

Seminar of the Department of Algebra
December 2, 2008 15:00, Moscow, Steklov Mathematical Institute, Room 540 (8 Gubkina)






Equisingularity theory of complex analytic families
A. Rangachev^{ab} ^{a} Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
^{b} Massachusetts Institute of Technology

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Abstract:
A fundamental goal of complex geometry is to describe the
structure of singular sets. If the set is a member of an analytic
family, then it is easier to predict when the set's structure is
similar to that of most members. In general, the problem seems to be
hard. The first step was made by Bernard Teissier (1972) who studied
the case of analytic families of hypersurface germs with isolated
singularities. Later on, Terry Gaffney and Steven Kleiman (1999)
extended Teissier's work to analytic families of germs of isolated
completeintersection singularities or ICIS germs. They managed to
develop algebraic methods to describe standard equisingularity
conditions in terms of numerical invariants of individual members,
rather than the family. These numerical invariants are certain
Buchsbaum–Rim multiplicities that arise from the column space of
the Jacobian module of the ICIS germ.
Gaffney and Kleiman studied various equisingularity conditions on
families of ICIS germs. Specifically, they studied the Thom
condition $A_{f}$ and the Whitney condition $W_{f}$ for a fixed
function $f$ on the total space $X$. They proved that the constancy
of two sequences of Milnor numbers, and the constancy of a single
Buchsbaum–Rim multiplicity are necessary and sufficient conditions
for the Thom and the Whitney equisingularity conditions to hold.
This talk will report on a current joint work of Steven Kleiman
(MIT) and the speaker, which aims to generalize previous results to
the case of arbitrary isolated singularities.
Language: English

