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Annals of Mathematics Studies #153: Rigidity and Quasi-Rigidity of Extremal Cycles in Hermitian Symmetric Spaces

Annals of Mathematics Studies #153: Rigidity and Quasi-Rigidity of Extremal Cycles in Hermitian Symmetric Spaces Cover

 

Synopses & Reviews

Publisher Comments:

This book investigates the geometry of complex subvarieties of compact Hermitian symmetric spaces, particularly the complex Grassmannians, which are central to Schubert calculus and its applications to enumerative algebraic geometry. To do so, Robert Bryant employs a combination of Hermitian differential geometry, calibrations, and classical moving frame constructions.

The main result is that, for Hermitian symmetric spaces M of rank greater than 1, there are homology classes c (called extremal) such that the complex varieties V in M that represent c display rigidity in unexpected ways. There are other cycles that display a weaker form of this sort of rigidity, but whose moduli space of representing cycles can still be described in terms of the geometry of subvarieties of related complex projective spaces.

These results have applications to other problems in algebraic geometry. For example, for a holomorphic bundle E over a compact complex manifold M that is generated by its sections, the Schur polynomials in its Chern classes are known to be non-negative. The above results allow one to give a complete description of such bundles in several cases where one of these Schur polynomials actually vanishes. The book, which will interest researchers and graduate students in complex algebraic geometry or differential geometry, contains a thorough exposition of the geometry of Hermitian symmetric spaces and their Schubert cycles and characteristic classes as well as other preparatory material needed to obtain the results.

About the Author

Robert L. Bryant is Juanita M. Kreps Professor of Mathematics at Duke University. He is co-author of "Exterior Differential Systems".

Product Details

ISBN:
9780691096292
Author:
Bryant, Robert L.
Publisher:
Princeton University Press
Location:
Princeton
Subject:
General
Subject:
Advanced
Subject:
Mathematics
Subject:
Mathematics - General
Subject:
Geometry - Algebraic
Copyright:
Edition Description:
Hardcover
Series:
Annals of Mathematics Studies (Hardcover)
Series Volume:
(AM-153)
Publication Date:
February 2010
Binding:
Hardback
Grade Level:
College/higher education:
Language:
English
Illustrations:
6 line illus.
Pages:
144
Dimensions:
9 x 6 in

Related Subjects

Science and Mathematics » Mathematics » General

Annals of Mathematics Studies #153: Rigidity and Quasi-Rigidity of Extremal Cycles in Hermitian Symmetric Spaces
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Product details 144 pages Princeton University Press - English 9780691096292 Reviews:
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