Beilinson–Bloch Height Pairing

• Beilinson–Bloch height pairing is a bilinear arithmetic intersection theory that assigns quantitative heights to homologically trivial cycles on smooth projective varieties.
• It unifies cycle-theoretic constructions, regulator maps, and Hodge-theoretic methods to connect arithmetic intersection theory with conjectures on motivic L-functions.
• The pairing exhibits key properties such as positivity, index theorems, and compatibility with classical height pairings for curves and abelian varieties.

The Beilinson–Bloch height pairing is a bilinear, arithmetic intersection-theoretic object designed to endow the group of homologically trivial algebraic cycles on a smooth projective variety over a global field (function field or number field) with a theory of “quantitative” heights analogous to those for divisors (Néron–Tate, Arakelov, etc.). Its definition, properties, and extensions unify intersection theory, regulator constructions in cohomology, Hodge-theoretic biextensions, and special value conjectures for motivic $L$-functions.

1. Cycle-theoretic and Regulator Construction

Let $X$ be a smooth projective variety of dimension $d$ over the function field $K = k(B)$ of a smooth projective curve $B$ over a perfect field $k$, or over a number field. The foundational cycle group in this context is the subgroup of “admissible” cycles

$CH^i(X)^{(0)} \subset CH^i(X)$

consisting of those cycles numerically (or algebraically) trivial on all fibers of a regular (proper) model $\mathcal{X} \to B$ (Kahn et al., 2020). For $i=1,d$, this subgroup coincides with numerically trivial divisors and degree-zero $0$-cycles, respectively.

The refined Beilinson–Bloch height pairing is constructed as a bilinear map

$$(\, ,\,): CH^i(X)^{(0)} \times CH^{d+1-i}(X)^{(0)} \to CH^1(B)\otimes \mathbb{Q}$$

in the category of additive groups modulo isogeny (Kahn et al., 2020). The pairing is defined by choosing extensions of cycles on $X$ to the model $\mathcal{X}$ satisfying admissibility, then intersecting them on $\mathcal{X}$ and pushing forward via $f_*:\mathcal{X} \to B$.

Compatibility diagrams relate the intersection-theoretic pairing to the cup product in $\ell$-adic or Betti cohomology: $$\begin{CD} CH^i(X)^{(0)} \times CH^{d+1-i}(X)^{(0)} @>(\, ,\,)>> CH^1(B)\otimes\mathbb{Q} \ @V\mathrm{cl}\times\mathrm{cl}VV @VV\deg\otimes\mathbb{Q}V \ H^{2i-1}(X_K,\mathbb{Q}(i)) \times H^{2(d-i)-1}(X_K,\mathbb{Q}(d-i+1)) @>\cup>> H^2(B,\mathbb{Q}(1)) \end{CD}$$ Here the cycle regulator $\mathrm{cl}$ targets Deligne/étale cohomology, and the bottom row is the cup product pairing followed by $f_*$ (Kahn et al., 2020, Rössler et al., 2020, Wisson, 11 Aug 2025). The pairing is conjectured to agree with the cup-product pairing under the Beilinson regulator.

2. Relationship to Classical Heights: Curves and Abelian Varieties

In cases with $i=1$, $CH^1(X)^{(0)} = \operatorname{Pic}^0(X_K)$ and $CH^d(X)^{(0)}$ is generated by degree-zero $0$-cycles. The refined height pairing

$$\operatorname{Pic}^0(X_K) \times A_0(X_K) \to CH^1(B)\otimes\mathbb{Q}$$

recovers the classical Beilinson–Bloch height pairing on homologically trivial cycles (Kahn et al., 2020, Bloch et al., 2022, Gil et al., 2020). In the case $X_K = A$ an abelian variety, this coincides up to sign conventions with Moret–Bailly’s geometric height and Schneider’s $l$-adic height (Kahn et al., 2020).

For smooth curves, the “height” pairing is equivalent to the Néron–Tate pairing, via explicit arithmetic local-global formulas involving Archimedean integration (regularized Néron differential periods) and finite-place intersection theory (Bloch et al., 2022). The self-pairing for a divisor $D$ is given by

$$\hat{h}(D) = \frac{1}{2}\langle D, D\rangle_{\mathrm{NT}} = \frac{1}{2}\sum_\nu \langle D,D\rangle_\nu$$

where local contributions $\langle D,D\rangle_\nu$ are defined for both Archimedean ($\nu|\infty$) and non-Archimedean ($\nu\nmid\infty$) places by integration of differentials of the third kind and intersection indices (Bloch et al., 2022).

3. Cohomological and Hodge-theoretic Interpretation

The Beilinson–Bloch height pairing admits a Hodge-theoretic formulation as the “height” of a biextension in the category of mixed Hodge structures (Bloch et al., 2022, Gil et al., 2024, Beilinson, 2022, Gil et al., 2020). For a pair of (complementary codimension) cycles intersecting properly and having homologically trivial classes, there exists a mixed Hodge structure $B_{Z,W}$ fitting in a three-step extension

$$0 \to \mathbb{Q}(1) \to W_{-2}B \to H \to 0, \quad 0 \to W_{-2}B \to W_{-1}B \to \mathbb{Q}(0) \to 0$$

where $H$ is the appropriate cohomology (e.g., $H^{2p-n-1}$ for higher cycles). The integral of suitable Green currents produces a real-valued height via the Deligne splitting (Gil et al., 2020, Gil et al., 2024).

In degeneration settings, the limit mixed Hodge structure associated to smoothing a singular fiber encodes the height pairing of cycles supported on the exceptional divisors of the resolution (Beilinson, 2022). Bloch’s conjecture, confirmed by Beilinson, asserts that the Beilinson–Bloch height equals the Hodge period of the limit mixed Hodge structure modulo $\mathbb{Q}\log|k^\times|$.

For higher Chow cycles, the archimedean height pairing takes the form

$$h_1(Z,W) = \Im \langle e_H^\vee, \delta_H(e_H) \rangle, \qquad h_2(Z,W) = \langle e_H^\vee, \delta_H(e_H) \rangle$$

where $H$ is a framed mixed Hodge structure attached to the intersection product of $Z$ and $W$ (Gil et al., 2024).

4. Degeneration and Asymptotics of the Pairing

In one-parameter degenerating families, the archimedean height pairing $\langle Z_t, W_t\rangle_\infty$ is closely controlled by the non-archimedean geometric intersection pairing of admissible liftings to the total space (Chen, 28 Dec 2025, Nakayama, 27 Dec 2025). The main conjecture, proved for algebraically trivial cycles and supported by explicit monodromy computations, is that

$$\langle Z_t,W_t\rangle_\infty + \mu_0 \log|t| \quad \text{extends continuously at } t=0, \qquad \mu_0 = \langle Z, W\rangle_{\mathrm{geom},\,0}$$

with $E_{\langle z, w\rangle}$ the algebraic height bundle on the base.

The archimedean local height pairing corresponds to the logarithm of the metric on the biextension line bundle in the Hodge-theoretic setting (Nakayama, 27 Dec 2025), and the leading $\log|t|$ term is naturally identified with the non-archimedean intersection multiplicity.

5. Positivity, Nondegeneracy, and Index Theorems

The Beilinson–Bloch height pairing satisfies positivity and index theorems reminiscent of the classical Hodge index theorem (Zhang, 2010, Kahn et al., 2020, Chen, 28 Dec 2025). On the space of homologically trivial cycles modulo numerical equivalence,

$Q(z, w) = \langle z, w\rangle_{BB}$

is positive semi-definite of rank 1; for $z$ primitive, $\langle z, z\rangle_{BB} \geq 0$ with equality if and only if $z$ is numerically trivial.

Explicitly, the self-pairing line bundle $L=E_{\langle z, z^\vee\rangle}$ carries a semipositive (biextension) metric, so that

$\deg\,L = \langle z, z^\vee\rangle_X \geq 0$

(Chen, 28 Dec 2025). Applications include explicit lower bounds for the Faltings height of curves, the Bogomolov conjecture, and non-negativity for Gross–Schoen cycles.

6. Generalizations: Higher Dimensions, Number Fields, and L-functions

The refined Beilinson–Bloch pairing is extended to higher-dimensional bases (e.g., $B$ an arbitrary smooth projective variety) by leveraging intersection products, regulator maps, and perverse sheaf techniques (Rössler et al., 2020, Wisson, 11 Aug 2025). Over algebraically closed fields the pairing takes values in $CH^1(B)_\mathbb{Q}$ or $H^2(B, \mathbb{Q}(1))$, projecting to $\mathbb{Q}$ via chosen ample classes.

For varieties defined over number fields, the pairing is realized in the arithmetic Chow groups of Gillet–Soulé via Green currents and Arakelov intersection theory, extending to $L$-height pairings using canonical arithmetic liftings aligned with canonical decompositions induced by Hodge-theoretic harmonic forms (Zhang, 2020).

Connections to special $L$-values (e.g., Gross–Zagier, Bloch–Beilinson conjectures) identify determinants or specific pairings with derivatives or leading terms of motivic $L$-functions (Lilienfeldt et al., 2024, Bajpai et al., 2022); see explicit formulas for generalized Heegner cycles, Picard modular motives, and compatibility with Eisenstein cohomology (Lilienfeldt et al., 2024, Bajpai et al., 2022).

7. Functorial Properties and Projection Formulas

The pairing is $\mathbb{Q}$-bilinear, symmetric when $i=d+1-i$, and functorial with respect to pushforward, pullback, and correspondences (Kahn et al., 2020, Wisson, 11 Aug 2025). For finite morphisms, it satisfies the projection formula

$$h_X(g_*\alpha, \beta) = f_* h_{X'}(\alpha, g^*\beta)$$

matching the behavior of intersection products and ensuring compatibility with the motivic and cohomological frameworks.

In summary, the Beilinson–Bloch height pairing and its refined versions provide a versatile and robust scheme for measuring arithmetic and geometric complexity of algebraic cycles, unifying intersection theory, Hodge structures, arithmetic Chow groups, and the conjectural structure of motivic L-functions (Kahn et al., 2020, Bloch et al., 2022, Rössler et al., 2020, Chen, 28 Dec 2025, Wisson, 11 Aug 2025, Beilinson, 2022, Gil et al., 2020, Gil et al., 2024, Zhang, 2010, Zhang, 2021, Zhang, 2020, Lilienfeldt et al., 2024, Bajpai et al., 2022).