A combinatorial description of the cup product for smooth complete toric varieties

For any smooth variety X, there exists an associated vector space of first-order deformations. This vector space can be interpreted using sheaf cohomology; it is the first cohomology group H^1(X,T_X), where T_X is the tangent sheaf. One can ask when it is possible to "combine" two first-order deformations. The cup product takes elements of H^1(X,T_X) x H^1(X,T_X) and maps to the obstruction space H^2(X, T_X), and the vanishing of the cup product tells us precisely when this is possible. In this thesis we give a combinatorial description of the cup product (on the level of Čech cohomology) when X is a smooth, complete, toric variety with an associated fan Σ. We also give an example of a smooth, complete, toric 3-fold for which the cup product is nonvanishing.