Let $R$ be an isolated hypersurface singularity, and let $M$ and $N$ be finitely generated $R$-modules. As $R$ is a hypersurface, the torsion modules of $M$ against $N$ are eventually periodic of period two (i.e., $\Tor_i^R(M,N) \iso \Tor_{i+2}^R(M,N)$ for $i >> 0$).
Since $R$ has only an isolated singularity, these torsion modules are of finite length for $i \gg 0$. The theta invariant of the pair $(M,N)$ is defined by Hochster to be $\length(\Tor_{2i}^R(M,N)) - \length(\Tor_{2i+1}^R(M,N))$ for $i >> 0$.
H. Dao has conjectured that the theta invariant is zero for all pairs $(M,N)$ when $R$ has even dimension and contains a field. We prove this conjecture under the additional assumption that $R$ is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of $R$ is odd, and relate it to a classical pairing on the smooth hypersurface $\Proj{R}$.