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Let R be an isolated hypersurface singularity, and let M and N be finitelygenerated R-modules. As R is a hypersurface, the torsion modules of M against Nare eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic toTor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolatedsingularity, these torsion modules are of finite length for i sufficientlylarge. The theta invariant of the pair (M,N) is defined by Hochster to belength(Tor_{2i}^R(M,N)) - length(Tor_{2i+1}^R(M,N)) for i sufficiently large.H. Dao has conjectured that the theta invariant is zero for all pairs (M,N)when R has even dimension and contains a field. This paper proves thisconjecture under the additional assumption that R is graded with its irrelevantmaximal ideal giving the isolated singularity. We also give a careful analysisof the theta pairing when the dimension of R is odd, and relate it to aclassical pairing on the smooth variety Proj(R).

Hochster's theta invariant and the Hodge-Riemann bilinear relations