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

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

Joseph Albert Wolf	Related
M. F. Atiyah	Related	Volume
S. S. Chern	Related	Volume
Wen-ling Huang	Related
Karin Reich	Related
Ernest Kuh	Related
Lizhen Ji	Related
Robert Everist Greene	Related
Manfredo P. do Carmo	Related
Jun Li	Related
Y. Ron Shen	Related
Shing-Tung Yau	Related
Alan Weinstein	Related
Isadore M. Singer	Related	Volume

@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},
}