Celebratio Mathematica

[1] F. Hirzebruch: Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen [On four-dimensional Riemann surfaces of many-valued analytic functions of two complex variables]. Ph.D. thesis, Westfälische Wilhelms-Universität Münster, 1950. Advised by H. A. Behnke and H. Hopf. Republished in Math. Ann. 126 (1953). phdthesis

Heinrich Adolph Behnke	Related
Heinz Hopf	Related

[2] F. Hirzebruch: “Über eine Klasse von einfach-zusammenhängenden komplexen Mannigfaltigkeiten” [On a class of simply connected complex manifolds], Math. Ann. 124 (1951), pp. 77–86. MR 45384 Zbl 0043.30302 article

[3] F. Hirzebruch: “Übertragung einiger Sätze aus der Theorie der algebraischen Flächen auf komplexe Mannigfaltigkeiten von zwei komplexen Dimensionen” [Transformation of some theorems from the theory of algebraic surfaces on complex manifolds of two complex dimensions], J. Reine Angew. Math. 1953 : 191 (1953), pp. 110–124. Dedicated to Otto Haupt on the occasion of his sixty-fifth birthday. MR 71092 Zbl 0052.19802 article

[4] F. Hirzebruch: “On Steenrod’s reduced powers, the index of inertia, and the Todd genus,” Proc. Natl. Acad. Sci. U.S.A. 39 : 9 (September 1953), pp. 951–956. MR 63670 Zbl 0051.14501 article

This note is a preliminary report on some results concerning the Steenrod reduced powers [Steenrod 1953] in oriented manifolds \( M^m \) and the index \( \tau \) of such a manifold.

Definition of \( \tau \): If the dimension \( m \) of \( M \) is not divisible by 4, then \( \tau(M) = 0 \). If \( m = 4k \), then \( \tau(M) \) is equal to the number of positive squares minus the number of negative squares of the normalized quadratic form defined by the cup product \( x \smile x \) where \( x \in H^{2k}(M^{4k},R) \).

We apply the results to almost complex manifolds, in particular we give a definition of the Todd genus [Todd 1937] of an almost complex manifold and state properties of this genus. Since a special algebraic formalism will be used throughout the note, we outline this formalism in the first section. All manifolds occurring in this note are compact, differentiable, and oriented unless stated to the contrary. Full details with further applications will appear elsewhere.

[5] F. Hirzebruch: “Über die quaternionalen projektiven Räume” [On quaternionic projective spaces], Sitzungsber., Bayer. Akad. Wiss., Math-Naturwiss. Kl. 1953 (1953), pp. 301–312. MR 65155 Zbl 0057.15503 article

[6] F. Hirzebruch: “Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen” [On four-dimensional Riemann surfaces of many-valued analytic functions of two complex variables], Math. Ann. 126 (1953), pp. 1–22. Republication of Hirzebruch’s doctoral thesis. MR 62842 Zbl 0093.27605 article

[7] F. Hirzebruch: On Steenrod reduced powers in oriented manifolds. Mimeographed manuscript, Princeton University, 1953. techreport

[8] F. Hirzebruch: Todd arithmetic genus for almost complex manifolds. Mimeographed manuscript, Princeton University, 1953. techreport

[9] F. Hirzebruch: “Arithmetic genera and the theorem of Riemann–Roch for algebraic varieties,” Proc. Natl. Acad. Sci. U.S.A. 40 : 2 (February 1954), pp. 110–114. MR 74086 Zbl 0055.38803 article

In a preceding note [1953] we posed the problem whether the arithmetic genus \( \Pi(V_n) \) of a (non-singular) algebraic variety coincides with the Todd genus \( T(V_n) \). The purpose of the present note is to prove that this is actually the case. Moreover we prove a main theorem which gives a formula for the Euler–Poincaré characteristic of \( V_n \) with respect to the cohomology of \( V_n \) with coefficients in the sheaf (faisceau) of local holomorphic cross-sections of any complex analytic bundle \( W \) over \( V_n \) which has the complex vector space \( C_q \) as fibre and the linear group \( GL(q,C) \) as structure group. The main theorem expresses this Euler–Poincaré characteristic as a polynomial in the Chern classes of the tangential bundle of \( V_n \) and in the Chern classes of the tangential bundle of \( V_n \) and in the Chern classes of the bundle \( W \). As special cases one gets the Todd formula \( \Pi(V_n) = T(V_n) \) and the “Theorem of Riemann–Roch for arbitrary dimensions.”

[10] F. Hirzebruch: “Some problems on differentiable and complex manifolds,” Ann. Math. (2) 60 : 2 (September 1954), pp. 213–236. MR 66013 Zbl 0056.16803 article

A conference with the title Fiber bundles and differential geometry was held at Cornell University from May 3 to May 7, 1953. The purpose of the present paper is to record those problems presented at the conference which concern differentiable, almost-complex, complex, and algebraic manifolds. Several of the problems have been solved since the conference. These solutions gave rise to new problems which we will discuss.

This report is divided into three sections: differentiable manifolds, almost-complex manifolds, complex manifolds. The last part contains also the problems on algebraic varieties. Since some problems of the first two sections were motivated by theorems on algebraic varieties, there are occasional references in Sections 1 and 2 to Section 3.1.

[11] F. Hirzebruch: “On the characteristic cohomology classes of differentiable manifolds,” pp. 5 in Colloque Henri Poincaré [Henri Poincaré colloquium] (Paris, 18–27 October 1954). Textes des Conférences. Secrétariat Mathématique (Paris), 1954. incollection

[12] F. Hirzebruch: “The theorem of Riemann–Roch: Application to special classes of algebraic manifolds,” pp. 3–4 in Colloque Henri Poincaré [Henri Poincaré colloquium] (Paris, 18–27 October 1954). Textes des Conférences. Secrétariat Mathématique (Paris), 1954. incollection

[13] F. Hirzebruch: “Der Satz von Riemann–Roch in Faisceau-theoretischer Formulierung: Einige Anwendungen und offene Fragen” [The Riemann–Roch theorem in sheaf-theoretic formulation: Some applications and open questions], pp. 457–473 in Proceedings of the International Congress of Mathematicians 1954 (Amsterdam, 2–9 September 1954), vol. 3. North-Holland (Amsterdam), 1956. MR 87187 Zbl 0074.36701 incollection

[14] F. Hirzebruch: Neue topologische Methoden in der algebraischen Geometrie [New topological methods in algebraic geometry]. Ergebnisse der Mathematik und ihrer Grenzgebiete 9. Springer (Berlin), 1956. Republished in 1962. MR 82174 Zbl 0070.16302 book

[15] F. Hirzebruch: “Der Satz von Riemann–Roch und das Toddsche arithmetische Geschlecht für algebraische Mannigfaltigkeiten” [The Riemann–Roch theorem and Todd arithmetic genus for algebraic manifolds], pp. 232–233 in Proceedings of the International Congress of Mathematicians 1954 (Amsterdam, 2–9 September 1954), vol. 2. North-Holland (Amsterdam), 1956. incollection

[16] F. Hirzebruch and K. Kodaira: “On the complex projective spaces,” J. Math. Pures Appl. (9) 36 (1957), pp. 201–216. MR 92195 Zbl 0090.38601 article

[17] S. S. Chern, F. Hirzebruch, and J.-P. Serre: “On the index of a fibered manifold,” Proc. Am. Math. Soc. 8 : 3 (1957), pp. 587–596. MR 87943 Zbl 0083.17801 article

S. S. Chern	Related	Volume
Jean-Pierre Serre	Related

[18] F. Hirzebruch and G. Scheja: Garben- und Cohomologietheorie [Sheaf theory and cohomology theory]. Ausarbeitungen Mathematischer und Physikalischer Vorlesungen 20. Aschendorff (Münster), 1957. MR 105100 Zbl 0088.15201 book

[19] F. Hirzebruch: “Automorphe Formen und der Satz von Riemann–Roch” [Automorphic forms and the Riemann–Roch theorem], pp. 129–144 in Symposium internacional de topología algebraica [International symposium on algebraic topology] (Mexico City, 1958). Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris), 1958. MR 103280 Zbl 0129.29801 incollection

[20] A. Borel and F. Hirzebruch: “Characteristic classes and homogeneous spaces, I,” Am. J. Math. 80 : 2 (April 1958), pp. 458–538. MR 102800 Zbl 0097.36401 article

It is known that the characteristic classes of a real or complex vector bundle may be interpreted as elementary symmetric functions in certain variables, which are 1, 2 or 4 dimensional cohomology classes. If we consider the tangent bundle of the coset space \( G/U \) of a compact connected Lie group modulo a closed subgroup, it turns out that these variables may be identified with certain roots of \( G \) (or their squares). Our first purpose is to establish this connection between roots and characteristic classes, which is the basis of this paper, and to compute the characteristic classes of certain well-known homogeneous spaces. These results are then applied in particular to \( G/T \) (\( T \) maximal torus of \( G \)), and to other algebraic homogeneous spaces, where they lead to relations between characteristic classes, Betti numbers, the Riemann–Roch theorem and representation theory; they are also used to discuss multiplicative properties of the Todd genus and other genera in fibre bunldes with \( G/U \) as fibre. As an application, we get a divisibility property of the Chern class of a complex vector bundle over an even dimensional sphere which yields some information about certain homotopy groups of Lie groups.

[21] F. Hirzebruch and H. Hopf: “Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten” [Fields of surface elements on 4-dimensional manifolds], Math. Ann. 136 : 2 (1958), pp. 156–172. MR 100844 Zbl 0088.39403 article

[22] F. Hirzebruch: “Characteristic numbers of homogeneous domains,” pp. 92–104 in Semininars on analytic functions (Princeton, NJ, 2–14 September 1957), vol. 2. Institute for Advanced Study (Princeton, NJ), 1958. Zbl 0123.13805 incollection

[23] M. F. Atiyah and F. Hirzebruch: “Riemann–Roch theorems for differentiable manifolds,” Bull. Am. Math. Soc. 65 : 4 (1959), pp. 276–281. MR 110106 Zbl 0142.40901 article

The Riemann–Roch Theorem for an algebraic variety \( Y \) (see [Hirzebruch 1956]) led to certain divisibility conditions for the Chern classes of \( Y \). It was natural to ask whether these conditions held more generally for any compact almost complex manifold. This question, and various generalizations of it, were raised in [Hirzebruch 1954] and most of these have since been answered in the affirmative in [Borel and Hirzebruch 1958] and [Milnor 1960].

More recently Grothendieck has obtained [Borel and Serre 1958] a more general Riemann–Roch Theorem for a map \( f: Y\to X \) of algebraic varieties. This reduces to the previous Riemann–Roch Theorem on taking \( X \) to be a point. Grothendieck’s Theorem implies many conditions on characteristic classes, and again it is natural to ask if these conditions hold more generally for almost complex or even differentiable manifolds. The purpose of this note is to enunciate certain differentiable analogues of Grothendieck’s Theorem. These “differentiable Riemann–Roch Theorems” yield, as special cases, the divisibility conditions mentioned above and also certain new homotopy invariance properties of Pontrjagin classes. As an application of the latter we get a new proof (and slight improvement) of the result of Kervaire–Milnor [1960] on the stable \( J \)-homomorphism.

Another differentiable Riemann–Roch Theorem, with applications to embeddability problems of differentiable manifolds, will be found in [Atiyah and Hirzebruch 1959].

The proofs of our theorems rely heavily on the Bott periodicity of the classical groups [Bott 1957, 1958, 1959], and are altogether different from the earlier methods of [Borel and Hirzebruch 1958] and [Milnor 1960], which were based on Thom’s cobordism theory and Adams’ spectral sequence.

[24] M. F. Atiyah and F. Hirzebruch: “Quelques théorèmes de non-plongement pour les variétés différentiables” [Some non-immersion theorems for differentiable manifolds], Bull. Soc. Math. France 87 (1959), pp. 383–396. MR 114231 Zbl 0196.55903 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958], [Borel and Hirzebruch 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espace euclidien (ou ce qui est équivalent, une sphère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui étaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott [Borel and Hirzebruch 1958/59; Bott 1958] qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

[25] A. Borel and F. Hirzebruch: “Characteristic classes and homogeneous spaces, II,” Am. J. Math. 81 : 2 (April 1959), pp. 315–382. MR 110105 article

[26] F. Hirzebruch: “A Riemann–Roch theorem for differentiable manifolds,” pp. 129–149 in Séminaire Bourbaki: 1958/59, exposés 169–186 [Bourbaki seminar: 1958/59, presentations 169–186]. Société Mathématique de France (Paris), 1959. volume 5, exposé no. 177. MR 1603462 Zbl 0129.15406 incollection

[27] A. Borel and F. Hirzebruch: “Characteristic classes and homogeneous spaces, III,” Am. J. Math. 82 : 3 (July 1960), pp. 491–504. MR 120664 article

[28] F. Hirzebruch: “Komplexe Mannigfaltigkeiten” [Complex manifolds], pp. 119–136 in Proceedings of the International Congress of Mathematicians 1958 (Edinburgh, 14–21 August 1958). Edited by J. A. Todd. Cambridge University Press (New York), 1960. MR 123733 Zbl 0132.19704 incollection

[29] M. F. Atiyah and F. Hirzebruch: “Vector bundles and homogeneous spaces,” pp. 7–38 in Differential geometry (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. Republished in Algebraic topology: A student’s guide (1972). MR 139181 Zbl 0108.17705 incollection

Carl Barnett Allendoerfer	Related
M. F. Atiyah	Related	Volume

[30] M. F. Atiyah and F. Hirzebruch: “Cohomologie-Operationen und charakteristische Klassen” [Cohomology operations and characteristic classes], Math. Z. 77 : 1 (1961), pp. 149–187. Dedicated to Friedrich Karl Schmidt on the occasion of his sixtieth birthday. MR 156361 Zbl 0109.16002 article

M. F. Atiyah	Related	Volume
Friedrich Karl Schmidt	Related

[31] M. F. Atiyah and F. Hirzebruch: “Bott periodicity and the parallelizability of the spheres,” Proc. Camb. Philos. Soc. 57 : 2 (April 1961), pp. 223–226. MR 126282 Zbl 0108.35902 article

The theorems of Bott [1958, 1959a] on the stable homotopy of the classical groups imply that the sphere \( S^n \) is not parallelizable for \( n \neq 1 \), \( 3, 7 \). This was shown independently by Kervaire [1958] and Milnor [1958; Bott and Milnor 1958]. Another proof can be found in [Borel and Hirzebruch 1959, §26.11]. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that \( S^n \) with any (perhaps extraordinary) differentiable structure is not parallelizable if \( n \neq 1 \), \( 3, 7 \). Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in [Bott 1958, 1959a]. The purpose of this note is to show how the refined form of Bott’s results given in [Bott 1959b] leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the folowing theorem due to Milnor [1958] which implies the non-parallelizability.

There exists a real vector bundle \( \xi \) over the sphere \( S^n \) with \( w_n(\xi) \neq 0 \) only for \( n = 1 \), \( 2,4 \) or 8

[32] M. F. Atiyah and F. Hirzebruch: “Charakteristische Klassen und Anwendungen” [Characteristic classes and applications], Enseignement Math. (2) 7 : 1 (1961), pp. 188–213. MR 154294 Zbl 0104.39801 article

[33] F. Hirzebruch: “Zur Theorie der Mannigfaltigkeiten” [On the theory of manifolds], Int. Math. Nachr., Wien 15 : 67 (May 1961), pp. 50. Brief research announcement dated 1960-11-18. article

[34] M. F. Atiyah and F. Hirzebruch: “The Riemann–Roch theorem for analytic embeddings,” Topology 1 : 2 (April–June 1962), pp. 151–166. MR 148084 Zbl 0108.36402 article

In [Borel and Serre 1958] Grothendieck formulated and proved a generalization of the Riemann–Roch theorem which we shall refer to as GRR. This theorem is concerned with a proper morphism \( f:Y\to X \) of algebraic manifolds (any ground field) and reduces to the version (HRR) given in [Hirzebruch 1956] when \( X \) is a point (and the ground field is \( \mathbb{C} \)). It is not known whether GRR or even HRR holds for arbitrary complex manifolds. However the proof of GRR given in [Borel and Serre 1958] breaks up into two separate cases:

\( f \) is an embedding,
\( f \) is a projection \( X\times P_N \to X \), where \( P_N \) is a projective space,

and the main purpose of this paper is to give a proof of GRR in case (i) for arbitrary complex manifolds. This proof is quite different from, and in many ways simpler than, that of [Borel and Serre 1958] and, for the complex algebraic case, it gives a new proof of GRR.

[35] M. F. Atiyah and F. Hirzebruch: “Analytic cycles on complex manifolds,” Topology 1 : 1 (January–March 1962), pp. 25–45. MR 145560 Zbl 0108.36401 article

Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \)-dimensional complex analytic subspace of \( X \). Then \( Y \) defines or “carries” a \( 2k \)-dimensional integral homology class \( y \) of \( X \), although the precise definition of \( y \) presents technical difficu1ties. A finite formal linear combination \( \sum n_iY_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_iY_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \). These conditions are that certain cohomology operations should vanish on \( u \), for example \( \mathrm{Sq}^3u = 0 \): they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.

[36] F. Hirzebruch: Neue topologische Methoden in der algebraischen Geometrie [New topological methods in algebraic geometry], 2nd edition. Ergebnisse der Mathematik und ihrer Grenzgebiete 9. Springer (Berlin), 1962. An English translation was published in 1966. A Russian translation was published in 1973. MR 137706 Zbl 0101.38301 book

[37] F. Hirzebruch: “The topology of normal singularities of an algebraic surface,” pp. 129–137 in Séminaire Bourbaki: 1962/63, exposés 247–252 [Bourbaki seminar: 1962/63, presentations 247–252]. Société Mathématique de France (Paris), 1964. Volume 8, exposé No. 250. After an article by D. Mumford. MR 1611536 Zbl 0126.16903 incollection

[38] F. Hirzebruch: “Bericht über Arbeiten am Mathematischen Institut der Universität Bonn” [Report on work at the Mathematical Institute of the University of Bonn], pp. 311–326 in Landesamt für Forschung, Jahrbuch 1965 [State Office for Research, yearbook 1965]. Westdeutscher Verlag (Köln and Opladen), 1965. incollection

[39] F. Hirzebruch: Introduction to the theory of vector bundles and \( K \)-theory. Mimeographed manuscript, University of Amsterdam, 1965. Lecture notes taken by M. Hazewinkel und D. Erle in 1964. techreport

Dieter Erle	Related
Michiel Hazewinkel	Related

[40] F. Hirzebruch: Topological methods in algebraic geometry, 3rd, enlarged edition. Grundlehren der Mathematischen Wissenschaften 131. Springer (New York), 1966. With an appendix by R. L. E. Schwarzenberger and an additional section by A. Borel. English translation of 2nd German edition (1962). MR 202713 Zbl 0138.42001 book

Armand Borel	Related
Rolph Ludwig Edward Schwarzenberger	Related

[41] F. Hirzebruch: “Über Singularitäten komplexer Flächen” [On singularities of complex surfaces], Rend. Mat. e Appl. (5) 25 (1966), pp. 213–232. Also published in Simposio internazionale di geometria algebrica (1967). MR 217819 Zbl 0165.24101 article

[42] F. Hirzebruch: “Elliptische Differentialoperatoren auf Mannigfaltigkeiten” [Elliptic differential operators on manifolds], pp. 583–608 in Festschrift zur Gedächtnisfeier für Karl Weierstraß, 1815–1965 [Festschrift in commemoration of Karl Weierstrass, 1815–1965]. Edited by H. Behnke and K. Kopfermann. Wissenschaftliche Abhandlungen der Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen 33. Westdeutscher Verlag (Cologne), 1966. Also published in Über den Vorbereitungssatz von Weierstraß. Elliptische Differentialoperatoren auf Mannigfaltigkeiten (1966). A Russian translation was published in Uspekhi Mat. Nauk 23:1(139) (1968). MR 210151 Zbl 0181.10302 incollection

Karl Theodor Wilhelm Weierstraß	Related
Heinrich Adolph Behnke	Related
Klaus Kopfermann	Related

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	AUTHOR = {Hirzebruch, Friedrich},
	TITLE = {Elliptische {D}ifferentialoperatoren
		 auf {M}annigfaltigkeiten [Elliptic differential
		 operators on manifolds]},
	BOOKTITLE = {Festschrift zur {G}ed\"achtnisfeier
		 f\"ur {K}arl {W}eierstra{\ss}, 1815--1965
		 [Festschrift in commemoration of {K}arl
		 {W}eierstrass, 1815--1965]},
	EDITOR = {Behnke, Heinrich and Kopfermann, Klaus},
	SERIES = {Wissenschaftliche Abhandlungen der Arbeitsgemeinschaft
		 f\"ur Forschung des Landes Nordrhein-Westfalen},
	NUMBER = {33},
	PUBLISHER = {Westdeutscher Verlag},
	ADDRESS = {Cologne},
	YEAR = {1966},
	PAGES = {583--608},
	NOTE = {Also published in \textit{\"Uber den
		 Vorbereitungssatz von Weierstra{\ss}.
		 Elliptische Differentialoperatoren auf
		 Mannigfaltigkeiten} (1966). A Russian
		 translation was published in \textit{Uspekhi
		 Mat. Nauk} \textbf{23}:1(139) (1968).
		 MR:210151. Zbl:0181.10302.},
	ISSN = {0078-9410},
	ISBN = {9783663156970},
}

[43] F. Hirzebruch: “Elliptische Differentialoperatoren auf Mannigfaltigkeiten” [Elliptic differential operators on manifolds], pp. 33–60 in H. Cartan and F. Hirzebruch: Über den Vorbereitungssatz von Weierstraß. Elliptische Differentialoperatoren auf Mannigfaltigkeiten [On the Weierstrass preparation theorem. Elliptic differential operators on manifolds]. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen 157. Westdeutscher Verlag (Cologne), 1966. Also published in Festschrift zur Gedächtnisfeier für Karl Weierstraß, 1815–1965 (1966). A Russian translation was published in Uspekhi Mat. Nauk 23:1(139) (1968). MR 222911 Zbl 0237.58016 incollection

@incollection {key222911m,
	AUTHOR = {Hirzebruch, Friedrich},
	TITLE = {Elliptische {D}ifferentialoperatoren
		 auf {M}annigfaltigkeiten [Elliptic differential
		 operators on manifolds]},
	BOOKTITLE = {\"Uber den {V}orbereitungssatz von {W}eierstra{\ss}.
		 {E}lliptische {D}ifferentialoperatoren
		 auf {M}annigfaltigkeiten [On the {W}eierstrass
		 preparation theorem. {E}lliptic differential
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	SERIES = {Arbeitsgemeinschaft f\"ur Forschung
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	NUMBER = {157},
	PUBLISHER = {Westdeutscher Verlag},
	ADDRESS = {Cologne},
	YEAR = {1966},
	PAGES = {33--60},
	DOI = {10.1007/978-3-322-98555-2_4},
	NOTE = {Also published in \textit{Festschrift
		 zur Ged\"achtnisfeier f\"ur Karl Weierstra{\ss},
		 1815--1965} (1966). A Russian translation
		 was published in \textit{Uspekhi Mat.
		 Nauk} \textbf{23}:1(139) (1968). MR:222911.
		 Zbl:0237.58016.},
	ISSN = {0365-2254},
	ISBN = {9783322979728},
}

[44] F. Hirzebruch: “Bericht über Arbeiten am Mathematischen Institut der Universit\”t Bonn” [Report on work at the Mathematics Institute of Bonn University], pp. 311–326 in Landesamt für Forschung: Jahrbuch [National Office for Research: Yearbook]. Westdeutscher Verlag (Cologne), 1966. incollection

[45] F. Hirzebruch: “Über Singularitäten komplexer Flächen” [On singularities of complex surfaces], pp. 213–232 in Simposio internazionale di geometria algebrica [International symposium on algebraic geometry] (Rome, 30 September–5 October 1965). Cremonese (Rome), 1967. Also published in Rend. Mat. e Appl. (5) 25 (1966). incollection

[46] F. Hirzebruch: “Involutionen auf Mannigfaltigkeiten” [Involutions on manifolds], pp. 148–166 in Proceedings of the conference on transformation groups (New Orleans, LA, 8 May–2 June 1967). Edited by P. S. Mostert. Springer (New York), 1968. Dedicated to Heinrich Behnke on the occasion of his seventieth birthday. MR 250339 Zbl 0177.26401 incollection

Paul Stallings Mostert	Related
Heinrich Adolph Behnke	Related

[47] F. Hirzebruch: “Singularities and exotic spheres,” pp. 13–32 in Séminaire Bourbaki: 1966/67, exposés 313–330 [Bourbaki seminar: 1966/67, presentations 313–330]. Société Mathématique de France (Paris), 1968. volume 10, exposé no. 314. MR 1610436 Zbl 0213.47701 incollection

Brieskorn has proved [1966a] that the \( n \)-dimensional affine algebraic variety \[ z_0^3 + z_1^2 + \cdots + z_n^2 = 0 \] (\( n \) odd, \( n \geq 1 \)) is a topological manifold though the variety has an isolated singular point (which is normal for \( n \geq 2 \)). Such a phenomenon cannot occur for normal singularities of 2-dimensional varieties, as was shown by Mumford [1961], [Hirzebruch 1962–1963]). Brieskorn’s result stimulated further research on the topology of isolated singularities ([Brieskorn 1966b], [Milnor 1966] and [Hirzebruch 1968]). Brieskorn [1966b] uses the paper of F. Pham [1965], whereas the speaker studied certain singularities from the point of view of transformation groups using results of [Bredon 1965a, 1965b], [Hsiang and Hsiang 1966] and [Janich 1966].

[48] F. Hirzebruch: “Elliptic differential operators on manifolds,” Uspekhi Mat. Nauk 23 : 1 (139) (1968), pp. 191–209. Russian translation of original German article published in Über den Vorbereitungssatz von Weierstraß. Elliptische Differentialoperatoren auf Mannigfaltigkeiten (1966). MR 222912 article

[49] F. Hirzebruch and K. H. Mayer: \( \operatorname{O}(n) \)-Mannigfaltigkeiten, exotische Sphären und Singularitäten [\( \operatorname{O}(n) \)-Manifolds, exotic spheres and singularities] (Berlin). Lecture Notes in Mathematics 57. Springer, 1968. MR 229251 Zbl 0172.25304 book

[50] F. Hirzebruch and K. Jänich: “Involutions and singularities,” pp. 219–240 in Algebraic geometry (Bombay, 16–23 January 1968). Edited by S. S. Abhyankar. Studies in Mathematics 4. Oxford University Press (London), 1969. MR 259948 Zbl 0221.57024 incollection

Shreeram Shankar Abhyankar	Related
Klaus Jänich	Related

[51] F. Hirzebruch: “The signature of ramified coverings,” pp. 253–265 in Global analysis: Papers in honor of K. Kodaira. Edited by D. C. Spencer and S. Iyanaga. Princeton Mathematical Series 29. Princeton University Press, 1969. MR 258060 Zbl 0208.51802 incollection

Shōkichi Iyanaga	Related
Donald Clayton Spencer	Related
Kunihiko Kodaira	Related

[52] M. Atiyah and F. Hirzebruch: “Spin-manifolds and group actions,” pp. 18–28 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)] (Geneva, 26–28 March 1969). Edited by R. Narasimhan and A. Haefliger. Springer (New York), 1970. MR 278334 Zbl 0193.52401 incollection

M. F. Atiyah	Related	Volume
Georges de Rham	Related
Raghavan Narasimhan	Related
André Haefliger	Related

[53] F. Hirzebruch: “Lösung einer Aufgabe von H. Hasse” [Solution to a problem of H. Hasse], Jahresber. Dtsch. Math.-Ver. 72 (1970), pp. 29–32. article

[54] F. Hirzebruch: “Free involutions on manifolds and some elementary number theory,” pp. 411–419 in Teoria dei modelli; Geometria [Model theory; Geometry] (Rome, 17–20 November 1969 and 13–16 April 1970). Symposia Mathematica 5. Academic Press (London), 1971. MR 287574 Zbl 0206.52404 incollection

[55] F. Hirzebruch: “Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces,” pp. 207–212 in Singularities–Symposium II (Liverpool, UK, September 1969–August 1970). Edited by C. T. C. Wall. Lecture Notes in Mathematics 209. Springer (Berlin), 1971. MR 336752 Zbl 0217.49301 incollection

[56] 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

Lars Hörmander	Related
John W. Milnor	Related
Jean-Pierre Serre	Related
Isadore M. Singer	Related	Volume

[57] F. Hirzebruch, W. D. Neumann, and S. S. Koh: Differentiable manifolds and quadratic forms. Lecture Notes in Pure and Applied Mathematics 4. Marcel Dekker (New York), 1971. With an appendix by W. Scharlau. MR 341499 Zbl 0226.57001 book

Winfried Scharlau	Related
Sebastian Su Koh	Related
Walter D. Neumann	Related	Volume

[58] F. Hirzebruch: “The Hilbert modular group, resolution of the singularities at the cusps and related problems,” pp. 275–288 in Séminaire Bourbaki: 1970/71 Exposés 382–399 [Bourbaki seminar: 1970/71 Presentations 382–399]. Lecture Notes in Mathematics 244. Springer (Berlin), 1971. exposé 396. MR 417187 Zbl 0232.10017 incollection

[59] F. Hirzebruch and W. Scharlau: Einführung in die Funktionalanalysis [Introduction to functional analysis]. Hochschultaschenbücher 296. Bibliographisches Institut (Mannheim), 1971. Republished in 1991. MR 463864 Zbl 0219.46001 book

[60] 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

Lars Hörmander	Related
John W. Milnor	Related
Jean-Pierre Serre	Related
Isadore M. Singer	Related	Volume

[61] F. Hirzebruch: “The signature theorem for differentiable manifolds and some elementary number theory,” pp. 78–92 in Geometry of manifolds (Kyoto, 21–23 February 1972). Edited by K. Daigaku and S. K. Kenkyujo. R.I.M.S. Seminar Reports 158. Kyoto University, 1972. Notes by S. Morita. MR 407314 Zbl 0387.57005 incollection

Sūri Kaiseki Kenkyūjo	Related
Shigeyuki Morita	Related
Kyōto Daigaku	Related

[62] F. Hirzebruch: “Pontrjagin classes of rational homology manifold,” pp. 1–11 in Geometry of manifolds (Kyoto, 21–23 February 1972). Edited by K. Daigaku and S. K. Kenkyujo. R.I.M.S. Seminar Reports 158. Kyoto University, 1972. Notes by S. Morita. MR 394694 Zbl 0387.57006 incollection

Sūri Kaiseki Kenkyūjo	Related
Shigeyuki Morita	Related
Kyōto Daigaku	Related

[63] M. F. Atiyah and F. Hirzebruch: “Vector bundles and homogeneous spaces,” pp. 197–222 in Algebraic topology: A student’s guide. Edited by J. F. Adams. London Mathematcial Society Lecture Note Series 4. Cambridge University Press, 1972. Dedicated to Professor Marston Morse. Republished from Differential geometry (1961). incollection

M. F. Atiyah	Related	Volume
Harold Calvin Marston Morse	Related
John Frank Adams	Related

[64] F. Hirzebruch: “Lectures on \( K \)-theory,” pp. 223–238 in Algebraic topology: A student’s guide (Seattle, 1963). Edited by J. F. Adams. London Mathematical Society Lecture Note Series 4. Cambridge University Press, 1972. Notes prepared by Paul Baum. Lecture notes of the AMS Summer Topology Institute. incollection

Paul Baum	Related	Volume
John Frank Adams	Related

[65] H. Behnke and F. Hirzebruch: “In memoriam Heinz Hopf,” Math. Ann. 196 : 1 (1972), pp. 1–7. MR 292635 Zbl 0226.01013 article

Heinrich Adolph Behnke	Related
Heinz Hopf	Related

[66] W. F. Hammond and F. Hirzebruch: “\( L \)-series, modular imbeddings, and signatures,” Math. Ann. 204 : 4 (1973), pp. 263–270. MR 374038 Zbl 0245.10017 article

For each cusp singularity of the Hilbert modular variety of a totally real algebraic number field there is associated an \( L \)-series which determines the corresponding “parabolic contribution” to the version of Selberg’s trace formula worked out by Shimizu.

For a real quadratic field Meyer [1957, 1967] has given an elementary algorithm by Dedekind sums for the computation of the parabolic contribution (see also Siegel [1965]) from which a simple formula for the parabolic contribution in terms of the continued fraction associated with the cyclic resolution of the corresponding cusp singularity can be derived [Hirzebruch 1971, 1972, 1976]. In fact, the parabolic contribution equals \[ \frac{1}{12}(3r - b_0 - b_1 - \dots - b_{r-1}) \] where \( b_0,\dots,b_{r-1} \) are the characteristic numbers of the dual normal bundles of the rational curves which form the exceptional fibre of the cyclic resolution.

The first purpose of this note is to give a simple formula for the total parabolic contribution, i.e., the sum of the parabolic contributions for the different cusps. This formula is deduced from a general identity satisfied by certain \( L \)-series in any real algebraic number field. In the quadratic case one obtains an expression for the total parabolic contribution in terms of the class numbers of imaginary quadratic fields. Further, in the quadratic case the total parabolic contribution vanishes if and only if the discriminant of the field is the sum of two squares and is negative otherwise.

The total parabolic contribution is of additional interest in the quadratic case for two reasons: first, it is one-fourth the signature (adjusted for defects coming from quotient singularities in certain special cases) of the open rational homology 4-manifold obtained as the quotient of the product of two upper-half planes by the non-symmetric Hilbert modular group. This result, announced ot some extent in [Hirzebruch 1971], will be proved here. Thus, from [Hammond 1966] the vanishing of this (adjusted) signature is a necessary and sufficient condition for the existence of modular imbeddings. Second, it will be shown that this signature is twice the difference of the arithmetic genera for the ordinary and mixed Hilbert modular groups.

[67] F. Hirzebruch: “The Hilbert modular group and some algebraic surfaces,” pp. 59–72 in Proceedings of the international conference on number theory (Moscow, 14–18 September 1971). Edited by I. M. Vinogradov, A. A. Karacuba, K. I. Oskolkov, and A. N. Paršin. Proceedings of the Steklov Institute of Mathematics 132. American Mathematical Society (Providence, RI), 1973. MR 344255 Zbl 0305.14005 incollection

Anatoly Alexeyevich Karatsuba	Related
Ivan Matveevich Vinogradov	Related
Konstantin I. Oskolkov	Related
A. N. Paršin	Related

[68] F. E. P. Hirzebruch: “Hilbert modular surfaces,” Enseignement Math. (2) 19 : 3–4 (1973), pp. 183–281. MR 393045 Zbl 0285.14007 article

[69] F. E. P. Hirzebruch: Hilbert modular surfaces (Tokyo, February–March 1972). Monographies de L’Enseignement Mathématique 21. Université de Genève, 1973. This is also number 4 of Série des Conférences de l’Union Mathématique Internationale. MR 389921 book

[70] F. Hirzebruch: Topologiceskie metody v algebraiceskoi geometrii [Topological methods in algebraic geometry]. Mir (Moscow), 1973. with appendices by R. L. E. Schwarzenberger and A. Borel. Russian translation of 2nd German edition (1962). Zbl 0266.14011 book

Armand Borel	Related
Rolph Ludwig Edward Schwarzenberger	Related

[71] F. Hirzebruch: “Kurven auf den Hilbertschen Modulflächen und Klassenzahlrelationen” [Curves on Hilbert modular surfaces and class number relations], pp. 75–93 in Classification of algebraic varieties and compact complex manifolds. Edited by H. Popp. Lecture Notes in Mathematics 412. Springer (Berlin), 1974. MR 376694 Zbl 0301.14010 incollection

[72] F. Hirzebruch and D. Zagier: The Atiyah–Singer theorem and elementary number theory. Mathematics Lecture Series 3. Publish or Perish (Boston), 1974. MR 650832 Zbl 0288.10001 book

[73] F. Hirzebruch and A. Van de Ven: “Hilbert modular surfaces and the classification of algebraic surfaces,” Invent. Math. 23 : 1 (1974), pp. 1–29. MR 364262 Zbl 0296.14020 article

[74] F. Hirzebruch: “Mathematik, Studium und Forschung” [Mathematics, study and research], pp. 450–467 in Mathematiker über die Mathematik [Mathematicians on mathematics]. Edited by T. Mies and G. Schubring. Wissenschaft und Öffentlichkeit. Springer (Berlin), 1974. incollection

[75]Proceedings of the International Conference on Number Theory (Moscow, September 14–18, 1971). Edited by I. M. Vinogradov, A. A. Karacuba, K. T. Oskolkov, and A. N. Paršin. Proceedings of the Steklov Institute of Mathematics 132. American Mathematical Society (Providence, R.I.), 1975. MR 0369228

Ivan Matveevich Vinogradov	Related
Anatoly Alexeyevich Karatsuba	Related
Konstantin I. Oskolkov	Related
A. N. Paršin	Related

[76] F. Hirzebruch: “The Hilbert modular group for the field \( \textbf{Q}(\sqrt{5}) \), and the cubic diagonal surface of Clebsch and Klein,” Uspehi Mat. Nauk 31 : 5(191) (1976), pp. 153–166. An English translation was published in Russ. Math. Surv. 31:5 (1976). MR 498397 Zbl 0335.14007 article

[77] F. Hirzebruch: “Hilbert modular surfaces and class numbers,” pp. 151–164 in Colloque analyse et topologie: En l’honneur de Henri Cartan [Colloquium on analysis and topology: In honor of Henri Cartan] (Orsay, France, 17–20 June 1974). Astérisque 32–33. Société Mathématique de France (Paris), 1976. MR 447255 Zbl 0334.14017 incollection

[78] F. Hirzebruch and D. Zagier: “Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus,” Invent. Math. 36 : 1 (1976), pp. 57–113. To Jean-Pierre Serre. A short summary was published in Delange–Pisot–Poitou seminar: Number theory (1977). MR 453649 Zbl 0332.14009 article

Don Bernard Zagier	Related
Jean-Pierre Serre	Related

[79] F. Hirzebruch: “Hilbert’s modular group of the field \( \mathbb{Q}(\sqrt{5}) \) and the cubic diagonal surface of Clebsch and Klein,” Russ. Math. Surv. 31 : 5 (1976), pp. 96–110. English translation of an article published in Uspehi Mat. Nauk 31:5(191) (1976). Zbl 0356.14010 article

In this paper we establish an isomorphism between a certain model of the modular surface belonging to a congruence subgroup \( \Gamma \) of Hilbert’s modular group of the field \( \mathbb{Q}(\sqrt{5}) \) on the one hand, and the famous cubic surface which Clebsch [1871] called diagonal and which Klein [1922] described by means of the icosahedron. From this isomorphism there follows theorems on the structure of the ring of Hilbert’s modular forms for \( \Gamma \) and for the whole Hilbert modular group of \( \mathbb{Q}(\sqrt{5}) \), which are then compared with the results of Gundlach [1963] and Resnikoff [1974]. For the general theory of the Hilbert modular surfaces the reader is referred to [Hirzebruch 1973] and the references quoted there.

[80] F. Hirzebruch and D. Zagier: “Classification of Hilbert modular surfaces,” pp. 43–77 in Complex analysis and algebraic geometry: A collection of papers dedicated to K. Kodaira. Edited by W. L. Baily, Jr. and T. Shioda. Iwanami Shoten (Tokyo), 1977. MR 480356 Zbl 0354.14011 incollection

Kunihiko Kodaira	Related
Tetsuji Shioda	Related
Don Bernard Zagier	Related
Walter Lewis Baily, Jr.	Related

[81] F. Hirzebruch: “The ring of Hilbert modular forms for real quadratic fields of small discriminant,” pp. 287–323 in Modular functions of one variable (Bonn, 2–14 July 1976), vol. VI: Proceedings of the second international conference. Edited by J. P. Serre and D. B. Zagier. Lecture Notes in Mathematics 627. Springer (Berlin), 1977. MR 480355 Zbl 0369.10017 incollection

Don Bernard Zagier	Related
Jean-Pierre Serre	Related

[82] F. Hirzebruch: “Positive definite binary quadratic forms and Hilbert modular surfaces (joint work with D. Zagier),” pp. 229 in Journées arithmétiques de Caen 1976 [Caen arithemetic days, 1976] (Caen, France, 3–8 May 1976). Astérisque 41–42. Société Mathématique de France (Paris), 1977. Zbl 0345.14008 incollection

Let \( p \) be a prime \( \equiv 1 \pmod 4 \) and \( \mathscr{O} \) the ring of integers of \( \mathbb{Q}(\sqrt{p}) \). The group \( SL_2(\mathscr{O}) \) operates on \( H^2 \). The Hilbert modular surface \[ X = H^2/SL_2(\mathscr{O}) \] is a non-compact complex surface with finitely many singularities. On \( X \) we define a series of “modular” curves \( F_1, F_2 \), \( F_3,\dots \). All intersections of \( F_M \) and \( F_N \) (\( M \neq N \)) are transversal and occur at certain distinguished points of \( X \) which we call special. To each special point \( z \in X \) is associated a positive definite binary quadratic form \( \phi_z \), and \( F_M \) and \( F_N \) meet in \( z \) if and only if \( \phi_z \) represents both \( M \) and \( N \) primitively. The following questions will be answered: How often does a given form occur as the form \( \phi_z \) for \( z \in X \)? How does one calculate the intersection number of \( F_M \) and \( F_N \)? These questions led to elementary problems concerning the representation of numbers by positive definite binary forms and the representation of forms by forms, which will be discussed. The intersection numbers are related to coefficients of modular forms for \( \Gamma_0(p) \), cf. the lecture by D. Zagier.

[83] F. Hirzebruch: “Intersection numbers of curves on Hilbert modular surfaces,” pp. 1–3 in Séminaire Delange–Pisot–Poitou: Théorie des nombres [Delange–Pisot–Poitou seminar: Number theory], fascicule 1: Exposés 1–22. 1977. Exposé 22. This is a short summary of the work presented in an article published in Invent. Math. 36:1 (1976). Zbl 0352.14018 incollection

[84] F. Hirzebruch: “Überlagerungen der projektiven Ebene und Hilbertsche Modulflächen” [Superpositions of the projective plane and Hilbert modulular surfaces], Enseign. Math. (2) 24 : 1–2 (1978), pp. 63–78. MR 498399 Zbl 0382.14010 article

[85] F. Hirzebruch: “Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe” [Modular surfaces and modular curves for the symmetric Hilbert modular group], Ann. Sci. École Norm. Sup. (4) 11 : 1 (1978), pp. 101–166. MR 498398 Zbl 0384.14011 article

[86] F. Hirzebruch: Topological methods in algebraic geometry, 3rd (2nd corrected printing) edition. Grundlehren der Mathematischen Wissenschaften 131. Springer (Berlin), 1978. with appendices by R. L. E. Schwarzenberger and A. Borel. Republication of 1966 English translation of 2nd German edition (1962). Reprinted in 1995. Zbl 0376.14001 book

Armand Borel	Related
Rolph Ludwig Edward Schwarzenberger	Related

[87] F. Hirzebruch and A. Van de Ven: “Minimal Hilbert modular surfaces with \( p_g=3 \) and \( K^2=2 \),” Am. J. Math. 101 : 1 (February 1979), pp. 132–148. MR 527829 Zbl 0414.14023 article

[88] 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

[89] F. Hirzebruch and G. van der Geer: Lectures on Hilbert modular surfaces. Séminaire de Mathématiques Supérieures 77. Presses de l’Université de Montréal, 1981. Based on notes taken by W. Hausmann and F. J. Koll. MR 639898 Zbl 0483.14009 book

Franz Josef Koll	Related
Wilfried Hausmann	Related
Gerard B. M. van der Geer	Related

[90] Leopoldina-symposion: Singularitäten (Thüringen, 3–7 May 1978). Edited by F. Hirzebruch and O.-H. Keller. Nova Acta Leopoldina (Neue Folge) 52. Barth (Leipzig), 1981. Zbl 0461.00012 book

[91] F. Hirzebruch: “Some examples of algebraic surfaces,” pp. 55–71 in Papers in algebra, analysis and statistics (Hobart, Australia, 12 January–6 February 1981). Edited by R. Lidl. Contemporary Mathematics 9. American Mathematical Society (Providence, RI), 1982. MR 655974 Zbl 0487.14009 incollection

[92] F. Hirzebruch: “Arrangements of lines and algebraic surfaces,” pp. 113–140 in Arithmetic and geometry: Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday, vol. II: Geometry. Edited by M. Artin and J. T. Tate. Progress in Mathematics 36. Birkhäuser (Boston), 1983. MR 717609 Zbl 0527.14033 incollection

John Torrence Tate, Jr.	Related
Michael Artin	Related
Igor Rostislavovich Shafarevich	Related

[93]B. L. van der Waerden: Zur algebraischen Geometrie: Selected papers. Springer (Berlin), 1983. With a foreword by F. Hirzebruch. MR 716328 Zbl 0552.14001

[94] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, A. Prestel, and R. Remmert: Zahlen [Numbers]. Edited by K. Lamotke. Grundwissen Mathematik 1. Springer (Berlin), 1983. Also with an introduction by K. Lamotke. Further German editions were published in 1988 and 1992. English translations were published in 1990 and 1991. MR 736150 Zbl 0543.00001 book

Reinhold Remmert	Related
Max Koecher	Related
Klaus Lamotke	Related
Klaus Mainzer	Related
Heinz-Dieter Ebbinghaus	Related
Alexander Prestel	Related
Hans Hermes	Related

[95] F. Hirzebruch: “Mannigfaltigkeiten und algebraische Topologie” [Manifolds and algebraic topology], pp. 489–506 in Forschung in der Bundesrepublik Deutschland Beispiele: Kritik, Vorschläge [Research in the Federal Republic of Germany: Examples, criticism, suggestions]. Edited by C. Schneider. Verlag Chemie (Weinheim), 1983. incollection

[96] F. Hirzebruch: “Chern numbers of algebraic surfaces: An example,” Math. Ann. 266 : 3 (1984), pp. 351–356. MR 730175 Zbl 0504.14030 article

An algebraic surface has the Chern number \( c_2 \) (Euler–Poincaré characteristic or Euler number) and \( c_1^2 \) (self-intersection number of a canonical divisor). For a minimal surface of general type these numbers are positive and satisfy \[ c_1^2 \leq 3c_2 \] (see [Miyaoka 1977; Yau 1977]). If \( \{E_i\} \) is a finite set of disjoint smooth elliptic curves on a minimal surface of general type, then \( E_iE_i < 0 \) and \[ -\sum E_iE_i \leq 3c_2 - c_1^2 \] (see [Sakai 1980, §7]). We shall construct a sequence \( X_n \) (\( n=2 \), \( 3,\dots \)) of minimal surfaces of general type with the following properties \begin{align*} & c_2(X_n) = n^7 \\ & 3c_2(X_n) - c_1^2(X_n) = 4n^5. \end{align*}

[97] F. Hirzebruch: “Algebraic surfaces with extremal Chern numbers (report on the thesis of T. Höfer, Bonn, 1984),” Uspekhi Mat. Nauk 40 : 4(244) (1985), pp. 121–129. An English translation was published in Russ. Math. Surv. 40:4 (1985). MR 807793 article

[98] Arbeitstagung (Bonn, 15–22 June 1984). Edited by F. Hirzebruch, J. Schwermer, and S. Suter. Lecture Notes in Mathematics 1111. Springer (Berlin), 1985. MR 797412 Zbl 0547.00007 book

Joachim Schwermer	Related
Silke Suter	Related

[99] F. Hirzebruch: “Algebraic surfaces with extreme Chern numbers (report on the thesis of T. Höfer, Bonn 1984),” Russ. Math. Surv. 40 : 4 (1985), pp. 135–145. English translation of article published in Uspekhi Mat. Nauk 40:4(244). Zbl 0593.14023 article

For a smooth algebraic surface \( X \) the Chern numbers \( c_1^2(X) \) and \( c_2(X) \) are defined. Here \( c_2(X) \) is equal to the Euler–Poincaré characteristic of \( X \) and \( c_1^2(X) \) is the self-intersection number of a canonical divisor of \( X \). For a surface of general type they satisfy the Miyaoka–Yau inequality \( c_1^2 \leq 3c_2 \) (see [Miyaoka 1977] and [Yau 1977]), where the equality sign holds if and only if the universal cover of the surface is the unit ball \( |z_1|^2 + |z_2|^2 \) \( < 1 \) (see [Yau 1977] and [Miyaoka 1983, §2] for the difficult, and [Hirzebruch 1958] for the easy direction of this equivalence.) If \( c_1^2 = 3c_2 \), then automatically \( c_2 > 0 \), in fact \( c_2 \) is the volume of the surface (normalized by the Gauss–Bonnet form) with respect to the complex-hyperbolic metric induced from the ball.

We wish to construct surfaces of general type with extreme Chern numbers (\( c_1^2 = 3c_2 \)) that are ramified covers of the complex projective plane branched along lines. Thus, we continue the investigation of the paper [Hirzebruch 1983]. However, the much better developed theory and the new examples are due to Höfer [1985]. This note is a report on his work.

We have to omit many things. For example, we only consider the case when the surface is the quotient of the ball by a discrete group \( \Gamma \) of automorphisms, \( \Gamma \) operating freely with compact quotient. Höfer includes the case when \( \Gamma \) may have torsion and cusps (at infinity) and constructs many such ball quotients as ramified covers of the plane. His formulae (with suitable modifications) hold for this more general case.

[100] F. Hirzebruch and R. Leis: “Mathematik als Grundlagen- und Angewandte Wissenschaft” [Mathematics as fundamental and applied science], pp. 245–262 in Sonderforschungsbereiche 1969–1984 [Collaborative research centers, 1969–1984]. Edited by K. Stackmann and A. Streiter. VCH Verlagsgesellschaft (Weinheim), 1985. incollection

[101] F. Hirzebruch: “Singularities of algebraic surfaces and characteristic numbers,” pp. 141–155 in The Lefschetz centennial conference on algebraic geometry, algebraic topology and differential equations (Mexico City, 10–14 December 1984), part I. Edited by D. Sundararaman, A. Verjovsky, and S. Gitler. Contemporary Mathematics 58. American Mathematical Society (Providence, RI), 1986. MR 860410 Zbl 0601.14030 incollection

Solomon Lefschetz	Related
Santiago Alberto Verjovsky Sola	Related
Duraiswamy Sundararaman	Related
Samuel C. Gitler	Related

[102] F. Hirzebruch: “Algebraische Flächen als Quotienten symmetrischer Räume” [Algebraic surfaces as quotients of symmetric spaces], pp. 69–71 in Mathematiker-Kongress der DDR 1986 [GDR mathematicians’ congress 1986] (Rostock, Germany, 10–14 February 1986), vol. 1. Wilhelm-Pieck-Universität (Rostock), 1986. MR 845289 incollection

[103] F. Hirzebruch: Some examples of threefolds with trivial canonical bundle. Lecture notes SFB/MPI 85-86, Sonderforschungsbereich 40 and Max-Planck-Institut für Mathematik, 1986. Augmented 1985–1986 lecture notes taken by J. Werner. techreport

[104] F. Hirzebruch: Gesammelte Abhandlungen [Collected works], vol. I: 1951–1962. Springer (Berlin), 1987. Republished in 2013. MR 931775 Zbl 0627.01044 book

[105]H. Grauert, G. Harder, and R. Remmert: “Curriculum vitae mathematicae,” Math. Ann. 278 : 1–4 (1987), pp. iii–viii. Dual German and English texts. MR 909214

Hans Grauert	Related
Reinhold Remmert	Related
G. Harder	Related

[106]A. Marin: “Les sections secrètes des tores simples dans l’espace euclidien \( \mathbf{ R}^3 \) telles que Kuiper les dévoile à Hirzebruch,” Math. Ann. 278 : 1–4 (1987), pp. 211–217. MR 909226 Zbl 0624.57007

Nicolaas Hendrik Kuiper	Related
Alexis Marin	Related

[107] G. Barthel, F. Hirzebruch, and T. Höfer: Geradenkonfigurationen und Algebraische Flächen [Configurations of lines and algebraic surfaces]. Aspekte der Mathematik D4. Vieweg & Sohn (Braunschweig), 1987. MR 912097 Zbl 0645.14016 book

Thomas Höfer	Related
Gottfried Barthel	Related

[108] F. Hirzebruch: Gesammelte Abhandlungen [Collected works], vol. II: 1951–1962. Springer (Berlin), 1987. Republished in 2013. book

[109] F. Hirzebruch: “Elliptic genera of level \( N \) for complex manifolds,” pp. 37–63 in Differential geometrical methods in theoretical physics (Como, Italy, 24–29 August 1987). Edited by K. Bleuler and M. Werner. NATO ASI Series, Series C: Mathematical Physical Sciences 250. Springer (Berlin), 1988. MR 981372 Zbl 0667.32009 incollection

[110] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert: Zahlen [Numbers], 2nd edition. Edited by K. Lamotke. Grundwissen Mathematik 1. Springer (Berlin), 1988. Republication of 1983 German original. Zbl 0701.00001 book

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[111] F. Hirzebruch: “Laudatio auf Jacques Tits” [Laudatio for Jacques Tits], pp. 15–19 in Mathematische Betrachtungen [Mathematical considerations]. Bonner Akademische Reden 68. Bouvier (Bonn), 1988. incollection

[112] F. Hirzebruch: “Codierungstheorie und ihre Beziehung zu Geometrie und Zahlentheorie” [Coding theory and its relation to geometry and number theory], pp. 7–33 in F. Hirzebruch and D. Zagier: Codierungstheorie und ihre Beziehung zu Geometrie und Zahlentheorie/Primzahlen: Theorie und Anwendung [Coding theory and its relation to geometry and number theory/Prime numbers: Theory and application] (Düsseldorf, 5 November 1986 and 7 June 1989). Rheinisch-Westfälische Akademie der Wissenschaften: Natur-, Ingenieur- und Wirtschaftswissenschaften 370. 1989. MR 1151176 Zbl 0697.94010 incollection

@incollection {key1151176m,
	AUTHOR = {Hirzebruch, Friedrich},
	TITLE = {Codierungstheorie und ihre {B}eziehung
		 zu {G}eometrie und {Z}ahlentheorie [Coding
		 theory and its relation to geometry
		 and number theory]},
	BOOKTITLE = {Codierungstheorie und ihre {B}eziehung
		 zu {G}eometrie und {Z}ahlentheorie/{P}rimzahlen:
		 {T}heorie und {A}nwendung [Coding theory
		 and its relation to geometry and number
		 theory/{P}rime numbers: {T}heory and
		 application]},
	SERIES = {Rheinisch-Westf\"alische Akademie der
		 Wissenschaften: Natur-, Ingenieur- und
		 Wirtschaftswissenschaften},
	NUMBER = {370},
	YEAR = {1989},
	PAGES = {7--33},
	NOTE = {(D\"usseldorf, 5 November 1986 and 7
		 June 1989). MR:1151176. Zbl:0697.94010.},
	ISSN = {0066-5754},
	ISBN = {9783531083704},
}

[113] “Friedrich Hirzebruch,” C. R. Acad. Sci. Paris Sér. Gén. Vie Sci. 6 : 6 (1989), pp. 493–494. MR 1063963 article

[114] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert: Numbers. Graduate Texts in Mathematics 123. Springer (New York), 1990. With an introduction by K. Lamotke and a preface by J. H. Ewing. English translation of 1983 German original. A 2nd corrected English edition was published in 1991. MR 1066206 Zbl 0705.00001 book

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[115] Ein Jahrhundert Mathematik 1890–1990. Festschrift zum Jubiläum der DMV [A century of mathematics 1890–1990: Festschrift for the DMV jubilee]. Edited by G. Fischer, F. Hirzebruch, W. Scharlau, and W. Törnig. Dokumente zur Geschichte der Mathematik 6. Vieweg & Sohn (Braunschweig), 1990. MR 1085961 Zbl 0706.01002 book

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[116] F. Hirzebruch and T. Höfer: “On the Euler number of an orbifold,” Math. Ann. 286 : 1–3 (1990), pp. 255–260. Dedicated to Hans Grauert on his sixtieth birthday. MR 1032933 Zbl 0679.14006 article

Thomas Höfer	Related
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[117]J. G. Bliss, R. V. Moody, and A. Pianzola: “Appendix to: ‘Elliptic genera, involutions, and homogeneous spin manifolds’ by F. Hirzebruch and P. Slodowy,” Geom. Dedicata 35 : 1–3 (1990), pp. 345–351. MR 1066571

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[118] F. Hirzebruch and P. Slodowy: “Elliptic genera, involutions, and homogeneous spin manifolds,” Geom. Dedicata 35 : 1–3 (1990), pp. 309–343. Dedicated to Jacques Tits on the occasion of his sixtieth birthday. MR 1066570 Zbl 0712.57010 article

Jacques Tits	Related
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[119] F. Hirzebruch: “Axiome, Definitionen, Begriffe in der Mathematik” [Axioms, definitions, concepts in mathematics], Nova Acta Leopoldina (N.F.) 63 : 272 (1990), pp. 61–77. MR 1137444 Zbl 0743.00019 article

[120] F. Hirzebruch: “One hundred years of the Deutsche Mathematiker-Vereinigung,” Math. Intell. 13 : 2 (1991), pp. 8–11. This was featured in the journal’s regular “Years Ago” column. MR 1098215 Zbl 0718.01035 article

[121] F. Hirzebruch and W. Scharlau: Einführung in die Funktionalanalysis [Introduction to functional analysis]. Hochschultaschenbücher 296. Bibliographisches Institut (Mannheim), 1991. Republication of the 1971 original. MR 1183466 Zbl 0925.46001 book

[122]L. A. Zalcman: “Mathematicians sweep 1988 Wolf Prizes,” Pokroky Mat. Fyz. Astronom. 36 : 3 (1991), pp. 129–140. MR 1146850

[123] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert: Numbers, 2nd, corrected edition. Graduate Texts in Mathematics 123. Springer (New York), 1991. With an introduction by K. Lamotke and a preface by J. H. Ewing. The original edition English translation of the {1983 German original|MR:736150} was published in 1990. MR 1415833 Zbl 0741.00003 book

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[124] Miscellanea mathematica. Edited by P. Hilton, F. Hirzebruch, and R. Remmert. Springer (Berlin), 1991. Festschrift for Heinz Götze. MR 1131113 Zbl 0723.00017 book

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[125] F. Hirzebruch: “Centennial of the German Mathematical Society (Bremen, September 16th–22nd 1990),” pp. 177–194 in Miscellanea mathematica (Berlin). Edited by P. Hilton, F. Hirzebruch, and R. Remmert. Springer, 1991. Festschrift for Heinz Götze. MR 1131124 Zbl 0850.01019 incollection

Heinz Götze	Related
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[126] F. Hirzebruch: “Ansprache anläßlich der Jubiläumstagung der Deutschen Mathematiker-Vereinigung (DMV)” [Address on the occasion of the centennial of the German Mathematicians’ Society (DMV)], Mitt. Ges. Angew. Math. Mech. 1991 : 1 (1991), pp. 23–32. Zbl 0758.01015 article

[127] J.-P. Bourguignon: “Questions posed to Professor Hirzebruch,” Gaz. Math. 53 (1992), pp. 11–19. MR 1175535 Zbl 0772.01008 article

[128] F. Hirzebruch, T. Berger, and R. Jung: Manifolds and modular forms. Aspects of Mathematics E20. Vieweg & Sohn (Braunschweig), 1992. With appendices by Nils-Peter Skoruppa and by Paul Baum. MR 1189136 Zbl 0767.57014 book

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[129] F. Hirzebruch: “Mannigfaltigkeiten und Modulformen” [Manifolds and modular forms], pp. 20–38 in Jahresbericht der DMV: Jubiläumstagung 100 Jahre DMV [Annual report of the DMV: Anniversary meeting, 100 years DMV] (Bremen, Germany, 17–22 September 1990). Edited by W.-D. Geyer. Teubner (Stuttgart), 1992. Zbl 0752.57013 incollection

[130] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert, and K. Lamotke: Zahlen [Numbers], 3rd edition. Grundwissen Mathematik 1. Springer (Berlin), 1992. Republication of 1983 German original. Zbl 0753.00004 book

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[131] F. Hirzebruch: “Kombinatorik in der Geometrie” [Combinatorics in geometry], pp. 96–98 in Heidelberger Akademie der Wissenschaften für 1991 [Heidelberg Academy of Sciences yearbook 1991]. Heidelberger Akademie der Wissenschaften, 1992. From meeting 1991-11-30. incollection

[132] W. Ebeling: Lattices and codes: A course partially based on lectures by Friedrich Hirzebruch. Advanced Lectures in Mathematics. Vieweg & Sohn (Braunschweig), 1994. Republished in 2002 and 2013. MR 1280458 Zbl 0805.11048 book

[133] F. Hirzebruch: Topological methods in algebraic geometry, 3rd edition. Classics in Mathematics 131. Springer (Berlin), 1995. With appendices by R. L. E. Schwarzenberger and A. Borel, and a preface by the author and Schwarzenberger. Reprint of the 1978 edition. MR 1335917 Zbl 0843.14009 book

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[134] F. Hirzebruch and P. B. Cohen: “Book review: P. Deligne and G. D. Mostow, ‘Commensurabilities among lattices in \( \operatorname{PU}(1,n) \)’,” Bull. Am. Math. Soc. (N.S.) 32 : 1 (1995), pp. 88–105. Book by P. Deligne and G. D. Mostow (Princeton Univ. Press, 1993). MR 1568148 article

Paula Tretkoff	Related
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[135] M. Atiyah: “Friedrich Hirzebruch: An appreciation,” pp. 1–5 in Proceedings of the Hirzebruch 65 conference on algebraic geometry (Ramat Gan, Israel, 2–7 May 1993). Edited by M. Teicher. Israel Mathematical Conference Proceedings 9. Bar-Ilan University (Ramat Gan, Israel), 1996. Zbl 0834.01012 incollection

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[136] Proceedings of the Hirzebruch 65 conference on algebraic geometry (Ramat Gan, Israel, 2–7 May 1993). Edited by M. Teicher. Israel Mathematical Conference Proceedings 9. Bar-Ilan University (Ramat Gan, Israel), 1996. MR 1360492 Zbl 0828.00035 book

[137] F. Hirzebruch: “Regular polyhedra and the football,” Sogo Koen Kikaku Tokubetsu Koen Abusutorakuto 1996 (1996), pp. 1–23. A Japanese translation was published in Sugaku 49:2 (1997). article

[138] F. Hirzebruch: Hilbert modular surfaces and the icosahedron, 1996. Abstracts of Colloquium in Mathematics, Kyoto University, pp. 28–37. misc

[139] F. Hirzebruch, J.-P. Demailly, and J. Lannes: Journée en l’honneur de Henri Cartan [Day in honor of Henri Cartan] (Paris, 14 June 1997). Edited by J.-P. Demailly, F. Hirzebruch, and J. Lannes. SMF Journée Annuelle 1997. Société Mathématique de France (Paris), 1997. MR 1492594 Zbl 0932.00086 book

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[140] F. Hirzebruch: “Learning complex analysis in Münster, Paris, Zürich and Princeton from 1945 to 1953,” pp. 1–2 in Journée en l’honneur de Henri Cartan [Day in honor of Henri Cartan] (Paris, 14 June 1997). Edited by J.-P. Demailly, F. Hirzebruch, and J. Lannes. SMF Journée Annuelle 1997. Société Mathématique de France (Paris), 1997. Reprinted in Mitt. Dtsch. Math.-Ver. 1998:2 (1998) and Gaz. Math. 74 (1997). MR 1492595 Zbl 1071.01500 incollection

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[141]“Friedrich Hirzebruch, the second recipient of the Mathematical Society of Japan’s Takakazu Kan Prize,” Sūgaku 49 : 2 (1997), pp. 186. MR 1476662

[142] K. Ueno: “The work of Friedrich Hirzebruch, recipient of the Takakazu Kan Prize of the Mathematical Society of Japan,” Sūgaku 49 : 2 (1997), pp. 186–187. Zbl 0929.01039 article

[143] F. Hirzebruch: “The second Mathematical Society of Japan address upon receiving the Society’s educational unity prize: ‘Regular polyhedra and soccer balls’,” Sūgaku 49 : 2 (1997), pp. 173–181. Japanese translation of English original published in Sogo Koen Kikaku Tokubetsu Koen Abusutorakuto 1996 (1996). MR 1476659 Zbl 0936.01032 article

[144] F. Hirzebruch: “Learning complex analysis in Münster, Paris, Zürich and Princeton from 1945 to 1953,” Gaz. Math. 74 (1997), pp. 27–39. Reprinted from Journée en l’Honneur de Henri Cartan (1997). MR 1604723 Zbl 1024.01512 article

[145] F. Hirzebruch: “German-Russian cooperation in mathematics,” Mitt. Dtsch. Math.-Ver. 5 : 4 (April 1997), pp. 54–58. MR 1603634 Zbl 0966.01503 article

[146] A. Jackson: “Hirzebruch receives Lomonosov Medal,” Notices Am. Math. Soc. 44 : 9 (October 1997), pp. 1112. article

[147] “Friedrich Hirzebruch, the second recipient of the Mathematical Society of Japan’s Takakazu Kan Prize,” Sūgaku 49 : 2 (1997), pp. 186. Zbl 0929.01038 article

[148] F. Hirzebruch: “Kunihiko Kodaira: Mathematician, friend, and teacher,” Notices Am. Math. Soc. 45 : 11 (December 1998), pp. 1456–1462. Reprinted in Asian J. Math. 4:1 (2000). MR 1656896 Zbl 0908.01029 article

[149] F. Hirzebruch: “Learning complex analysis in Münster, Paris, Zürich and Princeton from 1945 to 1953,” Mitt. Dtsch. Math.-Ver. 6 : 2 (June 1998), pp. 5–11. Reprinted from Journée en l’Honneur de Henri Cartan (1997). MR 1631388 Zbl 1288.01019 article

[150] B. Branner: “An interview with Professor Friedrich Hirzebruch, Max-Planck-Institut für Mathematik, Bonn,” Eur. Math. Soc. Newsl. 1998 : 29 (1998), pp. 4–7. article

[151] Algebraic geometry: Hirzebruch 70 (Warsaw, 11–16 May 1998). Edited by P. Pragacz, M. Szurek, and J. Wiśniewski. Contemporary Mathematics 241. American Matheatmical Society (Providence, RI), 1999. MR 1718132 Zbl 0924.00033 book

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[152] F. Hirzebruch: “Complex cobordism and the elliptic genus,” pp. 9–20 in Algebraic geometry: Hirzebruch 70 (Warsaw, 11–16 May 1998). Edited by P. Pragacz, M. Szurek, and J. Wiśniewski. Contemporary Mathematics 241. American Mathematical Society (Providence, RI), 1999. Notes by Adrian Langer. MR 1718134 Zbl 0938.57025 incollection

Michał Szurek	Related
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Piotr Pragacz	Related
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[153] F. Hirzebruch: “Emmy Noether and topology,” pp. 57–65 in The heritage of Emmy Noether (Ramat Gan, Israel, 2–3 December 1996). Edited by M. Teicher. Israel Mathematical Conference Proceedings 12. Bar-Ilan University (Ramat Gan, Israel), 1999. MR 1665435 Zbl 0928.01012 incollection

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[154] F. Hirzebruch: “On the Euler characteristic of manifolds with \( c_1 = 0 \): A letter to V. Gritsenko,” Algebra i Analiz 11 : 5 (1999), pp. 126–129. An English translation was published in St. Petersbg. Math. J. 11:5 (2000). MR 1734349 article

[155]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

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[156] F. Hirzebruch: “Kunihiko Kodaira: Mathematician, friend, and teacher,” Asian J. Math. 4 : 1 (March 2000), pp. v–xiii. Reprinted from Notices Am. Math. Soc. 45:11 (1998). MR 1803727 Zbl 0994.01014 article

[157] F. Hirzebruch: “The Atiyah–Bott–Singer fixed point theorem and number theory,” pp. 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

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Shing-Tung Yau	Related
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[158]E. Brieskorn: “Singularities in the work of Friedrich Hirzebruch,” pp. 17–60 in Surveys in differential geometry. Surv. Differ. Geom. VII. Int. Press (Somerville, MA), 2000. MR 1919421 Zbl 1084.32506

[159]R. Langevin: “Interview: Friedrich Hirzebruch,” pp. 1229–1239 in Development of mathematics 1950–2000. Birkhäuser (Basel), 2000. MR 1796875

[160] Wolf prize in mathematics, vol. 1. Edited by S. S. Chern and F. Hirzebruch. World Scientific (River Edge, NJ), 2000. MR 1834907 Zbl 0972.01032 book

[161] R. Langevin: “Interview: Friedrich Hirzebruch,” pp. 1229–1239 in Development of mathematics 1950–2000, vol. 1. Edited by J.-P. Pier. Birkhäuser (Basel), 2000. Zbl 0963.01022 incollection

Rémi Langevin	Related
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[162] F. Hirzebruch: “On the Euler characteristic of manifolds with \( c_1 = 0 \): A letter to V. Gritsenko,” St. Petersbg. Math. J. 11 : 5 (2000), pp. 805–807. English translation of Russian article published in Algebra i Analiz 11:5 (1999). Zbl 0972.14014 article

[163] F. Hirzebruch: “Response for Lars Hörmander and Friederich Hirzebruch,” pp. 389 in Wolf Prize in Mathematics, vol. 1. Edited by S. S. Chern and F. Hirzebruch. World Scientific (River Edge, NJ), 2000. Wolf Prize Ceremony, The Knesset, Jerusalem, May 12, 1988. incollection

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[164] Applications of algebraic geometry to coding theory, physics and computation (Eilat, Israel, 25 February–1 March 2001). Edited by C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher. NATO Science Series II: Mathematics, Physics and Chemistry 36. Kluwer Academic (Dordrecht), 2001. MR 1866890 Zbl 0971.00013 book

Rick Miranda	Related
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[165] Wolf prize in mathematics, vol. 2. Edited by S. S. Chern and F. Hirzebruch. World Scientific (River Edge, NJ), 2001. MR 1863749 Zbl 1030.01001 book

[166] F. Hirzebruch: “Zum 100-jährigen Jubiläum der Berliner Mathematischen Gesellschaft” [On the 100-year jubilee of the Berlin Mathematical Society], pp. vii–x in 100 Jahre Berliner Mathematische Gesellschaft (1901–2001): Festschrift [100 years of the Berlin Mathematical Society (1901–2001): Festschrift]. Edited by M. Aigner, H. Begehr, H. Koch, and G. M. Ziegler. Berliner Mathematische Gesellschaft, 2001. Zbl 1090.01513 incollection

Heinrich Gottfried Werner Begehr	Related
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[167] W. Ebeling: Lattices and codes: A course partially based on lectures by Friedrich Hirzebruch, 2rd revised edition. Advanced Lectures in Mathematics. Vieweg & Sohn (Braunschweig), 2002. Republication of 1994 original. MR 1938666 Zbl 1030.11030 book

[168] 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

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[169] F. Hirzebruch: “Wir sind für uns erfolgreich” [We are successful for ourselves], pp. 105–112 in Management by mathematics: Erfahrungen und Erfolge von Executives und Politikern [Management by mathematics: Experiences and achievements of executives and politicians]. Edited by U. Hirsch and G. Dueck. Vieweg (Wiesbaden), 2003. incollection

Ulrich Hirsch	Related
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[170] F. Hirzebruch: “Traueransprache für Peter Slodowy, 27. November 2002” [Eulogy for Peter Slodowy, 27 November 2002], pp. 13–17 in Geometry and topology of caustics–Caustics ’02 (Warsaw, 17–29 June 2002). Edited by S. Janeczko and D. Siersma. Banach Center Publications 62. Polish Academy of Sciences (Warsaw), 2004. MR 2055866 Zbl 1072.01534 incollection

Stanisław Janeczko	Related
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[171] J. Arthur, E. Bombieri, K. Chandrasekharan, F. Hirzebruch, G. Prasad, J.-P. Serre, T. A. Springer, and J. Tits: “Armand Borel (1923–2003),” Notices Am. Math. Soc. 51 : 5 (May 2004), pp. 498–524. The various contributions were reprinted in Asian J. Math. 8:4 (2004). MR 2046057 Zbl 1052.01012 article

Tonny A. Springer	Related
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[172] J. Cuntz: “Laudatio auf Prof. Friedrich Hirzebruch” [Laudatio for Prof. Friedrich Hirzebruch], Mitt. Dtsch. Math.-Ver. 12 : 4 (2004), pp. 287. MR 2110687 article

[173]A. W. Knapp, J.-P. Serre, K. Chandrasekharan, E. Bombieri, F. Hirzebruch, T. A. Springer, J. Tits, J. Arthur, G. Prasad, and M. Goresky: “Armand Borel,” Asian J. Math. 8 : 4 (2004), pp. xlvii–xcvi. Reprinted from Notices Amer. Math. Soc. 51:5 (2004), 498–524. MR 2127939

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[174] F. Hirzebruch: “Armand Borel,” Asian J. Math. 8 : 4 (December 2004), pp. lxiii–lxviii. Reprinted from Notices Am. Math. Soc. 51:5 (2004). Zbl 1072.01535 article

[175] F. Hirzebruch: “The projective tangent bundles of a complex three-fold.” Edited by L. Ji, K. Liu, and S.-T. Yau. Pure Appl. Math. Q. 1 : 3 (2005), pp. 441–448. Dedicated to my friend Armand Borel in memory of the golden fifties in Princeton. MR 2201324 Zbl 1187.14009 article

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[176]W. Lück and V. A. Schmidt: “Interview mit Friedrich Hirzebruch,” Mitt. Dtsch. Math.-Ver. 13 : 3 (2005), pp. 183–187. MR 2185694

Vasco Alexander Schmidt	Related
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[177] A. Jackson: “Hirzebruch awarded Cantor Medal,” Notices Am. Math. Soc. 52 : 2 (February 2005), pp. 253. article

[178] W. Lück and V. A. Schmidt: “Interview mit Friedrich Hirzebruch (Bonn)” [Interview with Friedrich Hirzebruch (Bonn)], Eur. Math. Soc. Newsl. 2005 : 58 (December 2005), pp. 31–35. English translation of an interview published in Int. Math. Nachr., Wien 200 (2005). Zbl 1077.01515 article

Vasco Alexander Schmidt	Related
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[179] W. Lück and V. A. Schmidt: “Interview mit Friedrich Hirzebruch” [Interview with Friedrich Hirzebruch], Int. Math. Nachr., Wien 200 (2005), pp. 25–33. An English translation was published in Eur. Math. Soc. Newsl. 2005:58 (2005). Zbl 1176.01014 article

Vasco Alexander Schmidt	Related
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[180] F. Hirzebruch: “Gründungsgeschichte des Max-Planck-Instituts für Mathematik” [The history of the foundation of the Max-Planck-Institute for Mathematics], Mitt. Dtsch. Math.-Ver. 14 : 2 (2006), pp. 73–79. MR 2244583 Zbl 1244.01033 article

[181] F. Hirzebruch and U. Simon: “Nachruf auf Shiing-Shen Chern” [Obituary for Shiing-Shen Chern], Jahresber. Dtsch. Math.-Ver. 108 : 4 (2006), pp. 197–217. MR 2287091 Zbl 1156.01340 article

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[182] 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

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@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},
}

[183] F. Hirzebruch: “Eulerian polynomials,” Münster J. Math. 1 : 1 (2008), pp. 9–14. MR 2502493 Zbl 1255.11004 article

[184] J.-P. Bourguignon, R. Remmert, and F. Hirzebruch: “In memory of Henri Cartan,” Eur. Math. Soc. Newsl. 2008 : 70 (2008), pp. 5–7. Reprinted under a different title in Notices Am. Math. Soc. 57:8 (2010). article

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[185] J.-P. Bourguignon, R. Remmert, and F. E. P. Hirzebruch: “In memory of Henri Cartan,” Eur. Math. Soc. Newsl. 2008 : 70 (December 2008), pp. 5–7. Zbl 1167.01316 article

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[186] 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

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[187] L. Ji, K. Liu, and S.-T. Yau: “Preface,” Pure Appl. Math. Q. 5 : 2 (2009), pp. i–ii. MR 2508896 Zbl 1172.01325 article

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[188] R. Thiele: Van der Waerden in Leipzig. Edition Am Gutenbergplatz (Leipzig), 2009. With a foreword by Friedrich Hirzebruch. MR 2582147 Zbl 1180.01069 book

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[189] F. E. P. Hirzebruch and M. Kreck: “On the concept of genus in topology and complex analysis,” Notices Am. Math. Soc. 56 : 6 (June–July 2009), pp. 713–719. MR 2536790 Zbl 1171.57300 article

The English word “genus” hails from biology, where it is used to connote a grouping of organisms having common chacteristics. In mathematics the words is also used to group objects with common characteristics. The concept of genus arises in various mathematical contexts, such as number theory, as well as the areas we consider in this article, topology and complex analysis. Even within the latter two areas there are various notions of genus that historically originated with the genus of an oriented surface. We begin with these origins and afterward treat generalizations and modifications. We provide no detailed definitions and proofs; rather, our goal is to give the reader an intuitive feeling for the concept of genera.

[190] F. Hirzebruch: “Chern characteristic classes in topology and algebraic geometry,” pp. 17–30 in Oberwolfach Jahresbericht 2009 [Oberwolfach annual report 2009] (Oberwolfach, Germany, 10 October 2009). Mathematisches Forschungsinstitut Oberwolfach, 2009. incollection

[191] P. D. Lax, F. Hirzebruch, B. Mazur, L. Conlon, E. B. Curtis, H. M. Edwards, J. Huebschmann, and H. Shulman: “Raoul Bott as we knew him,” pp. 43–49 in A celebration of the mathematical legacy of Raoul Bott (Montreal, 9–13 June 2008). Edited by P. R. Kotiuga. CRM Proceedings & Lecture Notes 50. American Mathematical Society (Providence, RI), 2010. MR 2648884 Zbl 1195.01029 incollection

Johannes Huebschmann	Related
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[192] J.-P. Bourguignon, R. Remmert, and F. Hirzebruch: “Cartan, Europe, and human rights,” Notices Am. Math. Soc. 57 : 8 (September 2010), pp. 972–975. Reprinted (with different title) from Eur. Math. Soc. Newsl. 2008:70 (2008). MR 2667494 Zbl 1195.01052 article

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[193] F. Hirzebruch: “Bericht über meine Zeit in der Schweiz in den Jahren 1948–1950” [A report on my time in Switzerland in the years 1948–1950], pp. 303–315 in math.ch/100: Schweizerische Mathematische Gesellschaft 1910–2010 [math.ch/100: Swiss Mathematical Society 1910–2010]. Edited by B. Colbois, C. Riedtmann, and V. Schroeder. European Mathematical Society (Zürich), 2010. Zbl 1209.01018 incollection

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[194] A. Loos, T. Vogt, and T. Heidborn: “Interview mit Friedrich Hirzebruch” [Interview with Friedrich Hirzebruch], Mitt. Dtsch. Math.-Ver. 19 : 2 (2011), pp. 86–89. MR 2859063 article

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[195] F. Hirzebruch: “Arithmetic genera and the theorem of Riemann–Roch,” pp. 81–93 in Teorema di Riemann–Roch e questioni connesse [The Riemann–Roch theorem and related questions] (Varenna, Italy, 29 June–8 July 1955). Edited by F. Severi. C.I.M.E. Summer Schools 4. Springer (Heidelberg), 2011. MR 2867162 incollection

[196] 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

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@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},
}

[197] F. Hirzebruch: “Why do I like Chern? And why do I like Chern classes?,” pp. 61–67 in Frontiers of mathematical sciences: The inauguration of the Mathematical Sciences Center of Tsinghua University and the Tsinghua-Sanya International Mathematics Forum. Edited by B. Gu and S.-T. Yau. International Press (Somerville, MA), 2011. This was included as Hirzebruch’s contribution to a multi-author obituary for Chern in Notices Am. Math. Soc. 58:9 (2011). MR 3050817 incollection

S. S. Chern	Related	Volume
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[198] Friedrich Hirzebruch: Giant of German mathematics, 2 December 2011. Online 10 minute video interview, part of “Science Lives” series. Interview by Matthias Kreck. misc

[199] F. Hirzebruch: “Welcoming speech (extended),” pp. xi–xiii in Advances in geometric analysis (Warsaw, 6–8 April 2009). Edited by S. Janeczko, J. Li, and D. H. Phong. Advanced Lectures in Mathematics 21. International Press (Somerville, MA), 2012. Workshop in honour of Shing-Tung Yau’s 60th birthday. incollection

Stanisław Janeczko	Related
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Jun Li	Related
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[200] M. Sanz-Solé, M. Atiyah, C. Bär, G.-M. Greuel, Y. I. Manin, and J.-P. Bourguignon: “Friedrich Hirzebruch memorial session at the 6th European Congress of Mathematics: Kraków, July 5th, 2012,” Eur. Math. Soc. Newsl. 2012 : 85 (September 2012), pp. 12–20. MR 2986335 Zbl 1264.01020 article

Marta Sanz-Solé	Related
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[201] S. Xambó Descamps: “Friedrich Hirzebruch (1926–2012): Trazos de su influjo, in memoriam” [Friedrich Hirzebruch (1926–2012): Traces of his influence, in memoriam], Gac. R. Soc. Mat. Esp. 15 : 4 (2012), pp. 637–650. MR 3058103 article

[202] B. Schechter: “Friedrich Hirzebruch, mathematician, is dead at 84,” The New York Times (10 June 2012). article

[203] J. Schwermer: “Friedrich Hirzebruch (17. Oktober 1927–27. Mai 2012)” [Friedrich Hirzebruch (October 17th, 1927–May 27th, 2012)], Int. Math. Nachr., Wien 221 (2012), pp. 39–41. Zbl 1288.01062 article

[204] W. Ebeling: Lattices and codes: A course partially based on lectures by Friedrich Hirzebruch, 3rd revised edition. Advanced Lectures in Mathematics. Springer (Berlin), 2013. Republication of 1994 original. MR 2977354 Zbl 1257.11066 book

[205] F. Hirzebruch: Gesammelte Abhandlungen–Collected papers, vol. I: 1951–1962. Springer Collected Works in Mathematics. Springer (Heidelberg), 2013. Republication of the 1987 edition. MR 3184395 Zbl 1281.01012 book

[206] F. Hirzebruch: Gesammelte Abhandlungen–Collected papers, vol. II: 1963–1987. Springer Collected Works in Mathematics. Springer (Heidelberg), 2013. Republication of the 1987 edition. MR 3184930 book

[207] F. Hirzebruch: “The icosahedron,” pp. 656–661 in Gesammelte Abhandlungen–Collected papers, vol. 2: 1963–1987. Springer Collected Works in Mathematics. Springer (Heidelberg), 2013. Raymond and Beverly Sackler Distinguished Lectures in Mathematics, given at Tel Aviv University in March 1981. incollection

[208] M. Atiyah: “Friedrich Ernst Peter Hirzebruch: 17 October 1927–27 May 2012,” Biogr. Mem. Fellows R. Soc. 60 (December 2014). article

[209] M. Atiyah, D. Zagier, I. Manin, Yuri, G. van der Geer, J. Milnor, M. Teicher, K. Ueno, G. Segal, S. Janeczko, J.-P. Bourguignon, M. Kreck, and U. Schmickler-Hirzebruch: “Friedrich Hirzebruch (1927–2012),” Notices Am. Math. Soc. 61 : 7 (August 2014), pp. 706–727. article

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[210] F. Hirzebruch: “Singularities and exotic spheres,” pp. 35 in Special volume in honor of the life and mathematics of Egbert Brieskorn, published as J. Singul. 18. Issue edited by G.-M. Greuel, H. A. Hamm, and L. D. Tráng. Worldwide Center of Mathematics (Cambridge, MA), 2018. Translated from German original which was given at an Oberwolfach meeting on 1996-07-16 to commemorate the 60th birthday of Egbert Brieskorn. MR 3899532 Zbl 1453.14001 incollection

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Friedrich E. P. Hirzebruch

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