Positivity of Hochster Theta

Abstract

M. Hochster defines an invariant namely $\Theta(M,N)$ associated to two finitely generated module over a hyper-surface ring $R=P/f$, where $P=k\{x_0,...,x_n\}$ or $k[X_0,...,x_n]$, for $k$ a field and $f$ is a germ of holomorphic function or a polynomial, having isolated singularity at $0$. This invariant can be lifted to the Grothendieck group $G_0(R)_{\mathbb{Q}}$ and is compatible with the chern character and cycle class map, according to the works of W. Moore, G. Piepmeyer, S. Spiroff, M. Walker. They prove that it is semi-definite when $f$ is a homogeneous polynomial, using Hodge theory on Projective varieties. It is a conjecture that the same holds for general isolated singularity $f$. We give a proof of this conjecture using Hodge theory of isolated hyper-surface singularities when $k=\mathbb{C}$. We apply this result to give a positivity criteria for intersection multiplicty of proper intersections in the variety of $f$.