Incidence equivalence, a survey

Technical Summary: Incidence Equivalence, A Survey

Problem Statement
This survey addresses a fundamental question in the theory of algebraic cycles on smooth complex projective varieties, originally posed by P. Griffiths in the early 1970s. The core problem concerns the relationship between two equivalence relations defined on cycles of codimension $i$ that are algebraically equivalent to zero:

1. Incidence Equivalence ($\sim_{inc}$): A purely geometric relation defined via correspondences. A cycle $\alpha$ is incidence equivalent to zero if its "incidence divisors" on any smooth projective variety $Y$ (induced by integral correspondences of degree $1-i$) are rationally equivalent to zero.
2. Abel-Jacobi Equivalence ($\sim_{AJ}$): A transcendental relation defined via the kernel of the Abel-Jacobi map $\psi^i_{AJ}$, which maps homologically trivial cycles to Griffiths' intermediate Jacobians $J^i(X)$.

The central question, denoted $GC(X, i)$, asks whether these two relations coincide for cycles algebraically equivalent to zero, at least after tensoring with $\mathbb{Q}$. A positive answer would extend Abel's theorem for divisors to higher codimensions.

Methodology and Framework
The paper synthesizes classical results from the 1970s with more recent developments in Hodge theory and the theory of biextensions. The methodology relies on the following structural components:

• Correspondences: The survey utilizes the action of integral correspondences on Chow groups and cohomology to define incidence equivalence and to study the functoriality of the associated varieties.
• Intermediate Jacobians and Picard Varieties: The author distinguishes between Griffiths' intermediate Jacobians $J^i(X)$ (which are generally complex tori, not necessarily abelian varieties) and the "generalized Picard varieties" $Pic^i(X)$, defined as the quotient of algebraically trivial cycles by incidence equivalence. The problem $GC(X, i)$ is reformulated as the question of whether the image of the Abel-Jacobi map, $J^i_{alg}(X)$, is isogenous to $Pic^i(X)$.
• Standard Conjectures: The analysis is deeply intertwined with Grothendieck's standard conjectures, specifically:
  • Conjecture $B(X)$: The Lefschetz operator $\Lambda$ is induced by an algebraic correspondence.
  • Conjecture $D(X)$: Homological equivalence coincides with numerical equivalence.
  • The Generalized Hodge Conjecture ($GHC$): Relating the coniveau filtration on cohomology to Hodge substructures.
• Biextensions: The survey incorporates S. Müller-Stach's work utilizing Bloch's biextensions. This approach connects the splitting of certain $\mathbb{C}^\times$-torsors (Bloch-biextensions) to incidence equivalence, providing a bridge between geometric and transcendental properties.
• Lefschetz Decomposition: A significant portion of the technical argument involves analyzing the primitive components of the Lefschetz decomposition of cohomology classes within the coniveau filtration.

Key Contributions and Results
The paper provides a comprehensive overview of known results and establishes several implications regarding $GC(X, i)$:

1. Equivalence Reformulation: It establishes that $GC(X, i)$ holds if and only if $J^i_{alg}(X)$ is isogenous to the generalized Picard variety $Pic^i(X)$.
2. Implications of Standard Conjectures:
   • If the Generalized Hodge Conjecture $GHC(X, 2i-1, i)$ holds, then $GC(X, i)$ is true.
   • If Conjecture $B(X)$ holds, then all primitive components of the Lefschetz decomposition of algebraic classes are themselves algebraic. This condition, denoted $(\Lambda, X, i)$, implies $GC(X, i)$.
   • Consequently, $GC(X, i)$ is known to hold for abelian varieties, as $B(X)$ is known to hold for them.
3. Specific Cases:
   • Codimension 1 and $d$: $GC(X, 1)$ and $GC(X, d)$ hold (recovering classical results for divisors and 0-cycles via the Picard and Albanese varieties).
   • Codimension 2: The paper reviews J. Murre's proof for $i=2$. It notes that Murre's original geometric argument regarding the non-degeneracy of the cup-product pairing on $H^3_{alg}(X)$ appeared incomplete. The author provides a corrected and more general proof using the Hodge $*$-operator and Weil's results on the Lefschetz decomposition, confirming $GC(X, 2)$.
   • Odd Dimensions and Complete Intersections: For a variety $X$ of dimension $d=2m+1$ where the odd Betti numbers $b_{2k+1}(X)$ vanish for $k \neq m$ (e.g., smooth complete intersections in projective space), $GC(X, m)$ holds.
4. Duality: The survey clarifies the duality between $J^i_{alg}(X)$ and $J^j_{alg}(X)$ (where $i+j=d+1$) and shows that if $GC(X, i)$ holds, then $GC(X, j)$ holds under specific duality conditions.

Significance and Motivation
The author states that the motivation for this survey stems from a recent observation by Z. Chen linking Griffiths' question to the asymptotic behavior of the archimedean height pairing in a geometric setting. Specifically, recent work by Hain and Brosnan–Pearlstein on the asymptotic behavior of height pairings for admissible variations of mixed Hodge structures suggests that if $GC(X, i)$ has a positive answer, a specific coefficient in the asymptotic expansion corresponds to a local geometric height pairing.

The paper aims to consolidate older literature that has become difficult to locate and to clarify arguments that rely on knowledge common in the 1970s but potentially forgotten by newer generations. It serves as a reference connecting the classical theory of incidence equivalence to modern Hodge-theoretic developments, particularly regarding the behavior of classes of Hodge level $\leq 1$ in odd-rank cohomology. The survey does not claim to resolve the general case of $GC(X, i)$ for arbitrary varieties but rather delineates the precise conditions (such as $B(X)$ or $GHC$) under which the equivalence is known to hold.