UCLA

The focus of this thesis is $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$, which are $\mathbb{Q}$-vector spaces $V$ equipped with a decomposition into $n$-dimensional complex subspaces

$V\otimes _\mathbb{Q}\mathbb{C}=V^{w,0}\oplus V^{0,w}$

such that the two subspaces $V^{w,0}$ and $V^{0,w}$ are conjugate to each other. In this first part of the thesis, we investigate the possible Hodge groups of simple polarizable $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$. In particular, we generalize work of Moonen-Zarhin, Ribet, and Tankeev to completely determine the possible Hodge groups of such Hodge structures when $n$ is equal to $1$, $4$, or a prime $p$. In addition, we determine, under certain conditions on the endomorphism algebra, the possible Hodge groups when $n=2p$, for $p$ an odd prime. A consequence of these results is that, for all powers of a simple complex $2p$-dimensional abelian variety whose endomorphism algebra is of a particular specified type, both the Hodge and General Hodge Conjectures hold. In the second part of the thesis, we investigate the geometry of a particular class of examples of smooth projective varieties whose rational cohomology realizes a $\mathbb{Q}$-Hodge structure with Hodge numbers $(n,0,\ldots,0,n)$.