE Singularities

For du Val surface singularities:

• $A_n$: $\mathrm{LCT}(A_n) = \frac{n+1}{n}$ (even $n$), $\frac{n+2}{n+1}$ (odd $n$)
• $D_4$: $\mathrm{LCT}(D_4) = 4/3$ etc. (Watanabe, 2023)

The method is resolution-theoretic, based on explicit point blowups and discrepancy computations.

7. Advanced Notions: Potential and Global Log Canonical Thresholds

The potential log canonical threshold (plct), introduced by Choi–Jang (Choi et al., 2022), extends lct to pairs $(X, A)$ where $-(K_X+A)$ is only pseudoeffective, refining the notion for settings lacking strict nefness: $\plct(X, A; D) = \sup\{ t \mid (X, A + tD) \text{ is potentially lc for } t \leq t_{\text{eff}} \}$ The set of plct satisfies the ascending chain condition (ACC), generalizing the Hacon–McKernan–Xu theorem for lct-ACC.

For Fano varieties, the global log canonical threshold determines the existence of Kähler–Einstein metrics and constants in birational rigidity. In smooth Fano complete intersections of index 1 and suitable numerical invariants, $\glct=1$, the maximal possible value, guaranteeing the existence of Kähler–Einstein metrics (Eckl et al., 2014).

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The log canonical threshold is thus a unifying metric for singularity severity, deformation theory, vanishing theorems, birational geometry, and complex and arithmetic invariants. Its diverse characterizations and explicit formulas are intimately connected to specialized techniques—log resolutions, multiplier ideals, jet schemes, Newton polygons, weighted blowups, and analytic estimates—making it an indispensable tool in contemporary algebraic geometry and beyond.