A generalization of the Kuga-Satake construction
نویسنده
چکیده
The Kuga-Satake construction [3] associates to a polarized Hodge structure H of weight 2 with h2,0 = 1 an abelian variety A which satisfies the property that H is a sub-Hodge structure of Hom (H1(A),H1(A)). The construction is very tricky and intriguing geometrically: one first associates to the lattice (H,<,>) its Clifford algebra C(H), which is again a lattice. Then one constructs a complex structure on C(H) ⊗ R, using the rank 1 subspace H2,0 ⊂ H ⊗ C defining the Hodge structure on H. Thus the quotient C(H)⊗ R C(H)
منابع مشابه
Kuga-satake Varieties and the Hodge Conjecture
Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of Kuga-Satake varieties. The Hodge conjecture is discussed in section 2. An excellent survey of the Hodge conjecture for abelian varieties is [...
متن کاملEffective Computation of Picard Groups and Brauer-manin Obstructions of Degree Two K3 Surfaces over Number Fields
Using the Kuga-Satake correspondence we provide an effective algorithm for the computation of the Picard and Brauer groups of K3 surfaces of degree 2 over number fields.
متن کاملAbelian Varieties of Weil Type and Kuga-satake Varieties
We analyze the relationship between abelian fourfolds of Weil type and Hodge structures of type K3, and we extend some of these correspondences to the case of arbitrary dimension.
متن کاملHalf Twists of Hodge Structures of Cm-type
To a Hodge structure V of weight k with CM by a field K we associate Hodge structures V −n/2 of weight k + n for n positive and, under certain circumstances, also for n negative. We show that these ‘half twists’ come up naturally in the Kuga-Satake varieties of weight two Hodge structures with CM by an imaginary quadratic field.
متن کاملThe Hodge Conjecture for Self-products of Certain K3 Surfaces
We use a result of van Geemen [vG4] to determine the endomorphism algebra of the Kuga–Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of P which are ramified along six lines.
متن کامل