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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
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Complex Analytic Desingularization (Hardcover, 1st ed. 2018)
Loot Price: R3,680
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Complex Analytic Desingularization (Hardcover, 1st ed. 2018)
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[From the foreword by B. Teissier] The main ideas of the proof of
resolution of singularities of complex-analytic spaces presented
here were developed by Heisuke Hironaka in the late 1960s and early
1970s. Since then, a number of proofs, all inspired by Hironaka's
general approach, have appeared, the validity of some of them
extending beyond the complex analytic case. The proof has now been
so streamlined that, although it was seen 50 years ago as one of
the most difficult proofs produced by mathematics, it can now be
the subject of an advanced university course. Yet, far from being
of historical interest only, this long-awaited book will be very
rewarding for any mathematician interested in singularity theory.
Rather than a proof of a canonical or algorithmic resolution of
singularities, what is presented is in fact a masterly study of the
infinitely near "worst" singular points of a complex analytic space
obtained by successive "permissible" blowing ups and of the way to
tame them using certain subspaces of the ambient space. This taming
proves by an induction on the dimension that there exist finite
sequences of permissible blowing ups at the end of which the worst
infinitely near points have disappeared, and this is essentially
enough to obtain resolution of singularities. Hironaka's ideas for
resolution of singularities appear here in a purified and geometric
form, in part because of the need to overcome the globalization
problems appearing in complex analytic geometry. In addition, the
book contains an elegant presentation of all the prerequisites of
complex analytic geometry, including basic definitions and theorems
needed to follow the development of ideas and proofs. Its epilogue
presents the use of similar ideas in the resolution of
singularities of complex analytic foliations. This text will be
particularly useful and interesting for readers of the younger
generation who wish to understand one of the most fundamental
results in algebraic and analytic geometry and invent possible
extensions and applications of the methods created to prove it.
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