Math @ Duke

Publications [#320301] of Robert Bryant
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 Bryant, RL, Rigidity and quasirigidity of extremal cycles in Hermitian symmetric
spaces
(March, 2001) [math.DG/0006186]
(last updated on 2019/08/21)
Author's Comments: I use local differential geometric techniques to prove
that
the algebraic cycles in certain extremal homology
classes in
Hermitian symmetric spaces are either rigid (i.e.,
deformable
only by ambient motions) or quasirigid (roughly
speaking,
foliated by rigid subvarieties in a nontrivial way).
These rigidity results have a number of applications:
First,
they prove that many subvarieties in Grassmannians
and other
Hermitian symmetric spaces cannot be smoothed (i.e.,
are not
homologous to a smooth subvariety). Second, they
provide
characterizations of holomorphic bundles over compact
Kahler
manifolds that are generated by their global sections
but
that have certain polynomials in their Chern classes
vanish
(for example, c_2 = 0, c_1c_2  c_3 = 0, c_3 = 0, etc.).
Abstract: I use local differential geometric techniques to prove that the algebraic
cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasirigid (roughly
speaking, foliated by rigid subvarieties in a nontrivial way).
These rigidity results have a number of applications: First, they prove that
many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot
be smoothed (i.e., are not homologous to a smooth subvariety). Second, they
provide characterizations of holomorphic bundles over compact Kahler manifolds
that are generated by their global sections but that have certain polynomials
in their Chern classes vanish (for example, c_2 = 0, c_1c_2  c_3 = 0, c_3 = 0,
etc.).


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