# P7: Hodge conjecture

- ID: `P7`
- Reference: `hodge-conjecture`
- Page: https://theoremdb.org/statements/P7
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(X\) be a smooth projective complex variety and let \(p\ge 0\). Every class in \(H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)\) is a rational linear combination of cohomology classes of algebraic cycles of codimension \(p\).

### Context

The conjecture asks which topological classes on projective varieties arise from algebraic subvarieties.

### Problem setup

- **Definition (A rational Hodge class of degree 2p).** A rational Hodge class of degree 2p is a class in rational cohomology whose complexification lies in the (p,p) summand of the Hodge decomposition.
- **Definition (An algebraic cycle of codimension p).** An algebraic cycle of codimension p is a finite integer linear combination of codimension-p algebraic subvarieties; its cycle class lies in degree-2p cohomology.
- **Remark.** The conjecture asks which topological classes on projective varieties arise from algebraic subvarieties.

### What counts as a solution

- Prove the stated rational cycle-class assertion for every smooth projective complex variety and every p, or give a smooth projective complex variety with a rigorously verified rational (p,p) class outside the rational span of algebraic cycle classes.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Clay Mathematics Institute continues to list the rational Hodge conjecture as unsolved and notes known special cases, including varieties of complex dimension below four. Exact unresolved remainder: Prove the rational cycle-class assertion for every smooth projective complex variety and codimension, or give a rigorously verified rational Hodge class outside the rational span of algebraic cycle classes. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Clay Mathematics Institute continues to list the rational Hodge conjecture as unsolved and notes known special cases, including varieties of complex dimension below four. Exact unresolved remainder: Prove the rational cycle-class assertion for every smooth projective complex variety and codimension, or give a rigorously verified rational Hodge class outside the rational span of algebraic cycle classes.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Clay Mathematics Institute continues to list the rational Hodge conjecture as unsolved and notes known special cases, including varieties of complex dimension below four.

Exact unresolved remainder: Prove the rational cycle-class assertion for every smooth projective complex variety and codimension, or give a rigorously verified rational Hodge class outside the rational span of algebraic cycle classes.

### Background and intake notes

- Original intake status: The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- Status and formulation were checked against the Clay Mathematics Institute page on 2026-07-22.
- Consult the cited official problem description for the integral, rational, and primitive formulations and known cases.

### Open directions

- **Route 1** (reported): Prove the stated rational cycle-class assertion for every smooth projective complex variety and every p, or give a smooth projective complex variety with a rigorously verified rational (p,p) class outside the rational span of algebraic cycle classes. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hodge-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Clay Mathematics Institute, Hodge Conjecture, official Millennium Prize Problem page, checked 2026-08-01. Official Problem Description by Pierre Deligne; listed under Unsolved Millennium Prize Problems https://www.claymath.org/millennium/hodge-conjecture/
   - Also cited at official problem description, known special cases, and unsolved classification
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source use: original_summary
   - The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Provides the authoritative rational formulation and current status boundary.
   - Source named by the research packet.