2026-09-19T04:42:59Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/283122022-01-13T07:54:35Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Gang Tian.
Hou, Zuoliang, 1977-
Massachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2005-09-26T19:43:06Z
2005-09-26T19:43:06Z
2004
2004
http://hdl.handle.net/1721.1/28312
55635790
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
Includes bibliographical references (p. 59-61).
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.
by Zuoliang Hou.
Ph.D.
61 p.
2310260 bytes
2315816 bytes
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Massachusetts Institute of Technology
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Local complex singularity exponents for isolated singularities
Thesis
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<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.</Abstract>
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