Local complex singularity exponents for isolated singularities

Abstract

In this thesis, I studied the stability of local complex'singularity exponents (lcse) for holomorphic functions whose zero sets have only isolated singularities. For a given holomorphic function f defined on a neighborhood of the origin in C[to the power of]n, the lcse c[sub]0(f) is defined as the supremum of all positive real number [lambda] for which 1/[magnitude of]f[to the power of][2 lambda] is integrable on some neighborhood of the origin. It has been conjectured that c[sub]0(f) should not decrease if f is deformed small enough. Using J. Mather and S.S.T. Yau's result on the classification of isolated hypersurface singularities, together with a well known result on the stability of c[sub]0(f) when f is deformed in a finite dimension base space, I proved that if the zero set of f has only isolated singularity at the origin, then c[sub]0(g) >[or equal to][sub]0(f) for g close enough to f with respect to the C⁰ norm over a neighborhood of the origin, thus gave a partial solution to the conjecture. Using the stability results, I also computed the holomorphic invariant α(M) for some special Fano manifold M.