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F. Hirzebruch Career ([8],[9]) [10] , Web geometry, Bull. Amer. Math. Soc. ci 0 for i n r 1. For real differentiable man- (N.S.) 6 (1982), no. 1, 1–8. ifolds= such≥ questions− + are treated in the disserta- [11] S.-S. Chern and P. Griffiths, Abel’s theorem tion of Hopf’s student Stiefel [4], later a well-known and webs, Jahresber. Deutsch. Math.-Verein. 80 computer scientist. For a compact complex mani- (1978), no. 1-2, 13–110. fold X of dimension n, the n-dimensional products [12] S.-S. Chern and R. K. Lashof, On the total cur- of the Chern classes of the tangent bundle (all di- vature of immersed manifolds, Amer. J. Math. 79 (1957), 306–318. mensions complex) give the Chern numbers, when [13] S.-S. Chern and J. K. Moser, Real hypersur- integrated over X, for example cn[X] is the Euler- faces in complex manifolds, Acta Math. 133 Poincaré characteristic (Poincaré-Hopf theorem). (1974), 219–271. From 1950 to 1952 I was scientific assistant in [14] S.-S. Chern and R. Osserman, Complete min- Erlangen and wrote the paper [6] where ideas of imal surfaces in euclidean n-space, J. Analyse Hopf entered [2]. Some of the results could have Math. 19 (1967) 15–34. been generalized to higher dimensions. But the [15] S. Chern and J. Simons, Characteristic forms so-called “duality formula” was not yet proved. and geometric invariants, Ann. of Math. (2) 99 This formula says that the total Chern class 1 c1 (1974), 48–69. + + c2 of the direct sum of two complex vector [16] W. Dyck, Beiträge zur Analysis situs, Math. +··· Ann. 32 (1888), no. 4, 457–512. bundles equals the product of the total Chern [17] C. Ehresmann, Sur la topologie de certains es- classes of the summands. The paper [6] has a paces homogenes, Ann. of Math. (2) 35 (1934), remark written during proofreading that Chern no. 2, 396–443. and Kodaira told me that the “duality formula” [18] W. Fenchel, On total curvatures of Riemann- is proved in a forthcoming paper of Chern [7]. In ian manifolds: I, J. London Math. Soc. 15 (1940), the commentary to my paper [6] in volume 1 of 15–22. my Collected Papers (Springer 1987), I write that [19] N. Steenrod, The classification of sphere my knowledge about Chern classes increased with bundles, Ann. of Math. (2) 45 (1944), 294–311. the speed of a flash when I came to Princeton [20] A. Weil, Review on “S.-S. Chern, On integral in August 1952 as a member of the Institute for geometry in Klein spaces, Ann. of Math. 43 (1942), 178–189”, AMS Mathematical Reviews, Advanced Study and talked with K. Kodaira, D. C. MR0006075. Spencer, and, a little later, with A. Borel, who told [21] H. Whitney, Topological properties of differ- me about his thesis containing his theory about entiable manifolds, Bull. Amer. Math. Soc. 43 the cohomology of the classifying spaces of com- (1937), no. 12, 785–805. pact Lie groups. For the unitary group U(n), this [22] S.-T. Yau, Calabi’s conjecture and some new implies that the Chern class ci can be considered results in algebraic geometry, Proc. Nat. Acad. in a natural way as the ith elementary symmetric Sci. U.S.A. 74 (1977), no. 5, 1798–1799. function in certain variables x1,x2,...,xn. My two years (1952–54) at the Institute for Ad- vanced Study were formative for my mathematical F. Hirzebruch career ([8],[9]). I had to study and develop funda- mental properties of Chern classes, introduced the Why Do I Like Chern, and Why Do I Like Chern character, which later (joint work with M. F. Chern Classes? Atiyah) became a functor from K-theory to rational In 1949–50 I studied for three semesters at the ETH cohomology. I began to publish my results in 1953. in Zurich and learned a lot from Heinz Hopf and The main theorem is announced in [10]. It concerns Beno Eckmann [1], also about Chern classes, their the Euler number of a projective algebraic variety applications, and their relations to Stiefel-Whitney V with coefficients in the sheaf of holomorphic sec- classes ([2], [3], [4], [5]). Chern classes are defined tions of a complex analytic vector bundle W over V . for a complex vector bundle E over a reasonable Chern classes everywhere! I quote from [10]: “The n space X with fiber C . They are elements of the main theorem expresses this Euler-Poincaré char- cohomology ring of X. The ith Chern class of E acteristic as a polynomial in the Chern classes of is an element of H2i (X, Z) where 0 i n and ≤ ≤ the tangential bundle of V and in the Chern classes c0 1. They are used for the investigation of fields = of the bundle W .” of r-tuples of sections of the vector bundle, in par- The Chern classes accompanied me throughout ticular if X is a compact complex manifold and E all my mathematical life; for example: In 2009 I the tangent bundle of X. Then we have the basic gave the annual Oberwolfach lecture about Chern fact: If there exists an r-tuple of sections which classes [11]. are linearly independent in every point of X, then My fiancée joined me in Princeton in November 1952. We married. A “marriage tour” was orga- F. Hirzebruch is professor emeritus at the Universität nized, for which Spencer gave me some support Bonn and director emeritus of the Max-Planck- from his Air Force project. I lectured in seven Institut für Mathematik in Bonn. His email address is places during this trip, including Chicago, where [email protected]. we met the great master Shiing-Shen Chern and his October 2011 Notices of the AMS 1231 charming wife. He was forty-one, I was twenty-five. Berkeley sufficiently attractive to deserve your seri- For me he was a gentleman advanced in age. But ous consideration. Some disturbances are expected all shyness disappeared. He was interested in my but they need not concern you. I am going to sub- progress in Princeton about which I also talked mit to the NSF a new proposal for research support in my lecture. We must have spoken about his and will be glad to include you in the proposal.” In papers [3] and [7]. Chern begins in [3] with a study Bonn I was very involved in discussions with the of the Grassmannian H(n,N) of linear subspaces protesting students and expected to have a qui- of dimension n in the complex vector space of eter life in Berkeley as a new faculty member with dimension n N. He defines the Chern classes more time for mathematics. Finally I decided to of the n-dimension+ tautological bundle over the stay in Bonn. Chern was very disappointed. But the Grassmannian in terms of Schubert calculus. From invitations to Berkeley continued. The Cherns were here Chern comes to the definition using r-tuples always very helpful in many practical problems: of sections. For N , the Grassmannian be- picking us up at the airport, finding a house, lend- comes the classifying→ space ∞ of U(n), and we are ing us things useful for housekeeping, even lend- close to what I learned from Borel. For Hermit- ing us a car, depositing items in their house we ian manifolds Chern shows how to represent the had bought to be used during the next visit….We Chern classes by differential forms. enjoyed the Cherns’ hospitality in their beautiful The paper [7] has the following definition of home in El Cerrito, overlooking the Bay with the Chern classes: Let E be a complex vector bundle famous Bay Bridge, or in excellent Chinese restau- of dimension n over the base B. Let P be the rants in Berkeley and Oakland where the Cherns associated projective bundle with fiber Pn 1(C). were highly respected guests. There were always Let L be the tautological line bundle over P− and interesting conversations with the Cherns and the g c1(L). Then g restricted to the fiber of P is other dinner guests. = − 2 the positive generator of H (Pn 1(C), Z). Integra- In 1979 there was a conference, “The Chern n 1 m − tion of g − + over the fiber in P gives c¯m, the mth Symposium”, on the occasion of Chern’s retire- “dual” Chern class of E. The total “dual” Chern ment as a professor of the university. In the class c¯ 1 c¯1 c¯2 is defined by Proceedings [14] I. M. Singer writes: “The confer- = + + +··· ence also reflected Professor Chern’s personality, c c¯ 1. · = active yet relaxed, mixed with gentleness and good If B H(n,N), then c¯ is the total Chern class of humor. We wish him good health, a long life, hap- the= complementary N-dimensional tautological piness, and a continuation of his extraordinary bundle over B. deep and original contributions to mathematics.” Chern uses this to prove that the Chern classes This came also from my heart. are represented by algebraic cycles if everything Chern did not really retire. In 1981 he became happens in the projective algebraic category. the first director of the Mathematical Sciences Re- The Cherns invited my wife and me for dinner in search Institute in Berkeley. When the MSRI build- their home. For the first time we enjoyed the cook- ing was ready, I sometimes used Chern’s beautiful ing of Mrs. Chern. Many meals in Berkeley would office with a wonderful view. follow. The Chern family, with their two children In 1981 I nominated Chern for the “Alexander in 1950, can be seen in the photograph on page von Humboldt-Preis”.
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