R. Bott and S. S. Chern :
“Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections Acta Math.
114 : 1
(1965 ),
pp. 71–112 .
A Russian translation was published in Matematika 14 :2 (1970)
MR
0185607 Zbl
0148.31906 article
Abstract
People
BibTeX
At present a great deal is known about the value distribution of systems of meromorphic functions on an open Riemann surface. One has the beautiful results of Picard, E. Borel, Nevanlinna, Ahlfors, H. and J. Weyl and many others to point to. (See [Nevanlinna 1936; Ahlfors 1941; Weyl 1943].) The aim of this paper is to make the initial step towards an \( n \) -dimensional analogue of this theory.
@article {key0185607m,
AUTHOR = {Bott, Raoul and Chern, S. S.},
TITLE = {Hermitian vector bundles and the equidistribution
of the zeroes of their holomorphic sections},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {114},
NUMBER = {1},
YEAR = {1965},
PAGES = {71--112},
DOI = {10.1007/BF02391818},
NOTE = {A Russian translation was published
in \textit{Matematika} \textbf{14}:2
(1970). MR:0185607. Zbl:0148.31906.},
ISSN = {0001-5962},
}
R. Bott and S. S. Chern :
“Some formulas related to complex transgression 48–57
in
Essays on topology and related topics: Mémoires dédiés à Georges de Rham
[Essays on topology and related topics: Memoirs dedicated to Georges de Rham ].
Edited by A. Haefliger and R. Narasimhan .
Springer (New York ),
1970 .
MR
0264715 Zbl
0203.54202 incollection
Abstract
People
BibTeX
Let \( X \) be a complex manifold of complex dimension \( n \) and \( \pi:E\to X \) a holomorphic vector bundle whose fiber dimension is also \( n \) . On \( E \) we introduce a positive definite hermitian norm \( N \) and denote by
\[ B^*(E) = \{e\in E\mid 0 < N(e)\} \]
the subset of non-zero vectors of \( E \) . Let \( c_n(E) \) be the Chern form of the hermitian structure (to be described again below). In [1965] we proved the theorem: There exists a real-valued form \( \rho \) of type \( (n-1, \) \( n-1) \) on \( B^*(E) \) such that
\begin{equation*}\tag{1} \pi^* c_n(E) = \frac{dd^c}{4\pi}\rho. \end{equation*}
In this paper we wish to give a proof of (1) based on an explicit construction. We will follow and extend a formalism developed in the real case by H. Flanders [1953]. It is possible that this formalism will be useful for later purposes.
@incollection {key0264715m,
AUTHOR = {Bott, Raoul and Chern, Shiing S.},
TITLE = {Some formulas related to complex transgression},
BOOKTITLE = {Essays on topology and related topics:
{M}\'emoires d\'edi\'es \`a {G}eorges
de {R}ham [Essays on topology and related
topics: {M}emoirs dedicated to {G}eorges
de {R}ham]},
EDITOR = {Haefliger, Andre and Narasimhan, Raghavan},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1970},
PAGES = {48--57},
DOI = {10.1007/978-3-642-49197-9_5},
NOTE = {MR:0264715. Zbl:0203.54202.},
ISBN = {9783642491993},
}
M. Atiyah and S. S. Chern :
“Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections ,”
Matematika, Moskva
14 : 2
(1970 ),
pp. 117–154 .
Russian translation of an article in Acta Math. 114 :1 (1965)
Zbl
0208.35101 article
People
BibTeX
@article {key0208.35101z,
AUTHOR = {Atiyah, Michael and Chern, S. S.},
TITLE = {Hermitian vector bundles and the equidistribution
of the zeros of their holomorphic sections},
JOURNAL = {Matematika, Moskva},
VOLUME = {14},
NUMBER = {2},
YEAR = {1970},
PAGES = {117--154},
NOTE = {Russian translation of an article in
\textit{Acta Math.} \textbf{114}:1 (1965).
Zbl:0208.35101.},
}
R. Bott :
“On a topological obstruction to integrability ,”
pp. 127–131
in
Global analysis
(Berkeley, CA, 1–26 July 1968 ).
Edited by S.-S. Chern and S. Smale .
Proceedings of Symposia in Pure Mathematics 16 .
American Mathematical Society (Providence, RI ),
1970 .
MR
0266248 Zbl
0206.50501 incollection
People
BibTeX
@incollection {key0266248m,
AUTHOR = {Bott, Raoul},
TITLE = {On a topological obstruction to integrability},
BOOKTITLE = {Global analysis},
EDITOR = {Chern, Shiing-Shen and Smale, Stephen},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {16},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1970},
PAGES = {127--131},
NOTE = {(Berkeley, CA, 1--26 July 1968). MR:0266248.
Zbl:0206.50501.},
ISSN = {0082-0717},
ISBN = {9780821814161},
}
P. Baum and R. Bott :
“Singularities of holomorphic foliations J. Diff. Geom.
7 : 3–4
(1972 ),
pp. 279–342 .
To S. S. Chern and D. C. Spencer on their 60th birthdays.
MR
377923 Zbl
0268.57011 article
Abstract
People
BibTeX
The purpose of this note is twofold. First we give a simpler and more natural proof of our meromorphic vector-field theorem of [1970]; and second, we give a theorem on singularities of holomorphic foliations which includes the meromorphic vector-field theorem as a special case. We have tried to make the exposition as elementary and self-contained as possible.
@article {key377923m,
AUTHOR = {Baum, Paul and Bott, Raoul},
TITLE = {Singularities of holomorphic foliations},
JOURNAL = {J. Diff. Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {7},
NUMBER = {3--4},
YEAR = {1972},
PAGES = {279--342},
DOI = {10.4310/jdg/1214431158},
NOTE = {To S. S. Chern and D. C. Spencer on
their 60th birthdays. MR:377923. Zbl:0268.57011.},
ISSN = {0022-040X},
}
R. Bott :
“On the Gelfand–Fuks cohomology ,”
pp. 357–364
in
Differential geometry
(Stanford, CA, 30 July–17 August 1973 ),
part 1 .
Edited by S.-S. Chern and R. Osserman .
Proceedings of Symposia in Pure Mathematics 27 .
American Mathematical Society (Providence, RI ),
1975 .
MR
0380820 Zbl
0324.57013 incollection
Abstract
People
BibTeX
Roughly four years ago Gelfand and Fuks had the inspired idea of considering the continuous cohomology of the Lie-algebra of \( C^{\infty} \) -vector fields on a manifold ([1969, 1970a, 1970b]). They showed that the answer was finite dimensional for all compact manifodls and that for \( S^1 \) this cohomology is a tensor product of an exterior algebra in dimension 3 and a polynomial ring with generator in dimension 2:
\[ H_c(\mathcal{L}S^1) = E(\omega)\otimes \mathbf{R}[y]. \]
In this report I would like to present a new point of view towards their work which arose, on the one hand, out of the new insights into cohomology due to Guillemin and Trauber, and on the other out of conversations between G. Segal, A. Haefliger and myself during the past year. I will finally formulate a conjecture which would completely settle the homotopy theoretic nature of this cohomology. Although one finds evidence for its validity on every side, its proof has so far eluded us. I should also mention that when I recently showed this conjecture to Gelfand he told me that it had also, quite independently, been formulated by Fuks.
@incollection {key0380820m,
AUTHOR = {Bott, Raoul},
TITLE = {On the {G}elfand--{F}uks cohomology},
BOOKTITLE = {Differential geometry},
EDITOR = {Chern, Shiing-Shen and Osserman, Robert},
VOLUME = {1},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {27},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1975},
PAGES = {357--364},
NOTE = {(Stanford, CA, 30 July--17 August 1973).
MR:0380820. Zbl:0324.57013.},
ISSN = {0082-0717},
ISBN = {9780821802472},
}
R. Bott :
“Equivariant Morse theory and the Yang–Mills equation on Riemann surfaces 11–22
in
The Chern symposium, 1979: Proceedings of the international symposium on differential geometry in honor of S.-S. Chern
(Berkeley, CA, June 1979 ).
Edited by W.-Y. Hsiang, S. Kobayashi, I. M. Singer, A. Weinstein, J. Wolf, and H.-H. Wu .
Springer (New York ),
1980 .
MR
609555 Zbl
0492.58011 incollection
People
BibTeX
@incollection {key609555m,
AUTHOR = {Bott, Raoul},
TITLE = {Equivariant {M}orse theory and the {Y}ang--{M}ills
equation on {R}iemann surfaces},
BOOKTITLE = {The {C}hern symposium, 1979: {P}roceedings
of the international symposium on differential
geometry in honor of {S}.-{S}. {C}hern},
EDITOR = {Hsiang, W.-Y. and Kobayashi, S. and
Singer, I. M. and Weinstein, A. and
Wolf, J. and Wu, H.-H.},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {11--22},
DOI = {10.1007/978-1-4613-8109-9_2},
NOTE = {(Berkeley, CA, June 1979). MR:609555.
Zbl:0492.58011.},
ISBN = {9780387905372},
}
R. Bott :
“Lectures on Morse theory, old and new ,”
pp. 169–218
in
Proceedings of the 1980 Beijing symposium on differential geometry and differential equations
(Beijing, 1980 ),
vol. 1 .
Edited by S. S. Chern and W.-T. Wu .
Science Press (Beijing ),
1982 .
See also Bull. Am. Math. Soc. 7 :2 (1982)The mathematical heritage of Henri Poincaré (1983)
MR
714336 Zbl
0521.58019 incollection
People
BibTeX
@incollection {key714336m,
AUTHOR = {Bott, Raoul},
TITLE = {Lectures on {M}orse theory, old and
new},
BOOKTITLE = {Proceedings of the 1980 {B}eijing symposium
on differential geometry and differential
equations},
EDITOR = {Chern, S. S. and Wu, Wen-Ts\"un},
VOLUME = {1},
PUBLISHER = {Science Press},
ADDRESS = {Beijing},
YEAR = {1982},
PAGES = {169--218},
NOTE = {(Beijing, 1980). See also \textit{Bull.
Am. Math. Soc.} \textbf{7}:2 (1982)
and \textit{The mathematical heritage
of Henri Poincar\'e} (1983). MR:714336.
Zbl:0521.58019.},
ISBN = {9780677164205},
}
R. Bott :
“For the Chern volume ,”
pp. 106–108
in
S. S. Chern: A great geometer of the twentieth century
(Los Angeles, 1990 ).
Edited by S.-T. Yau .
International Press (Cambridge, MA ),
1992 .
MR
1201350 incollection
People
BibTeX
@incollection {key1201350m,
AUTHOR = {Bott, Raoul},
TITLE = {For the {C}hern volume},
BOOKTITLE = {S.~{S}. {C}hern: {A} great geometer
of the twentieth century},
EDITOR = {Yau, Shing-Tung},
PUBLISHER = {International Press},
ADDRESS = {Cambridge, MA},
YEAR = {1992},
PAGES = {106--108},
NOTE = {(Los Angeles, 1990). MR:1201350.},
ISBN = {9781571460981},
}
