Sunto
Nella prima parte di questa nota ricordiamo la congettura generalizzata di Hodge (G.H.C.) e la sua correzione formulata da Grothendieck (G.G.H.C.). Dopo aver dimostrato una stima sulla dimensione del sottotoro di Hodge massimale della jacobiana intermedia di una threefold abeliana, tale stima viene applicata per verificare la (G.G.H.C.) per il generico elemento di alcune famiglie di threefolds abeliane caratterizzate geometricamente. In particolare (G.G.H.C.) viene verificata per tutti i tripli prodotti di una curva ellittica.
Summary
In the first section of this note we recall the generalized Hodge conjecture (G.H.C.) and its correction as stated by Grothendieck (G.G.H.C.).
After proving an estimate on the dimension of the maximal Hodge subtorus of the intermediate jacobian of an abelian threefold, we apply it to check (G.G.H.C.) for the generic element of some geometrically defined families of abelian threefolds.
In particular (G.G.H.C.) is checked here for all triple products of an elliptic curve.
This is a preview of subscription content, log in via an institution to check access.
References
[Ba.]Bardelli F.,On Grothendieck’s generalized Hodge conjecture for a family of threefolds with geometric genus one. Proceedings Conf. Algebraic Geometry Berlin 1985, Teubner Texte zur Math. Band. 92.
[Co-Mu]Conte A., Murre J.,The Hodge conjecture for fourfolds admitting a covering by rational curves. Math. Ann. 238 (1978).
[De]Deligne P.,Theorie de Hodge III. Publ. Math. I.H.E.S. 44 (1974).
[Gr]Grothendieck A.,Hodge’s general conjecture is false for trivial reasons. Topology 8 (1969).
[Ha.1]Hartshorne R.,Ample subvarieties of algebraic varieties. Lecture Notes in Math 156 Springer Verlag 1970.
[Ha.2]Hartshorne R.,Algebraic Geometry. Springer Grad. Text in Math. Band 52, 1977.
[Li]Liebermann D.,Higher Picard varieties. Ann. Journ. of Math., 90 (1968).
[Ko]Konno K. et al.—References on Hodge Theory. Tohoku University 1987.
[Ra]Ran Z.,Cycles on Fermat hypersurfaces. Compositio Math. 42 (1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bardelli, F. A footnote to a paper by A. Grothendieck. Seminario Mat. e. Fis. di Milano 57, 109–124 (1987). https://doi.org/10.1007/BF02925045
Issue Date:
DOI: https://doi.org/10.1007/BF02925045