M. F. Atiyah :
“A note on the tangents of a twisted cubic ,”
Proc. Cambridge Philos. Soc.
48
(1952 ),
pp. 204–205 .
MR
0048079
Zbl
0046.14604
article

Abstract
BibTeX

Consider a rational normal cubic \( C_3 \) . In the Klein representation of the lines of \( S_3 \) by points of a quadric \( \Omega \) in \( S_5 \) , the tangents of \( C_3 \) are represented by the points of a rational normal quartic \( C_4 \) . It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves.

@article {key0048079m,
AUTHOR = {Atiyah, M. F.},
TITLE = {A note on the tangents of a twisted
cubic},
JOURNAL = {Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {48},
YEAR = {1952},
PAGES = {204--205},
NOTE = {MR:0048079. Zbl:0046.14604.},
ISSN = {0305-0041},
}
W. Hodge and M. Atiyah :
“Formes de seconde espèce sur une variété algébrique ”
[Forms of the second kind on an algebraic variety ],
C. R. Acad. Sci. Paris
239
(1954 ),
pp. 1333–1335 .
MR
0068869
article

People
BibTeX
@article {key0068869m,
AUTHOR = {Hodge, William and Atiyah, Michael},
TITLE = {Formes de seconde esp\`ece sur une vari\'et\'e
alg\'ebrique [Forms of the second kind
on an algebraic variety]},
JOURNAL = {C. R. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus de l'Academie des Sciences
-- Series I: Mathematics},
VOLUME = {239},
YEAR = {1954},
PAGES = {1333--1335},
NOTE = {MR:0068869.},
ISSN = {0764-4442},
}
M. F. Atiyah :
“Complex fibre bundles and ruled surfaces ,”
Proc. London Math. Soc. (3)
5
(1955 ),
pp. 407–434 .
MR
0076409
Zbl
0174.52804
article

Abstract
BibTeX

Although much work has been done in the topological theory of fibre bundles, very little appears to be known on the complex analytic side. In this paper we propose to study certain types of complex fibre bundle, showing how they can be classified. The methods we shall employ will be based on the theory of stacks.

@article {key0076409m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Complex fibre bundles and ruled surfaces},
JOURNAL = {Proc. London Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society, third series},
VOLUME = {5},
YEAR = {1955},
PAGES = {407--434},
DOI = {10.1112/plms/s3-5.4.407},
NOTE = {MR:0076409. Zbl:0174.52804.},
ISSN = {0024-6115},
}
W. V. D. Hodge and M. F. Atiyah :
“Integrals of the second kind on an algebraic variety ,”
Ann. Math. (2)
62
(1955 ),
pp. 56–91 .
MR
0074082
Zbl
0068.34401
article

People
BibTeX
@article {key0074082m,
AUTHOR = {Hodge, W. V. D. and Atiyah, M. F.},
TITLE = {Integrals of the second kind on an algebraic
variety},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics, second series},
VOLUME = {62},
YEAR = {1955},
PAGES = {56--91},
DOI = {10.2307/2007100},
NOTE = {MR:0074082. Zbl:0068.34401.},
ISSN = {0003-486X},
}
M. Atiyah :
“On the Krull–Schmidt theorem with application to sheaves ,”
Bull. Soc. Math. France
84
(1956 ),
pp. 307–317 .
MR
0086358
Zbl
0072.18101
article

Abstract
BibTeX

It is well-known that many standard algebraic results in the theory of groups, rings, modules, etc., can be proved more generally for suitable categories, in the sense of Eilenberg–Maclane [1945]. This has the usual advantages of abstraction. It singles out those features of a given algebraic structure which are essential to the results in question, and by so doing it extends the validity of these results to other domains. In this Note we shall be concerned with the Krull–Schmidt theorem for modules, which asserts under suitable conditions the existence and essential uniqueness of a direct decomposition into indecomposable factors. It is clear that if such a theorem is to have a meaning in some general category, then such notions as kernel, image and direct sum must be defined in the category, and must possess the usual properties. Such a category, called an exact category, has been considered by Buchsbaum [1955]. Basing ourselves on his paper we then have at our disposal all the necessary notions with the usual properties. Our purpose will be to investigate conditions under which the Krull–Schmidt theorem holds in an exact category. This categorical formulation will then enable us to obtain a Krull–Schmidt theorem for suitable categories of sheaves. This is of special interest in algebraic geometry, and it was this case of the theorem which provided our motivation.

@article {key0086358m,
AUTHOR = {Atiyah, M.},
TITLE = {On the {K}rull--{S}chmidt theorem with
application to sheaves},
JOURNAL = {Bull. Soc. Math. France},
FJOURNAL = {Bulletin de la Societe Mathematique
de France},
VOLUME = {84},
YEAR = {1956},
PAGES = {307--317},
URL = {http://www.numdam.org/item?id=BSMF_1956__84__307_0},
NOTE = {MR:0086358. Zbl:0072.18101.},
ISSN = {0037-9484},
}
M. F. Atiyah :
“Vector bundles over an elliptic curve ,”
Proc. London Math. Soc. (3)
7 : 1
(1957 ),
pp. 414–452 .
MR
0131423
Zbl
0084.17305
article

Abstract
BibTeX

The primary purpose of this paper is the study of algebraic vector bundles over an elliptic curve (defined over an algebraically closed field \( k \) ). The interest of the elliptic curve lies in the fact that it provides the first non-trivial case, Grothendieck [1955] having shown that for a rational curve every vector bundle is a direct sum of line-bundles.

@article {key0131423m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Vector bundles over an elliptic curve},
JOURNAL = {Proc. London Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society, third series},
VOLUME = {7},
NUMBER = {1},
YEAR = {1957},
PAGES = {414--452},
DOI = {10.1112/plms/s3-7.1.414},
NOTE = {MR:0131423. Zbl:0084.17305.},
ISSN = {0024-6115},
}
M. F. Atiyah :
“Complex analytic connections in fibre bundles ,”
Trans. Amer. Math. Soc.
85 : 1
(1957 ),
pp. 181–207 .
See also Symposium internacional de topología algebraica (1958) .
MR
0086359
Zbl
0078.16002
article

Abstract
BibTeX

In the theory of differentiable fibre bundles, with a Lie group as structure group, the notion of a connection plays an important role. In this paper we shall consider complex analytic connections in complex analytic fibre bundles. The situation is then radically different from that in the differentiable case. In the differentiable case connections always exist, but may not be integrable; in the complex analytic case connections may not exist at all. In both cases we are led therefore to certain obstructions, an obstruction to the integrability of a connection in the differentiable case, an obstruction to the existence of a connection in the complex analytic case. It is a basic theorem that, if the structure group is compact, the obstruction in the differentiable case (the curvature) generates the characteristic cohomology ring of the bundle (with real coefficients). What we shall show is that, in a large class of important cases, the obstruction in the complex analytic case also generates the characteristic cohomology ring. Using this fact we can then give a purely cohomological definition of the characteristic ring. This has a number of advantages over the differentiable approach: in the first place the definition is a canonical one, not depending on an arbitrary choice of connection; secondly we remain throughout in the complex analytic domain, our characteristic classes being expressed as elements of cohomology groups with coefficients in certain analytic sheaves; finally the procedure can be carried through without change for algebraic fibre bundles.

@article {key0086359m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Complex analytic connections in fibre
bundles},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {85},
NUMBER = {1},
YEAR = {1957},
PAGES = {181--207},
DOI = {10.2307/1992969},
NOTE = {See also \textit{Symposium internacional
de topolog\'ia algebraica} (1958). MR:0086359.
Zbl:0078.16002.},
ISSN = {0002-9947},
}
M. F. Atiyah :
“On analytic surfaces with double points ,”
Proc. Roy. Soc. London. Ser. A
247 : 1249
(1958 ),
pp. 237–244 .
MR
0095974
Zbl
0135.21301
article

Abstract
BibTeX

It is shown that the non-singular model of an algebraic surface, lying in complex projective 3-space and possessing only ordinary double points, is differentiably homeomorphic to any non-singular surface of the same degree. This result does not hold in any other dimension.

@article {key0095974m,
AUTHOR = {Atiyah, M. F.},
TITLE = {On analytic surfaces with double points},
JOURNAL = {Proc. Roy. Soc. London. Ser. A},
FJOURNAL = {Proceedings of the Royal Society A --
Mathematical, Physical \& Engineering
Sciences},
VOLUME = {247},
NUMBER = {1249},
YEAR = {1958},
PAGES = {237--244},
DOI = {10.1098/rspa.1958.0181},
NOTE = {MR:0095974. Zbl:0135.21301.},
ISSN = {0962-8444},
}
M. F. Atiyah :
Some examples of complex manifolds .
Bonner Mathematische Schriften 6 .
Universität Bonn ,
1958 .
MR
0105718
Zbl
0080.37502
book

BibTeX
@book {key0105718m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Some examples of complex manifolds},
SERIES = {Bonner Mathematische Schriften},
NUMBER = {6},
PUBLISHER = {Universit\"at Bonn},
YEAR = {1958},
PAGES = {28},
NOTE = {MR:0105718. Zbl:0080.37502.},
ISSN = {0524-045X},
}
M. F. Atiyah :
“Complex analytic connections in fibre bundles ,”
pp. 77–82
in
Symposium internacional de topología algebraica
(Universidad Nacional Autónoma de México ).
Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris ),
1958 .
See also Trans. Amer. Math. Soc. 85 :1 (1957) .
MR
0098194
Zbl
0123.16503
incollection

BibTeX
@incollection {key0098194m,
AUTHOR = {Atiyah, M. F.},
TITLE = {Complex analytic connections in fibre
bundles},
BOOKTITLE = {Symposium internacional de topolog\'ia
algebraica},
PUBLISHER = {Universidad Nacional Aut\'onoma de M\'exico
and UNESCO},
ADDRESS = {Mexico City and Paris},
YEAR = {1958},
PAGES = {77--82},
NOTE = {(Universidad Nacional Aut\'onoma de
M\'exico). See also \textit{Trans. Amer.
Math. Soc.} \textbf{85}:1 (1957). MR:0098194.
Zbl:0123.16503.},
}
M. Atiyah :
Collected works ,
vol. 1: Early papers; general papers .
Oxford Science Publications .
The Clarendon Press and Oxford University Press (Oxford and New York ),
1988 .
MR
951892
Zbl
0935.01034
book

BibTeX
@book {key951892m,
AUTHOR = {Atiyah, Michael},
TITLE = {Collected works},
VOLUME = {1: Early papers; general papers},
SERIES = {Oxford Science Publications},
PUBLISHER = {The Clarendon Press and Oxford University
Press},
ADDRESS = {Oxford and New York},
YEAR = {1988},
PAGES = {xxiii+364},
NOTE = {MR:951892. Zbl:0935.01034.},
ISBN = {9780198532750},
}