Local complex singularity exponents for isolated singularities

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2004

Author dc:creator

• Hou, Zuoliang, 1977-

Advisor dc:contributor.advisor

• Gang Tian.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/28312

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/28312