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

## TL;DR

This paper investigates Hochster's theta invariant for isolated hypersurface singularities, proving Dao's conjecture in the graded case with even dimension and relating the odd-dimensional case to classical pairings.

## Contribution

It proves Dao's conjecture for graded hypersurface singularities of even dimension and analyzes the theta pairing in odd dimensions, connecting it to classical geometric pairings.

## Key findings

- Theta invariant is zero for even-dimensional graded hypersurfaces with isolated singularities.
- The paper establishes a relationship between the theta pairing and classical pairings on smooth projective varieties.
- Analysis of theta pairing in odd dimensions links it to geometric structures on Proj(R).

## Abstract

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) is isomorphic to Tor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolated singularity, these torsion modules are of finite length for i sufficiently large. 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 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 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 variety Proj(R).