Mathematics Colloquium- Eyal Markman- UMass

Colloquium Title: Abelian varieties with complex multiplication and the Hodge conjecture

Abstract: Complex abelian varieties are compact complex tori that can be embedded as complex subvarieties of the complex projective space. They are among the most fundamental algebraic varieties and their study goes back to Riemann. In the 1950s Hodge introduced a necessary condition for a cohomology class of a smooth algebraic variety to be represented by a linear combination of classes of algebraic subvarieties. He conjectured that the condition is sufficient, i.e., that Hodge classes are algebraic. The Hodge conjecture is known in dimension at most three, but until recently it was not known even for 4-dimensional abelian varieties. Furthermore, Weil identified Hodge classes on 4-dimensional abelian varieties with complex multiplication and proposed them in the late 70's as test cases for the Hodge conjecture. In the late 90's it was realized that the algebraicity of the Weil classes would imply the Hodge conjecture for abelian varieties of dimension 4 and 5. We will explain the terms  above and the further developments that were needed for the recent proof that the Weil classes on abelian 4-folds are algebraic.