Beilinson's Height Pairing

• Beilinson’s height pairing is a conjectural canonical pairing on homologically trivial algebraic cycles in complementary codimension, bridging arithmetic intersection theory and Hodge structures.
• It employs both intersection-theoretic and motivic methods to connect algebraic cycles with derivatives of automorphic L-functions and Lefschetz principles.
• Recent advances extend its framework to arithmetic Chow groups, mixed Hodge structures, and perverse cohomology, enhancing its role in modern arithmetic geometry.

Beilinson’s height pairing, often called the Beilinson–Bloch height pairing, is the conjectural canonical pairing on homologically trivial algebraic cycles in complementary codimension. In one standard formulation, for a smooth projective variety $X/\mathbf Q$ of dimension $d-1$, it is a pairing

$$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$

where

$$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$

In geometric settings over function fields, the complementary condition is usually written $p+q=d+1$ for cycles of codimensions $p$ and $q$ on a $d$-dimensional generic fiber (Deninger, 2010, Rössler et al., 2020). The pairing is expected to be nondegenerate, compatible with Lefschetz operators, and definite on primitive cycles with the Hodge-theoretic sign predicted by Bloch and Beilinson (Deninger, 2010). At the same time, many of its most concrete realizations are intersection-theoretic rather than abstractly motivic, and in special cases the pairing is identified with derivatives of automorphic $L$-functions, as on orthogonal Shimura varieties (Andreatta et al., 2015).

1. Definition and conjectural framework

The basic domain of the pairing is the group of homologically trivial cycles. In Deninger’s formulation, for $X/\mathbf Q$ smooth projective of dimension $d-1$0,

$d-1$1

and the expected pairing is

$d-1$2

(Deninger, 2010). This is the arithmetic analogue of a middle-dual intersection form, but restricted to cycles whose ordinary cohomological intersection vanishes.

The Bloch–Beilinson conjecture attached to this pairing has two principal parts. First, the pairing should be nondegenerate. Second, if $d-1$3 is the class of a hyperplane section and $d-1$4, then the Lefschetz operator

$d-1$5

should be an isomorphism, and on primitive cycles the twisted pairing

$d-1$6

should be definite of sign $d-1$7 (Deninger, 2010). This is the height-theoretic analogue of the Hodge–Riemann bilinear relations.

In the geometric function-field setting, Beilinson’s original construction was for the function field of a curve over an algebraically closed field. Rössler and Szamuely formulate the complementary-codimension condition as

$d-1$8

for a smooth projective $d-1$9-dimensional variety over $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$0, and identify this as the natural condition under which intersection on a model pushes down to codimension $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$1 on the base $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$2 (Rössler et al., 2020). The same codimension relation governs later refinements with values in $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$3 or $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$4.

2. Arithmetic-intersection realizations

Over number fields, the standard concrete realization of Beilinson’s height pairing is via arithmetic intersection theory. For a regular arithmetic model $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$5, one works with arithmetic cycles $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$6, where $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$7 is an algebraic cycle and $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$8 is a Green current satisfying that $$(\ ,\ )_{CH}: CH^i(X)^0 \times CH^{d-i}(X)^0 \to \mathbf R,$$9 is smooth. The arithmetic intersection product

$$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$0

followed by the arithmetic degree

$$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$1

gives the standard arithmetic model for the pairing (Goswami et al., 2017). In this framework the height decomposes heuristically into finite-place intersection terms and archimedean contributions from Green currents.

Burgos and Goswami extend this pattern to higher cycles by defining higher arithmetic Chow groups $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$2 for smooth projective varieties over a number field. Their height pairing is defined for higher cycles of complementary dimensions with trivial real regulator by

$$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$3

where $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$4 is the structural morphism and $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$5, $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$6 are higher arithmetic cycles. In degree $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$7, this recovers the classical Beilinson pairing through the identification with Gillet–Soulé arithmetic Chow groups (Burgos-Gil et al., 2017).

A particularly explicit arithmetic Chow-theoretic realization is the orthogonal Shimura variety formula of Howard and Madapusi Pera. Let $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$8 be the Shimura variety attached to $$CH^i(X)^0=\ker\!\big(CH^i(X)\otimes \mathbf Q \to H^{2i}(X(\mathbf C),\mathbf Q)\big).$$9 for a quadratic space of signature $p+q=d+1$0, let $p+q=d+1$1 be its regular integral model, let $p+q=d+1$2 be the arithmetic divisor attached to a harmonic weak Maass form $p+q=d+1$3, and let $p+q=d+1$4 be the CM stack defined by a negative definite rank-$p+q=d+1$5 sublattice. Their main theorem is

$p+q=d+1$6

where $p+q=d+1$7 is the metrized cotautological bundle and $p+q=d+1$8 is the central derivative of a Rankin–Selberg convolution (Andreatta et al., 2015). This theorem is explicitly presented as a Beilinson-type height formula in arithmetic Chow theory rather than in abstract motivic language. When $p+q=d+1$9, the result is a variant of the Gross–Zagier theorem.

3. Function fields and higher-dimensional bases

In the function-field setting, Zhang gives a concrete codimension-$p$0 model of Beilinson’s height pairing over a function field $p$1 of characteristic $p$2, where $p$3 is a smooth projective curve and $p$4 is a smooth projective model of relative dimension $p$5. For a codimension-$p$6 cycle $p$7 satisfying the vertical orthogonality condition and the Albanese-triviality condition

$p$8

the height is realized by the intersection number

$p$9

and the main theorem proves

$q$0

with equality characterized by numerical equivalence to the fiber of a horizontal divisor (Zhang, 2010). In this form, the codimension-$q$1 Beilinson pairing is reduced to the classical Hodge index theorem on the total space.

Kahn answers Beilinson’s question about higher-dimensional bases by constructing a refined pairing

$q$2

for a $q$3-dimensional regular proper variety $q$4 over the function field of a smooth variety $q$5. Here $q$6 is a saturated subgroup defined through admissible liftings to models and local orthogonality conditions at codimension-$q$7 points of $q$8. For $q$9, Kahn proves

$d$0

and when $d$1 is a curve, composition with $d$2 produces a numerical pairing closely related to Beilinson’s geometric height pairing (Kahn et al., 2020). The target $d$3 is the key refinement: over higher-dimensional bases there is no canonical scalar degree, so the natural output is a divisor class on $d$4.

Rössler and Szamuely give a cohomological higher-dimensional generalization in the $d$5-adic category. For $d$6 a smooth integral $d$7-scheme of finite type, $d$8, and $d$9 smooth projective of dimension $L$0, they construct

$L$1

and prove that when $L$2 is a smooth proper curve, composing with the trace isomorphism

$L$3

recovers Beilinson’s original pairing (Rössler et al., 2020). The target $L$4 is the natural cohomological receptacle for a codimension-$L$5 class on the base, and the paper conjectures that the pairing should come from a $L$6-valued construction.

4. Archimedean, Hodge-theoretic, and topological forms

At the archimedean place, Beilinson’s height pairing admits both topological and mixed-Hodge-theoretic realizations. For a smooth complex projective manifold $L$7 of complex dimension $L$8, Hain–Reid–Poonen–Shekhtman construct a topological Abel–Jacobi map for homologically trivial smooth cycles $L$9 and define a topological height pairing

$X/\mathbf Q$0

modulo periods, where $X/\mathbf Q$1 and $X/\mathbf Q$2 is the linking form associated to $X/\mathbf Q$3. They then show that for analytic cycles the corresponding complex-valued holomorphic height has real part given by this topological pairing and imaginary part equal to the classical Beilinson–Bloch–Gillet–Soulé archimedean height. The same construction yields a distinguished lift of the Abel–Jacobi data to the fiber of the Poincaré bundle (Caibar et al., 2011).

For higher Chow cycles, Burgos Gil, Goswami, and Pearlstein replace the classical biextension formalism by framed mixed Hodge structures. For an $X/\mathbf Q$4-framed mixed Hodge structure $X/\mathbf Q$5 they define two heights,

$X/\mathbf Q$6

and for properly intersecting refined higher cycles

$X/\mathbf Q$7

they construct a framed mixed Hodge structure $X/\mathbf Q$8 and define

$X/\mathbf Q$9

These pairings generalize the archimedean height pairing between ordinary cycles, recover the classical biextension picture in the previously understood cases, and yield explicit regulator-period formulas; the comparison with the earlier star-product-based archimedean height for $d-1$00 remains open (Gil et al., 2024).

Beilinson’s paper on nearby cycles gives a precise degeneration formula of a different kind. In a one-parameter degeneration with isolated singularities, he shows that the Hodge period of a three-step mixed Hodge structure extracted from the nearby-cycle limit mixed Hodge structure equals the Beilinson–Bloch height pairing of certain homologically trivial cycles on the blow-up $d-1$01 of the singular fiber. If $d-1$02 are Bloch cycles on the exceptional divisor and $d-1$03 are the corresponding cycle classes, then

$d-1$04

(Beilinson, 2022). This identifies the height pairing with a Hodge period of nearby cycles rather than with an intersection number on the generic fiber.

Chen studies a complementary degeneration problem for the archimedean height pairing in a family $d-1$05 over a complex curve. For fiberwise homologically trivial cycles $d-1$06 of complementary codimension, the archimedean height is

$d-1$07

and Brosnan–Pearlstein imply an asymptotic

$d-1$08

Chen conjectures that $d-1$09 is the local geometric Beilinson–Bloch height $d-1$10, and proves the corresponding identification of the Lear extension with the geometric height line bundle for algebraically trivial cycles, conditional on Griffiths’s conjecture on incidence equivalence (Chen, 28 Dec 2025).

5. Positivity, Lefschetz theory, and conjectural structures

The sign and nondegeneracy properties of Beilinson’s height pairing are traditionally viewed as an arithmetic Hodge index theorem. Deninger formulates the standard Bloch–Beilinson package as the nondegeneracy of $d-1$11, hard Lefschetz on $d-1$12, and the definiteness of

$d-1$13

on primitive cycles with sign $d-1$14 (Deninger, 2010). He then sketches a conjectural arithmetic cohomology theory with an $d-1$15-action, infinitesimal operator $d-1$16, and Hodge $d-1$17-operator such that the height pairing is identified with a positive definite cohomological pairing. In this formalism one expects

$d-1$18

which would imply the conjectural sign rule from positivity of the Hodge inner product (Deninger, 2010).

Zhang’s codimension-$d-1$19 theorem over function fields supplies an unconditional positivity result in one important case. After constructing a horizontal correction cycle $d-1$20 so that $d-1$21 is primitive, the classical Hodge index theorem on the total space gives

$d-1$22

hence

$d-1$23

for the original cycle (Zhang, 2010). This is explicitly identified as the codimension-$d-1$24 case of Beilinson’s Hodge index conjecture over function fields of characteristic $d-1$25.

Zhang’s later $d-1$26-height pairing is a different extension of the formalism. Under conjectural extensions of Grothendieck’s standard conjectures to degenerate fibers, he constructs canonical admissible arithmetic liftings $d-1$27 for all cycles on the generic fiber of a polarized arithmetic variety and defines

$d-1$28

The pairing extends Beilinson–Bloch from homologically trivial cycles to all algebraic cycles, and on the homologically trivial part it recovers the classical pairing exactly (Zhang, 2020). The key mechanism is a Lefschetz-theoretic splitting of the arithmetic Chow group into cohomological, homologically trivial, and vertical pieces, with the cohomological part isotropic.

“The Business of Height Pairings” places the classical Beilinson pairing inside a broader Bloch–Beilinson filtration picture. In particular, it treats Beilinson’s pairing as the $d-1$29 case of a more general family of pairings and states, under conjectural assumptions, the existence of

$d-1$30

The same paper also formulates positivity statements for these higher-graded pairings and connects them to Néron–Tate pairings on algebraically trivial cycles (Goswami et al., 2017).

6. Geometric cases and recent developments

A major unconditional existence theorem is Zhang’s construction of the Beilinson–Bloch height pairing for codimension-$d-1$31 cycles on a threefold of the form

$d-1$32

with $d-1$33 a smooth projective curve and $d-1$34 a smooth projective surface over a number field. He proves that $d-1$35 satisfies the Beilinson–Bloch condition under strict semistability assumptions, and then defines the genuine pairing

$d-1$36

For an embedding $d-1$37, he introduces an arithmetic diagonal cycle $d-1$38, generalizing the Gross–Schoen modified diagonal, and in the function-field case proves the explicit self-height formula

$d-1$39

When $d-1$40 and $d-1$41 is the diagonal, this recovers the Gross–Schoen situation (Zhang, 2021).

A recent structural study of the higher-dimensional geometric pairing is Wisson’s “Properties of the Beilinson Height Pairing.” For $d-1$42 over the function field of a smooth integral base $d-1$43, Wisson constructs a new pairing on a regular projective model by means of perverse truncations of $d-1$44, compares it with the Rössler–Szamuely pairing, and shows that the discrepancy is a boundary term: $d-1$45 Under a “Beilinson extension” condition killing the boundary contribution, the model-theoretic pairing agrees with the Rössler–Szamuely pairing. The paper also proves the projection formula

$d-1$46

for generically finite base change (Wisson, 11 Aug 2025). This places the higher-dimensional geometric pairing much closer to ordinary intersection theory on models.

Taken together, these developments show that “Beilinson’s height pairing” is not a single construction in a single category. It is a coherent family of pairings and height-type invariants appearing in arithmetic Chow groups, in function-field intersection theory on models, in $d-1$47-adic perverse-cohomological extension classes, in mixed Hodge structures and nearby cycles, and in concrete automorphic formulas. The unifying structure is stable: the pairing is attached to homologically trivial cycles, requires archimedean or cohomological correction data, is governed by complementary codimension, and is expected to satisfy a Lefschetz-compatible sign rule. The main unresolved issues remain the full motivic construction, the comparison of different realizations in complete generality, and the unconditional positivity and nondegeneracy conjectures.