Let M be a simple hyperkähler manifold. Kuga-Satake construction gives an embedding of H 2 (M, C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H • (M, C) → H •+l (T, C) for some l, which is compatible with the Hodge structures and the Poincaré pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl(2)-triples on M .

The purpose of this note is to give an account of a well-known folklore result: the Hodge structure on the second cohomology of a compact hyperkähler manifold uniquely determines Hodge structures on all higher cohomology groups. We discuss the precise statement and its proof, which are somewhat difficult to locate in the literature.

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a b s t r a c t Let (M, I, J, K ) be a hyperkähler manifold, and Z ⊂ (M, I) a complex subvariety in (M, I).We say that Z is trianalytic if it is complex analytic with respect to J and K , and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures (M, I, J ′ , K ′ ) containing I. For a generic complex structure I on M, all complex subvarieties of (M, I) are absolutely trianalytic. It is known that the normalization Z ′ of a trianalytic subvariety is smooth; we prove thatTo study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold X is a k-dimensional space R of closed 2-forms on X which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in R is a smooth, non-degenerate quadric hypersurface in R.We consider absolutely trianalytic tori in a hyperkähler manifold M of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where k = b 2 (M). We show that the tangent bundle TX of a k-symplectic manifold is a Clifford module over a Clifford algebra Cl(k − 1). Then an absolutely trianalytic torus in a hyperkähler manifold M with b 2 (M) 2r + 1 is at least 2 r−1 -dimensional.

Let $X_1$ and $X_2$ be deformation equivalent projective hyperkähler manifolds. We prove that the André motive of $X_1$ is abelian if and only if the André motive of $X_2$ is abelian. Applying this to manifolds of $\mbox {K3}^{[n]}$, generalized Kummer and OG6 deformation types, we deduce that their André motives are abelian. As a consequence, we prove that all Hodge classes in arbitrary degree on such manifolds are absolute. We discuss applications to the Mumford–Tate conjecture, showing in particular that it holds for even degree cohomology of such manifolds.