Supersingular Tate conjecture for irreducible symplectic varieties
Let be a supersingular irreducible symplectic variety over a field of positive characteristic. The conjecture predicts that satisfies the supersingular Tate conjecture: the expected cycle-class maps from algebraic cycles to the entire -adic and crystalline cohomology ring are surjective; equivalently, the corresponding Tate-type Chow-motive properties hold. In particular, for every finite product of such varieties, every Tate class in the -adic or crystalline cohomology of should be algebraic.
References
Primary source
Additional references
- Supersingular Tate conjecture for irreducible symplectic varieties of known types: I — arXiv — Lie Fu, Xuanlin Huang, Zhiyuan Li
Progress summary
A new preprint proves the prediction for several important families, but the full statement for all such varieties remains open.
The conjecture predicts the expected cycle and motive properties for every supersingular irreducible symplectic variety in positive characteristic. The available results cover important constructions but do not establish the universal statement.
Known results
- Moduli spaces of sheaves on supersingular surfaces: under numerical and coprimality hypotheses, specified varieties are birational to Hilbert schemes and have Tate-type rational Chow motive.
- Albanese fibers of moduli spaces on supersingular abelian surfaces: under and general-polarization hypotheses, the supersingular abelian-motive and related conjectures hold.
- O’Grady sixfolds: for , the specified construction satisfies equivalent supersingularity, unirationality, Tate-motive, and cohomological conclusions.
- Unirationality remains open for generalized Kummer varieties of dimension at least .
September 30, 2026 development
Lie Fu, Xuanlin Huang, and Zhiyuan Li report results for additional specified deformation types, including Tate Chow-motive or supersingular abelian-motive conclusions and consequences for products. The preprint imposes lifting, characteristic, coprimality, and Artin-invariant restrictions; it explicitly does not prove the full conjecture.
Current status (as of October 2026): The conjecture is established or claimed for several specified families, while the general case for all supersingular irreducible symplectic varieties remains open.
Sources
- irma.math.unistra.fr
- ar5iv.labs.arxiv.org
- arxiv.org
- arxiv.org
- hal.science
- kskedlaya.org
- mathoverflow.net
- math.stackexchange.com
- arxiv.org
- www-cdn.anthropic.com
- quantamagazine.org
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
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