Archimedean Height Pairing

• Archimedean height pairing is a bilinear measure defined via Green currents and biextension line bundles, linking algebraic cycles with Hodge theory.
• It connects complex analysis with non-archimedean intersection theory through degeneration formulas and combinatorial interpretations on dual graphs.
• The pairing underpins positivity results and effective bounds in arithmetic intersection theory, influencing Diophantine analysis and heights on abelian varieties.

An Archimedean height pairing is a fundamental object in arithmetic geometry, providing a functorial, bilinear, and geometric measure of arithmetic intersection for algebraic cycles, particularly on abelian varieties and more general algebraic varieties. It is central both to Diophantine analysis and to the study of periods, normal functions, and degeneration phenomena, linking the Arakelov theory on arithmetic varieties with the Hodge-theoretic geometry of their complex points and influencing degeneration theory through its comparison with non-archimedean or local height pairings.

1. Definition and Local-Global Formalism

Let $X$ be a smooth projective complex variety of dimension $d$. For cycles $Z \in Z^p(X)$, $W \in Z^q(X)$ with $p+q=d+1$, homologically trivial (i.e., vanishing in $H^{2p}(X,\mathbb{Q})$ and $H^{2q}(X,\mathbb{Q})$ respectively, and with disjoint supports), the Archimedean height pairing $\langle Z, W \rangle_\mathrm{arch}$ is defined via Green currents $g_W$: $\langle Z, W \rangle_\mathrm{arch} = -\int_Z g_W,$ where $g_W$ satisfies the equation $dd^c g_W + \delta_W = \omega_W$, with $\omega_W$ a smooth representative of the class of $W$ in the Bott–Chern cohomology and $\delta_W$ the Dirac current along $W$.

Alternatively, if $\nu_Z : \mathrm{pt} \to J^p(X)$ and $\nu_W^\vee : \mathrm{pt} \to J^q(X)$ are the corresponding Griffiths normal-functions in intermediate Jacobians, then $\langle Z, W \rangle_\mathrm{arch}$ is the logarithm of the norm of the distinguished trivializing section of the pullback of the Poincaré biextension line bundle to $(\nu_Z, \nu_W^\vee)$ with its biextension metric: $$\langle Z, W \rangle_\mathrm{arch} = -\log\|1_{Z,W}\|.$$ This encoding of the pairing through biextension line bundles reflects its deep links with Hodge theory and period maps (Chen, 28 Dec 2025).

2. Degenerations and Relation to Non-Archimedean Pairings

For a one-parameter semistable degeneration $X \to \Delta$, with smooth projective fibers $X_t$ over $\Delta^*$ and an appropriate extension over all of $\Delta$, the Archimedean height pairing of specializations $h(t) = \langle Z_t, W_t \rangle_\mathrm{arch}$ exhibits a singularity at the boundary $t=0$: $h(t) + \mu_0 \log|t| \in C^0(\Delta),$ where the coefficient $\mu_0$ is conjecturally equal to the purely non-Archimedean height pairing of the specializations $$\langle \mathrm{sp}(Z), \mathrm{sp}(W) \rangle_{na}$$ on the generic fiber $X_\eta$ over $\mathbb{C}((t))$ (Chen, 28 Dec 2025). This “limit = non-archimedean height” relation connects the asymptotics of the complex (archimedean) height pairing with the intersection-theoretic height on a regular model over the disc (non-archimedean, or “geometric height pairing”).

This result is foundational in degeneration theory: in the case of curves, the leading coefficient is precisely the local non-archimedean Néron height pairing (Holmes et al., 2013). The positive-definite character of these limits plays a key role in positivity theorems for arithmetic cycle heights.

3. Explicit Formulas and Computations

For curves, the non-archimedean (or, in the context of degeneration, the limiting archimedean) height pairing is computed in terms of the intersection theory on a semistable model $\mathcal{X}$ over a discrete valuation ring: $$\langle D, D' \rangle_v = (D + \phi(D)) \cdot (D' + \phi(D')),$$ where $\phi(D)$ is a vertical $\mathbb{Q}$-divisor correcting $D$ to ensure orthogonality to all irreducible components of the special fiber. This is Zhang’s admissible pairing (Chen, 28 Dec 2025, Holmes et al., 2013).

There is an equivalent combinatorial description in terms of the reduction (dual) graph $\Gamma$ of the special fiber. The pairing is given by the Moore–Penrose inverse $\mathrm{Green}^+$ of the Laplacian of $\Gamma$: $$\langle D, D' \rangle_v = \mathrm{Green}^+(\overline{D}, \overline{D'}),$$ where $\overline{D}$ represents the reduction of $D$ to the graph.

For principally polarized abelian varieties, the non-archimedean pairing in the context of degenerations coincides with the local Néron–Tate canonical height, computable via Berkovich skeletons and tropical theta functions (Jong et al., 2024, Chen, 28 Dec 2025).

4. Comparison Theorems and Relation to Beilinson–Bloch Heights

Comparison theorems—under hypotheses such as the Griffiths incidence equivalence (that the map from the algebraic intermediate Jacobian to the higher Picard variety is an isogeny)—show that three candidate height line bundles on the base of a semistable degeneration coincide:

• The geometric height line bundle (non-archimedean geometric height),
• The Lear extension of the analytic biextension bundle associated to normal functions,
• The Néron–Tate height line bundle on higher Picard varieties.

Consequently, the non-archimedean geometric height pairing gives the leading term in the degeneration of the archimedean pairing, and both inherit positivity properties from the semipositivity of the biextension metric (Chen, 28 Dec 2025).

A key implication is a geometric interpretation of the asymptotics and positivity of Beilinson–Bloch heights over function fields. For example, on curves, the pairing recovers Deligne’s or Zhang’s admissible pairings, and on principally polarized abelian varieties, it corresponds to the Néron–Tate height, with explicit expressions via skeleta (Jong et al., 2024, Holmes et al., 2013).

5. Consequences: Positivity, Effectivity, and Applications

The identification of the limiting term of the archimedean pairing with the non-archimedean geometric height pairing yields substantial consequences:

• Positivity: By Brosnan–Pearlstein’s semipositivity results, the degree of the relevant line bundles is non-negative, giving $\langle z, z^\vee \rangle_X \geq 0$, i.e., a positivity statement for the global Beilinson–Bloch height pairing over function fields (Chen, 28 Dec 2025).
• Effectivity: In the context of moduli, this framework is used to prove that the “height jumping” divisor related to the degeneration of normal functions is effective, as conjectured by Hain, by interpreting jumps as a sum of local Green’s function values which are non-negative (Holmes et al., 2013).
• Explicit Bounds: The detailed description of the height degeneration enables effective and algorithmic approaches to Diophantine problems and equidistribution (Jong et al., 2024).

6. Extension to Higher Codimension and Open Problems

For higher codimension cycles, the global archimedean height pairing via Green currents or the bi-extension formalism on complex varieties continues to admit degeneration and comparison to non-archimedean geometric (intersection-theoretic) height pairings, provided appropriate regular models and cycle liftings are available (Chen, 28 Dec 2025).

Many open problems remain, notably:

• The generalization of these results to more singular degenerations,
• Direct perverse-sheaf or motivic interpretations of the degenerating height pairings,
• The full geometric realization of Beilinson’s height pairings in terms of explicit Arakelov-theoretic and moduli-theoretic data.

7. Summary Table: Key Aspects

Aspect	Archimedean Setting	Degeneration/Limit	Non-Archimedean/Model Computation
Definition	$-\int_Z g_W$ (Green currents)	Limit of pairing on $X_t$	Intersection on regular model; skeletons
Hodge-theoretic Structure	Intermediate Jacobian, period	Monodromy, period map asymptotics	Special fiber; dual (reduction) graph
Leading Coefficient in Degenerations	$\mu_0$ in $h(t)+\mu_0\log|t|$	$$\mu_0 = \langle \mathrm{sp}(Z), \mathrm{sp}(W) \rangle_{na}$$	Zhang's admissible pairing, Green kernel
Positivity	Biextension metric semipositivity	Retained in limit	Non-negativity of combinatorial pairing
Applications	Arithmetic intersection theory	Height jumping, function field positivity	Algorithmic Diophantine analysis, equidistribution

The Archimedean height pairing and its limiting behavior encode deep arithmetic, Hodge-theoretic, and intersection-theoretic information, serving as a bridge between analysis on complex varieties, combinatorial reductions on models, and the global arithmetic of algebraic cycles, as systematically developed and compared in (Jong et al., 2024, Chen, 28 Dec 2025), and (Holmes et al., 2013).