  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 $f$.
