منابع مشابه
The Hodge Conjecture for General Prym Varieties
We work over C, the field of complex numbers. The Prym variety of a double cover C → D of a smooth connected projective curve D by a smooth connected curve C is defined (see [7]) as the identity component of the kernel of the norm homomorphism N : J(C) → J(D) between the Jacobians of the curves. This is an abelian variety polarised by the restriction of the canonical polarisation on J(C); we de...
متن کاملKuga-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 کاملThe Tate Conjecture for Certain Abelian Varieties over Finite Fields
In an earlier work, we showed that if the Hodge conjecture holds for all complex abelian varieties of CM-type, then the Tate conjecture holds for all abelian varieties over finite fields (Milne 1999b). In this article, we extract from the proof a statement (Theorem 1.1) that sometimes allows one to deduce the Tate conjecture for the powers of a single abelian variety A over a finite field from ...
متن کاملSome Remarks on the Hodge Conjecture for Abelian Varieties
Let X be a smooth complex projective variety of dimension g. A Hodge class of degree 2d on X is, by definition, an element of H(X,Q)∩H(X). The cohomology class of an algebraic subvariety of codimension d of X is a Hodge class of degree 2d. The classical Hodge conjecture states that any Hodge class on X is algebraic, i.e., a Q-linear combination of classes of algebraic subvarieties of X. Lefsche...
متن کاملA counterexample to the Hodge conjecture for Kähler varieties
H(X,C) = ⊕p+q=kH (X). A class α ∈ H(X,Q) is said to be a rational Hodge class if its image in H(X,C) belongs to H(X). As is well known, the classes which are Poincaré dual to irreducible algebraic subvarieties of codimension p of X are degree 2p Hodge classes. The Hodge conjecture asserts that any rational Hodge class is a combination with rational coefficients of such classes. In the case of a...
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 1979
ISSN: 0386-2194
DOI: 10.3792/pjaa.55.111