Positivity of Hochster theta over C

摘要

M. Hochster defines an invariant namely θ(M, N) associated to two finitely generated modules over a hypersurface 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)_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 hypersurface singularities when k = C. We apply this result to give a positivity criteria for intersection multiplicty of proper intersections in the variety of f.