Height Pairing on Higher Cycles and Mixed Hodge Structures II

Technical Summary: Height Pairing on Higher Cycles and Mixed Hodge Structures II

Problem Statement
The paper addresses the generalization of the archimedean component of the height pairing, originally defined by Beilinson for classical algebraic cycles homologous to zero, to the setting of Bloch's higher Chow cycles. Specifically, the authors consider pairs of refined higher cycles $Z \in Z_p(X, n)_{00}$ and $W \in Z_q(X, n)_{00}$ on a smooth projective variety $X$ of dimension $d$, which intersect properly and satisfy the complementary codimension condition $p + q = d + n + 1$ with $n \geq 1$.

A primary difficulty in extending the classical theory to higher cycles ($n \geq 1$) is that the cycle classes in de Rham cohomology vanish, meaning one cannot restrict to cycles homologous to zero. Furthermore, unlike the classical case ($n=0$), properly intersecting higher cycles do not necessarily have empty intersections; they may intersect in a finite set of points after pulling back to $X \times (\mathbb{P}^1)^{2n}$. This lack of a "local product situation" prevents the immediate application of standard duality arguments required to define mixed Hodge structures (MHS) and associated height pairings.

Methodology
The authors employ a multi-step construction combining algebraic geometry, Hodge theory, and the analysis of differential forms:

1. Framed Mixed Hodge Structures: The authors define a "framed mixed Hodge structure" as an MHS equipped with specific morphisms from and to Tate structures ($\mathbb{Q}(0)$ and $\mathbb{Q}(n+1)$). They introduce two distinct notions of height, $ht_1$ and $ht_2$, for such structures.

   • $ht_1$ is defined via the imaginary part of the pairing between Betti generators in the Deligne bigrading.
   • $ht_2$ is defined via the Deligne splitting map $\delta_H$.
   • These definitions are shown to recover known single-valued polylogarithms (Bloch-Wigner, Brown) when applied to polylogarithm variations.

2. Resolution of Singularities and Blow-ups: To handle the intersection of higher cycles, the authors construct a specific resolution of singularities. They blow up the space $X \times (\mathbb{P}^1)^{2n}$ along centers contained in the union of the cycles $Z$ and $W$. This process, utilizing Hironaka's resolution of singularities, produces a space $X_{Z,W}$ where the strict transforms of the cycles and the relevant divisors are in "local product situation." This allows for the definition of a dual pair of relative cohomology groups.

3. Differential Forms and Cohomology: A crucial technical innovation is the use of a complex of differential forms that are "slowly increasing" along one divisor ($A$) and "rapidly decreasing" along another ($B$), denoted $E^*_X(si A, rd B)$.

   • Unlike the standard complex of forms with logarithmic singularities vanishing along a divisor ($\Sigma_B E^*_X(\log A)$), this new complex allows for the representation of the exceptional cup-product and complex conjugation operations necessary for the construction.
   • The authors prove that this complex computes the relative cohomology $H^*(X \setminus A, B)$ and is compatible with pullbacks, blow-ups, and duality.

4. Construction of the Height Pairing:

   • The cycle classes $cl(Z)$ and $cl(W)$ are lifted to specific cohomology groups on the resolved space $X_{Z,W}$.
   • These classes define the framing morphisms $\phi_Z$ and $\psi_W$ for the MHS $H_{Z,W}$.
   • The height pairing $ht_i(Z, W)$ is then defined as the height of this framed MHS.

Key Contributions and Results

• Generalization to Higher Cycles: The paper successfully attaches a mixed Hodge structure to any pair of properly intersecting higher cycles in complementary codimensions, removing the restrictive assumptions required in their previous work ($n=1$).
• Two Height Pairings: The authors define and analyze two height pairings, $ht_1$ and $ht_2$. They prove these are independent of the choice of resolution of singularities and satisfy symmetry properties: $ht_i(W, Z) = (-1)^n ht_i(Z, W)$.
• Explicit Formulas: For $ht_1$, the authors derive an explicit integral formula involving differential forms $\eta_Z, \eta_W$ and Wang forms $\alpha_1, \alpha_2$:
  $$ht_1(Z, W) = \text{Im} \left( \frac{(-1)^{p+n}}{(2\pi i)^{p+q+2n}} \int_{X \times (\mathbb{P}^1)^{2n}} p_2^*\eta_W \wedge \alpha_1 \wedge \overline{p_1^*\eta_Z} \wedge \overline{\alpha_2} \right)$$
• Vanishing Results:
  • If $X, Z, W$ are defined over $\mathbb{R}$ and $n$ is odd, the height vanishes.
  • Dimensional Constraint: The paper proves that if $p, q > \dim(X)$, then $ht_1(Z, W) = 0$. This implies that for a non-zero height, one of the codimensions must be less than or equal to the dimension of the variety.
• Connection to Polylogarithms: The construction recovers the single-valued polylogarithms of Bloch, Wigner, and Brown as specific instances of these heights applied to the polylogarithm variation of MHS.

Significance and Claims
The authors state that this work generalizes the archimedean component of the height pairing to higher cycles, providing a Hodge-theoretic framework for these objects. They emphasize that while the construction yields two different heights ($ht_1$ and $ht_2$) which carry distinct information for general MHS, they coincide (up to normalization) for the specific biextensions arising in the classical and $n=1$ cases.

The paper positions these heights as the archimedean component of a global height pairing, analogous to the classical Beilinson height, though the global definition remains an open problem. The work is modest in its scope, focusing on the construction and properties of these local heights rather than claiming a complete solution to the Beilinson conjectures for higher cycles. The primary contribution is the rigorous establishment of the mixed Hodge structure and the associated height pairings in the presence of non-trivial intersections of higher cycles, facilitated by the novel use of slowly increasing/rapidly decreasing differential forms.