# On Hodge Cycles on Products of Certain Algebraic Varieties

#### Abstract

We say that two Hodge structures H1 and H2 are orthogonal if HomHS(H1, H2(m)) = 0 for any integer m. Let X1 and X2 be complex algebraic varieties. In some cases, if certain cohomology groups of X1 and X2 are orthogonal as Hodge structures, one can deduce that the Hodge conjecture holds for the product X1 × X2. We make this statement precise, showing which cohomology groups are required to be orthogonal for the Hodge conjecture to hold in the case when X1 and X2 are a curve and a threefold and in the case when X1 and X2 are two surfaces. We then prove that the required orthogonality conditions to imply the Hodge conjecture for X1 × X2 hold in a number of specific examples. We examine the product of a curve and a threefold, the product of a Fermat surface and a very general surface, and the product of two Abelian surfaces. For these classes of varieties, we are able to show that the Hodge conjecture holds in all cases when one of the factors is chosen outside at most a countable union of subvarieties of an appropriate moduli space. We also analyze products of two K3 surfaces and powers of Kummer surfaces and show how, after adding appropriate assumptions, similar results can be deduced in these scenarios as well.

#### Degree

Ph.D.

#### Advisors

Arapura, Purdue University.

#### Subject Area

Mathematics

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