F. Hirzebruch :
“The signature theorem: Reminiscences and recreation ,”
pp. 3–31
in
F. Hirzebruch, L. Hörmander, J. Milnor, J.-P. Serre, and I. Singer :
Prospects in mathematics
(Princeton, NJ, 16–18 March 1970 ).
Annals of Mathematics Studies 70 .
Princeton University Press ,
1971 .
MR
368023 Zbl
0252.58009 incollection
People
BibTeX
@incollection {key368023m,
AUTHOR = {Hirzebruch, F.},
TITLE = {The signature theorem: {R}eminiscences
and recreation},
BOOKTITLE = {Prospects in mathematics},
SERIES = {Annals of Mathematics Studies},
NUMBER = {70},
PUBLISHER = {Princeton University Press},
YEAR = {1971},
PAGES = {3--31},
NOTE = {(Princeton, NJ, 16--18 March 1970).
MR:368023. Zbl:0252.58009.},
ISSN = {0066-2313},
ISBN = {9780691080949},
}
F. Hirzebruch, L. Hörmander, J. Milnor, J.-P. Serre, and I. M. Singer :
Prospects in mathematics .
Annals of Mathematics Studies 70 .
Princeton University Press ,
1971 .
Zbl
0233.00003 book
People
BibTeX
@book {key0233.00003z,
AUTHOR = {Hirzebruch, Friedrich and H\"ormander,
Lars and Milnor, John and Serre, Jean-Pierre
and Singer, I. M.},
TITLE = {Prospects in mathematics},
SERIES = {Annals of Mathematics Studies},
NUMBER = {70},
PUBLISHER = {Princeton University Press},
YEAR = {1971},
NOTE = {Zbl:0233.00003.},
ISSN = {0066-2313},
ISBN = {9780691080949},
}
F. Hirzebruch :
“The canonical map for certain Hilbert modular surfaces 75–95
in
The Chern symposium 1979
(Berkeley, CA, June 1979 ).
Edited by W.-Y. Hsiang, S. Kobayashi, I. M. Singer, J. Wolf, H.-H. Wu, and A. Weinstein .
Springer (New York ),
1980 .
MR
609558 Zbl
0457.14019 incollection
Abstract
People
BibTeX
It was a real pleasure for me to participate in the symposium in honor of Shiing-shen Chern. In my lecture I intended to give a survey on Hilbert modular surfaces. But actually I discussed examples of such Hilbert modular surfaces for which specific information is available on their structure as algebraic surfaces. The paper presented here is an extended version of the talk.
Algebraic surfaces are often investigated by means of their pluricanonical maps (see for example Bombieri [1973]). The properties of the canonical map itself (given by the secctions of the canonical bundle or in the case of Hilbert modular surfaces by the cusp forms of weight 2) are relatively complicated (compare Beauville [1979]). In a certain range, namely for minimal surfaces of general type with
\[ 2p_g - 4 \leq K^2 \leq 2p_g - 2 ,\]
Horikawa’s results are available [1975, 1976a, 1976b, 1978, 1979]. To apply them, one has to prove that the surface being studied is minimal. For Hilbert modular surfaces this is a difficult problem, which was attacked first by van der Geer and Van de Ven [1977]. Van der Geer has obtained many results on the structure of special Hilbert modular surfaces [1978, 1979] including some of the surfaces studied here.
The rough classification of Hilbert modular surfaces according to rational, \( K3 \) , elliptic, and general type was considered in [Hirzebruch and Van de Ven 1974; Hirzebruch and Zagier 1977; Hirzebruch 1978]. The present paper tries to show that in some cases a finer classification of the surfaces of general type can be obtained.
@incollection {key609558m,
AUTHOR = {Hirzebruch, F.},
TITLE = {The canonical map for certain {H}ilbert
modular surfaces},
BOOKTITLE = {The {C}hern symposium 1979},
EDITOR = {Hsiang, W.-Y. and Kobayashi, S. and
Singer, I. M. and Wolf, J. and Wu, H.-H.
and Weinstein, A.},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1980},
PAGES = {75--95},
DOI = {10.1007/978-1-4613-8109-9_5},
NOTE = {(Berkeley, CA, June 1979). MR:609558.
Zbl:0457.14019.},
ISBN = {9781461381112},
}
Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer .
Edited by S.-T. Yau .
Surveys in Differential Geometry (supplement to Journal of Differential Geometry) 7 .
International Press (Somerville, MA ),
2000 .
MR
1919419 book
People
BibTeX
@book {key1919419m,
TITLE = {Papers dedicated to {A}tiyah, {B}ott,
{H}irzebruch, and {S}inger},
EDITOR = {Shing-Tung Yau},
SERIES = {Surveys in Differential Geometry (supplement
to {J}ournal of {D}ifferential {G}eometry)},
NUMBER = {7},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2000},
PAGES = {696},
NOTE = {MR:1919419.},
ISSN = {1052-9233},
ISBN = {9781571461780},
}
F. Hirzebruch :
“The Atiyah–Bott–Singer fixed point theorem and number theory 313–326
in
Papers dedicated to Atiyah, Bott, Hirzebruch and Singer
(Cambridge, MA, 14–16 May 1999 ).
Edited by S.-T. Yau .
Surveys in Differential Geometry 7 .
International Press (Somervile, MA ),
2000 .
MR
1919429 Zbl
1061.58022 incollection
People
BibTeX
@incollection {key1919429m,
AUTHOR = {Hirzebruch, F.},
TITLE = {The {A}tiyah--{B}ott--{S}inger fixed
point theorem and number theory},
BOOKTITLE = {Papers dedicated to {A}tiyah, {B}ott,
{H}irzebruch and {S}inger},
EDITOR = {Yau, Shing-Tung},
SERIES = {Surveys in Differential Geometry},
NUMBER = {7},
PUBLISHER = {International Press},
ADDRESS = {Somervile, MA},
YEAR = {2000},
PAGES = {313--326},
DOI = {10.4310/SDG.2002.v7.n1.a10},
NOTE = {(Cambridge, MA, 14--16 May 1999). MR:1919429.
Zbl:1061.58022.},
ISSN = {1052-9233},
ISBN = {9781571460691},
}
The founders of index theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer .
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2003 .
Republished in 2009 .
MR
2136846 Zbl
1072.01021 book
People
BibTeX
@book {key2136846m,
TITLE = {The founders of index theory: {R}eminiscences
of {A}tiyah, {B}ott, {H}irzebruch, and
{S}inger},
EDITOR = {Yau, S.-T.},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2003},
PAGES = {liv+358},
NOTE = {Republished in 2009. MR:2136846. Zbl:1072.01021.},
ISBN = {9781571461209},
}
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Republication of 2003 original .
MR
2547480 Zbl
1195.01090 book
People
BibTeX
@book {key2547480m,
TITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. Singer},
EDITOR = {Yau, S.-T.},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {lii+393},
NOTE = {Republication of 2003 original. MR:2547480.
Zbl:1195.01090.},
ISBN = {9781571461377},
}
S.-T. Yau, F. E. P. Hirzebruch, M. Atiyah, M. do Carmo, R. E. Greene, W.-l. Huang, K. Reich, L. Ji, J. Li, E. Kuh, Y. R. Shen, I. M. Singer, A. Weinstein, and J. A. Wolf :
“Shiing-Shen Chern (1911–2004) Notices Am. Math. Soc.
58 : 9
(October 2011 ),
pp. 1226–1249 .
Hirzebruch’s contribution to this article was also published in Frontiers of mathematical sciences (2011)
MR
2868099 Zbl
1228.01052 article
People
BibTeX
@article {key2868099m,
AUTHOR = {Yau, Shing-Tung and Hirzebruch, Friedrich
Ernst Peter and Atiyah, Michael and
do Carmo, Manfredo and Greene, Robert
E. and Huang, Wen-ling and Reich, Karin
and Ji, Lizhen and Li, Jun and Kuh,
Ernest and Shen, Y. Ron and Singer,
Isadore M. and Weinstein, Alan and Wolf,
Joseph A.},
TITLE = {Shiing-{S}hen {C}hern (1911--2004)},
JOURNAL = {Notices Am. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {58},
NUMBER = {9},
MONTH = {October},
YEAR = {2011},
PAGES = {1226--1249},
URL = {http://www.ams.org/notices/201109/rtx110901226p.pdf},
NOTE = {Hirzebruch's contribution to this article
was also published in \textit{Frontiers
of mathematical sciences} (2011). MR:2868099.
Zbl:1228.01052.},
ISSN = {0002-9920},
}
