Algebraicity of Hodge classes

On a non-singular complex projective variety, every Hodge class (a rational (p,p)-class in Betti cohomology) is a rational linear combination of classes of algebraic cycles.

OPEN gate update (Cycle 141): Bloch–Srinivas 1983 gives classical decomposition-of-the-diagonal scaffolding toward algeb —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 146): Goresky–MacPherson 1980/83 gives classical intersection-homology / duality scaffolding tow —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 151): Beilinson–Bernstein–Deligne 1982 (BBD) gives classical perverse-sheaf / decomposition / pu —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 156): de Cataldo–Migliorini 2005 gives classical geometric / relative Hard Lefschetz scaffolding —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 161): André 1996 gives classical motivated-cycle scaffolding toward Hodge; it does NOT prove Cla —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 166): Deligne–Mostow 1986 gives classical hypergeometric monodromy / ball-quotient period scaffo —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 171): Atiyah–Hirzebruch 1962 shows the integral Hodge conjecture fails (torsion); this does NOT —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 176): Clemens–Schmid 1977 gives classical degeneration / limiting MHS scaffolding; this does NOT —gate→ Algebraicity of Hodge classes

OPEN — Clemens 1983 (Publ. Math. IHÉS 58, 19–38) is classical Griffiths-group / non-finite-generation scaffolding; it do —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 181): Carlson–Green–Griffiths–Harris 1983 gives classical IVHS / infinitesimal period-map scaffo —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 186): Piatetski-Shapiro–Shafarevich 1971 gives classical polarized K3 Torelli / period-map scaff —gate→ Algebraicity of Hodge classes

Cycle 191 advances classical Beauville 1983 (c1=0 Kähler / Beauville–Bogomolov decomposition) as PROPOSED scaffolding un —gate→ Algebraicity of Hodge classes

GATE STATUS (honest, Cycle 195): mp_hodge and gt_hodge_alg remain OPEN. Zucker 1979 L²/VHS scaffolding is classical IMPO —gate→ Algebraicity of Hodge classes

GATE STATUS (honest, Cycle 199): mp_hodge and gt_hodge_alg remain OPEN. Kollár 1986 higher direct images of dualizing sh —gate→ Algebraicity of Hodge classes

GATE STATUS (honest, Cycle 203): mp_hodge and gt_hodge_alg remain OPEN. Grothendieck 1969 Standard Conjectures of Lefsch —gate→ Algebraicity of Hodge classes

GATE STATUS (honest, Cycle 207): mp_hodge and gt_hodge_alg remain OPEN. Clemens–Griffiths 1972 intermediate Jacobian / c —gate→ Algebraicity of Hodge classes

Cycle 211 Hodge Donagi 1983 generic Torelli for projective hypersurfaces scaffolding (Compositio Math. 50) does NOT clos —gate→ Algebraicity of Hodge classes

Cycle 215 Hodge Verbitsky 2013 global Torelli for hyperkähler manifolds scaffolding (Duke Math. J. 162) does NOT close C —gate→ Algebraicity of Hodge classes

Cycle 219 Hodge Nikulin 1979 K3 lattice / finite Aut scaffolding (Trudy Moskov. Mat. Obshch. 38; Trans. Moscow Math. Soc —gate→ Algebraicity of Hodge classes

Cycle 223 Hodge Kawamata–Viehweg 1982 vanishing scaffolding (Math. Ann. 261; J. Reine Angew. Math. 335) does NOT close C —gate→ Algebraicity of Hodge classes

Cycle 227 Hodge Siu 1983 every-K3-is-Kähler scaffolding (Invent. Math. 73) does NOT close Clay rational Hodge algebraici —gate→ Algebraicity of Hodge classes

Cycle 231 Hodge Cattani–Kaplan–Schmid 1986 degeneration/LMHS scaffolding (Ann. of Math. (2) 123) does NOT close Clay rat —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 235): Peters–Steenbrink 2008 gives classical Mixed Hodge Structures monograph scaffolding; this —gate→ Algebraicity of Hodge classes

OPEN gate update (Cycle 239): Steenbrink–Zucker 1985 gives classical Variation of mixed Hodge structure II scaffolding; —gate→ Algebraicity of Hodge classes