A conditional reduction for algebraic Kuga–Satake correspondences

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A conditional reduction for algebraic Kuga–Satake correspondences

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Recursos

Take one nonzero algebraic map from the primitive cohomology of a K3 surface to H² of some abelian variety. With it you can rebuild the whole Kuga–Satake correspondence, then carry that result across a polarized family. This class follows the reduction and points to the one step that is still a hypothesis.

The setup and the single input

For a polarized K3 surface S, the primitive cohomology V = h⊥ has dimension 21 and signature (2,19). The Kuga–Satake map κ_S : T → W ⊗ W ⊂ H²(A×A) is a Hodge morphism. Whether an algebraic cycle Γ_S induces it is open, and it is a case of the rational Hodge conjecture [0:21]. If Hg(V) = SO(V,q), three conditions are equivalent [0:49]:

Some cycle α gives a *nonzero map α: V → H²(B,Q)**, for any abelian variety B.
Some surface cover f: Y → S has wedges of one-forms that push forward to a nonzero two-form.
The full prescribed Kuga–Satake correspondence is algebraic.

Why one map recovers everything

Weight one forces spin [1:48]. The Hodge-circle generator D has eigenvalues ±1 on H¹(B). A piece with highest weight λ spreads its eigenvalues over a width of 4λ₁, so only the trivial piece and the spin representation Δ fit. This gives U_C = (Δ⊗M) ⊕ M₀.
One map fixes the shape [2:54]. V occurs exactly once in Δ⊗Δ, so α′_C = γ ⊗ t. The spin part is forced, and all the freedom sits in t.
Independent operations on the two slots [3:22]. Abelian homomorphisms act on each tensor factor separately, so the images (a⊗b)t span N₁⊗N₂.
Rational descent: the complex numbers only compute a span. You never need to split the Clifford algebra or descend a single spin copy. Because the maps and cycles are rational, Γ = Σ c_ij Γ_ij with c_ij ∈ Q.

Spreading, and what stays open

If the input holds very generally in a polarized component, Hilbert-scheme parameter spaces carry the cycle to every point [4:16]. Density of good points is not enough. You pick a very general point outside E (by Baire), and its components surject onto the whole disc. The seed itself is still unproved: a nonzero α must be an actual algebraic cycle, and absolute Hodge or motivated classes do not count [5:08].

When you read a reduction like this, trace three moves: name the single input, find the symmetry that forces it to grow, check how it spreads.

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