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.

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).