Speaker:
Title:
Abstract: Complex analytic geometry deals with curves, surfaces, and higher-dimensional objects
defined by analytic functions over the
complex numbers (i.e. by holomorphic functions).
Such objects are called analytic varieties and
may be smooth or singular.
This talk will focus on the case of a singular
complex analytic variety X contained in a larger
smooth variety M. A famous
theorem of Hironaka says that the singularities of
X may be resolved by a finite sequence of transformations
called blow-ups,
each of which replaces a smooth subvariety of the
singular set of X by a larger smooth variety.
A blow-up can also be defined using a coherent ideal sheaf.
A sheaf on an analytic variety is a structure which encodes local
information over the entire variety. Coherent sheaves are
locally finitely generated and are particularly useful. A
coherent ideal
sheaf on M determines a subvariety of M by encoding the
local defining functions for that subvariety. This talk
will describe
desingularization using a single blow-up along a coherent
ideal sheaf, which replaces a sequence of blow-ups along smooth
subvarieties, and will discuss some applications in the
construction of Chern forms and metrics for singular analytic
varieties.