Celebratio Mathematica

When I was a gradu­ate stu­dent of the Uni­versity of Tokyo, Pro­fess­or Kodaira of­ten told us his memor­ies of his time in the US. The most un­for­get­table one is the fol­low­ing.

A young Ger­man math­em­atician came to the In­sti­tute for Ad­vanced Study. He cal­cu­lated the Todd genus of sev­er­al al­geb­ra­ic man­i­folds. I wondered what he was really try­ing to prove. But sud­denly he proved the Riemann–Roch the­or­em for all al­geb­ra­ic man­i­folds. In that sum­mer I wrote a let­ter to J-P. Serre that Hirzebruch proved the Riemann–Roch the­or­em, while I could only prove that the Hodge man­i­folds were al­geb­ra­ic.

Kodaira had proved the Riemann–Roch The­or­em for al­geb­ra­ic threefolds by us­ing the the­ory of har­mon­ic in­teg­rals. He was try­ing to prove the the­or­em step-by-step as he could not fore­see that one could prove it in a single step. Hirzebruch used cobor­d­ism the­ory to prove it. This was a com­pletely new ap­proach and paved the way for the Atiyah–Sing­er in­dex the­or­em.

In Decem­ber 1971 Kodaira told us that Hirze bruch would vis­it Ja­pan the fol­low­ing Feb­ru­ary and de­liv­er a series of talks at the Uni­versity of Tokyo. So many times we had heard the name of Hirzebruch from Kodaira and also con­sul­ted his fam­ous book To­po­lo­gic­al Meth­ods in Al­geb­ra­ic Geo­metry, but we had nev­er ex­pec­ted that we would have a chance to meet him in Ja­pan. His lec­tures were the IMU lec­tures, which means that the IMU sup­por­ted his vis­it to Ja­pan. At that time the Ja­pan­ese eco­nomy was grow­ing but still not strong enough, so that the Ja­pan­ese gov­ern­ment sup­por­ted uni­versit­ies very little. We had the re­search grants of the Min­istry of Edu­ca­tion, but strangely enough it was for­bid­den to use them for either in­vit­ing for­eign schol­ars or for vis­it­ing for­eign in­sti­tutes. For that we had to ap­ply for an­oth­er grant, which was quite dif­fi­cult to get. This re­stric­tion was con­tin­ued for a long peri­od and only re­moved around fif­teen years ago. There­fore, at that time it was al­most im­possible to in­vite for­eign schol­ars with Ja­pan­ese funds.

In Janu­ary 1972 the title of his talks was an­nounced. To my sur­prise his talks were on the res­ol­u­tion of cusp sin­gu­lar­it­ies and Hil­bert mod­u­lar sur­faces. In Feb­ru­ary Hirzebruch came to Ja­pan. The lec­ture room was full of people. His talk was so clear and beau­ti­ful that I thought I un­der­stood every de­tail. Of course, this was his ma­gic, and later I real­ized that I had missed many im­port­ant points. In his lec­tures he posed sev­er­al ex­er­cises and prob­lems re­lated to the sub­ject. Since the clas­si­fic­a­tion the­ory of al­geb­ra­ic and ana­lyt­ic sur­faces was pop­u­lar among us, some of his prob­lems were not dif­fi­cult. After the lec­ture Kodaira in­tro­duced us to Hirzebruch. Be­fore his next talk I vis­ited him and showed him an­swers to some of his prob­lems. He was pleased and en­cour­aged me to study fur­ther. At that time I was in­vited to Man­nheim Uni­versity, and Hirzebruch was kind enough to give me sug­ges­tions. He asked me to at­tend the Arbeit­sta­gung in Bonn and prom­ised to send an in­vit­a­tion let­ter.

After Tokyo he vis­ited Kyoto and gave sev­er­al lec­tures. Many young act­ive Ja­pan­ese math­em­aticians at­ten­ded his lec­tures and solved sev­er­al prob­lems posed by him. He asked them to ap­ply to the SFB 40 in Bonn Uni­versity. Soon some of them got in­vit­a­tions to Bonn. At that time in Ja­pan there were sev­er­al pro­grams to vis­it for­eign uni­versit­ies as gradu­ate stu­dents but very few pos­sib­il­it­ies to vis­it for­eign coun­tries as re­search­ers, so that his ad­vice was very help­ful for young Ja­pan­ese math­em­aticians.

At the be­gin­ning of that March I went to Man­nheim Uni­versity. In June I re­ceived an in­vit­a­tion let­ter to the Arbeit­sta­gung from Hirzebruch. He nev­er for­got his prom­ise. The Arbeit­sta­gung was very in­ter­est­ing. I met there many math­em­aticians whose names I knew only from their pa­pers. In Oc­to­ber 1972 I was in­vited to the SFB 40 and stayed there half a year. Then I came back to Tokyo to get my Ph.D. and went back to Bonn the fol­low­ing spring.

At Bonn Uni­versity I got a room in the same build­ing where Hirzebruch had his of­fice. Al­most every day I saw him work­ing hard not only on ad­min­is­trat­ive works but also dis­cuss­ing math­em­at­ics with stu­dents and many math­em­aticians. He was busy enough, but he al­ways at­ten­ded im­port­ant sem­inars and col­loqui­um talks. Also, at teatime he came down to the tearoom and dis­cussed math­em­at­ics with us. He was very kind to an­swer our ques­tions and al­ways en­cour­aged us to do math­em­at­ics. If the ques­tions were not in his fields, he in­tro­duced us to the ap­pro­pri­ate math­em­aticians.

In the 1970s the only pos­sible way to in­vite for­eign schol­ars to Ja­pan was to use the JSPS (Ja­pan So­ci­ety for the Pro­mo­tion of Sci­ence) pro­gram. Hirzebruch’s second vis­it to Ja­pan was un­der this pro­gram. He stayed mainly in Kyoto and had dis­cus­sions with many young Ja­pan­ese math­em­aticians. After that he vis­ited Ja­pan sev­er­al times. He al­ways ad­vised young math­em­aticians to ap­ply to the SFB 40 and later the Max Planck In­sti­tute for Math­em­at­ics. Fol­low­ing his ad­vice, many young Ja­pan­ese math­em­aticians ap­plied to Bonn and many of them had chances to vis­it Bonn. They could not only con­cen­trate on their re­search but also col­lab­or­ate with for­eign math­em­aticians, of­ten in dif­fer­ent fields. Dur­ing the mid-1990s more than one hun­dred Ja­pan­ese math­em­aticians vis­ited Bonn.

In 1997 the Math­em­at­ic­al So­ci­ety of Ja­pan (MSJ) awar­ded the Seki–Kowa Prize to Hirzebruch for his out­stand­ing con­tri­bu­tion to the Ja­pan­ese math­em­at­ic­al com­munity in giv­ing many young Ja­pan­ese math­em­aticians the op­por­tun­ity to study and col­lab­or­ate with math­em­aticians from all over the world. At the same time the MSJ had ap­plied for the Or­der for Hirzebruch through the Min­istry of Edu­ca­tion. The Ja­pan­ese gov­ern­ment awar­ded him the Or­der of the Sac­red Treas­ure, Gold and Sil­ver Star, which was the highest or­der for for­eign­ers ex­cept politi­cians and dip­lo­mats. In Novem­ber 1997 the ce­re­mony was held at the Min­istry of Edu­ca­tion, and the vice min­is­ter awar­ded him the or­der. In the ce­re­mony Hirzebruch answered that he would ac­cept the or­der on be­half of all the Ja­pan­ese and Ger­man math­em­aticians who had once stayed in Bonn and col­lab­or­ated to­geth­er, the sec­ret­ar­ies and staff of the SFB 40 and the Max Planck In­sti­tute for Math­em­at­ics who helped their activ­it­ies, and the Deutsche Forschungs­ge­meinsch­aft and Max Planck Gesell­schaft for sup­port­ing them fin­an­cially. His speech deeply im­pressed of­fi­cials of the Min­istry of Edu­ca­tion who were present at the ce­re­mony.

Hirzebruch loved math­em­at­ic­al talks, and his talks were al­ways clear. Once when he vis­ited Kyoto, I asked him to give a lec­ture for high school stu­dents. At that time every two weeks I or­gan­ized a math­em­at­ics lec­ture for high school stu­dents at the Kyoto Uni­versity. He gave a beau­ti­ful lec­ture on the reg­u­lar ico­sa­hed­ron. The high school stu­dents en­joyed his talk and were im­pressed by how deeply he loved math­em­at­ics.

It is really sad that I can­not talk with him any­more. He al­ways talked with a gentle smile and nev­er failed to en­cour­age us to do math­em­at­ics. I am quite sure that his warm memory and his en­cour­age­ment to do math­em­at­ics will sur­vive in all math­em­aticians who once met him.