Celebratio Mathematica — Hirzebruch

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F. Hirzebruch: “The canonical map for certain Hilbert modular surfaces,” pp. 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

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.

Joseph Albert Wolf	Related
S. S. Chern	Related	Volume
Wu-yi Hsiang	Related
Shōshichi Kobayashi	Related	Volume
Alan Weinstein	Related
Isadore M. Singer	Related	Volume
Hung-hsi Wu	Related

F. Hirzebruch, N. Schappacher, B. Mazur, P. Cohen, S. Smale, J. Coates, D. Goldfeld, J. Jorgenson, P. Vojta, G. Lachaud, R. Howe, G. Wüstholz, J. Kramer, D. E. Rohrlich, M. J. Greenberg, S. Kobayashi, H.-H. Wu, L. A. Goldberg, M. Stanley, A. Jackson, J. Ewing, J. Gerver, and P. Duesberg: “Serge Lang, 1927–2005.” Edited by J. Jorgenson and S. G. Krantz. Notices Am. Math. Soc. 53 : 5 (2006), pp. 536–553. Hirzebruch’s contribution is on pp. 541–542. MR 2254400 Zbl 1138.01326 article

Paula Tretkoff	Related
Joseph Leonide Gerver	Related
Serge Lang	Related
Roger Evans Howe	Related
John H. Ewing	Related
Jay Alan Jorgenson	Related
Jürg Kramer	Related
Gilles Lachaud	Related
Marvin J. Greenberg	Related
Gisbert Wüstholz	Related
Barry C. Mazur	Related
John Henry Coates	Related
Hung-hsi Wu	Related
Norbert Schappacher	Related
Paul Alan Vojta	Related
Dorian Morris Goldfeld	Related
Steven Krantz	Related
Shōshichi Kobayashi	Related	Volume
David Ephraim Rohrlich	Related
Stephen Smale	Related
Allyn Jackson	Related

@article {key2254400m,
	AUTHOR = {Hirzebruch, Friedrich and Schappacher,
		 Norbert and Mazur, Barry and Cohen,
		 Paul and Smale, Steven and Coates, John
		 and Goldfeld, Dorian and Jorgenson,
		 Jay and Vojta, Paul and Lachaud, Gilles
		 and Howe, Roger and W\"ustholz, Gisbert
		 and Kramer, J\"urg and Rohrlich, David
		 E. and Greenberg, Marvin Jay and Kobayashi,
		 Shoshichi and Wu, Hung-Hsi and Goldberg,
		 Lisa A. and Stanley, Morgan and Jackson,
		 Allyn and Ewing, John and Gerver, Joseph
		 and Duesberg, Peter},
	TITLE = {Serge {L}ang, 1927--2005},
	JOURNAL = {Notices Am. Math. Soc.},
	FJOURNAL = {Notices of the American Mathematical
		 Society},
	VOLUME = {53},
	NUMBER = {5},
	YEAR = {2006},
	PAGES = {536--553},
	URL = {https://www.ams.org/notices/200605/fea-lang.pdf},
	NOTE = {Edited by J. Jorgenson
		 and S. G. Krantz. Hirzebruch's
		 contribution is on pp. 541--542. MR:2254400.
		 Zbl:1138.01326.},
	ISSN = {0002-9920},
}