Celebratio Mathematica

John and I had a long and very fruit­ful col­lab­or­a­tion that las­ted for more than 15 years and res­ul­ted in 20 pa­pers. I had met John while I was still a gradu­ate stu­dent at Har­vard, and he gave a sem­in­ar talk on his work on mixed Hodge struc­tures on the fun­da­ment­al group of a smooth al­geb­ra­ic vari­ety. After I came to Columbia, we col­lab­or­ated in par­al­lel on the mono­dromy of one para­met­er de­gen­er­a­tions of smooth vari­et­ies, but we only began to work to­geth­er in earn­est when I was vis­it­ing MSRI (now SLMath) in the spring of 1985.

It was an ex­cit­ing time for the study of 4-man­i­folds, and there was a spe­cial year at MSRI in 1984–85 on 3- and 4-di­men­sion­al to­po­logy. Don­ald­son’s pre­print on the fail­ure of the \( h \)-cobor­d­ism the­or­em in di­men­sion 4 had be­gun to cir­cu­late, and John asked me to ex­plain the al­geb­ra­ic geo­metry part in­volving stable holo­morph­ic vec­tor bundles on cer­tain al­geb­ra­ic sur­faces, the so-called Dol­gachev sur­faces \( X_{p,q} \). Dol­gachev sur­faces are spe­cial com­plex al­geb­ra­ic sur­faces with \( b_2^+ = 1 \) (or, equi­val­ently, with no nonzero holo­morph­ic 2-forms) in­dexed by two re­l­at­ively prime in­tegers \( p,q > 1 \). There is a holo­morph­ic fibra­tion \( X_{p,q} \to \mathbb{P}^1 \) whose gen­er­al fiber is a smooth el­lipt­ic curve (i.e., a com­pact Riemann sur­face of genus one) and with two mul­tiple fibers, of mul­ti­pli­cit­ies \( p \) and \( q \). The 4-man­i­folds \( X_{p,q} \) are all ho­mo­topy equi­val­ent to a ra­tion­al sur­face, namely \( \mathbb{C} P^2 \# 9 \overline{\mathbb{C} P}^2 \), or equi­val­ently to the blowup of \( \mathbb{C} P^2 \) at 9 points. Hence, by Mike Freed­man’s clas­si­fic­a­tion of simply con­nec­ted 4-man­i­folds, they are all homeo­morph­ic to \( \mathbb{C} P^2 \# 9 \overline{\mathbb{C} P}^2 \). Us­ing a gauge the­or­et­ic in­vari­ant \( \Gamma \) ad­ap­ted to simply con­nec­ted 4-man­i­folds with \( b_2^+ =1 \), Don­ald­son [e3] showed that \( X_{2,3} \) was not however dif­feo­morph­ic to \( \mathbb{C} P^2 \# 9 \overline{\mathbb{C} P}^2 \). The in­vari­ant \( \Gamma \) is com­puted in the case of \( X_{2,3} \), or more gen­er­ally \( X_{p,q} \), by study­ing the mod­uli space of stable rank two holo­morph­ic vec­tor bundles over \( X_{p,q} \) with \( c_1 =0 \) and \( c_2 = 1 \). The ana­lys­is of such bundles is fairly routine (al­though I had to go to the lib­rary to re­mind my­self of the defin­i­tion of a Mum­ford–Take­moto stable vec­tor bundle). We then de­scribed this pic­ture to Kirby, Taubes, Freed­man and maybe oth­ers. After re­tell­ing the same story many times, it even­tu­ally seemed worth­while to start think­ing about how to gen­er­al­ize these meth­ods and see what we could con­trib­ute. Our ini­tial goal was to study more gen­er­al Dol­gachev sur­faces \( X_{p,q} \), as well as their blowups. John’s key idea was to use the al­geb­ra­ic geo­metry of ra­tion­al sur­faces which were the blowup of \( \mathbb{C} P^2 \) along points on a smooth cu­bic curve to prove res­ults about fun­da­ment­al do­mains of cer­tain dis­crete groups act­ing on hy­per­bol­ic space and to ana­lyze cer­tain “cham­ber struc­tures” on those spaces. Us­ing these res­ults and the above-men­tioned de­scrip­tion of stable bundles, we were able to show that none of the \( X_{p,q} \) are dif­feo­morph­ic to \( \mathbb{C} P^2 \# 9 \overline{\mathbb{C} P}^2 \), and, moreover, that for \( q \) odd, the \( X_{2,q} \) provide an in­fin­ite fam­ily of pair­wise nondif­feo­morph­ic smooth struc­tures on the to­po­lo­gic­al 4-man­i­fold \( \mathbb{C} P^2 \# 9 \overline{\mathbb{C} P}^2 \), a res­ult also proved by Okonek and Van de Ven [e2]. This con­struc­tion gave the first ex­ample of a closed to­po­lo­gic­al man­i­fold which ad­mits an in­fin­ite num­ber of dif­fer­ent smooth struc­tures, a phe­nomen­on that can only hap­pen in di­men­sion 4. We were able to fur­ther gen­er­al­ize this pic­ture to blowups of Dol­gachev sur­faces, giv­ing an in­fin­ite fam­ily of pair­wise nondif­feo­morph­ic smooth struc­tures on the to­po­lo­gic­al 4-man­i­fold \( \mathbb{C} P^2 \# n \overline{\mathbb{C} P}^2 \) for all \( n \ge 9 \) [2], [3]. These meth­ods also showed that there are strong re­stric­tions on the self-dif­feo­morph­isms of the “stand­ard” smooth 4-man­i­fold \( \mathbb{C} P^2 \# n \overline{\mathbb{C} P}^2 \) provided that \( n \ge 10 \). More pre­cisely, an old res­ult of C.T.C. Wall showed that for \( n\le 9 \), every auto­morph­ism of the lat­tice \( H^2(\mathbb{C} P^2 \# n \overline{\mathbb{C} P}^2;\mathbb{Z}) \) (i.e., every iso­morph­ism

pre­serving the quad­rat­ic form defined by self-in­ter­sec­tion) is real­ized by a self-dif­feo­morph­ism of \( \mathbb{C} P^2 \# n \overline{\mathbb{C} P}^2 \). For \( n \ge 10 \), we char­ac­ter­ized the sub­group of auto­morph­isms of \( H^2(\mathbb{C} P^2 \# n \overline{\mathbb{C} P}^2;\mathbb{Z}) \) real­ized by self-dif­feo­morph­isms of \( \mathbb{C} P^2 \# n \overline{\mathbb{C} P}^2 \) and showed that this sub­group has in­fin­ite in­dex in the full auto­morph­ism group of the lat­tice. Prov­ing all of these res­ults ne­ces­sit­ated nu­mer­ous trips down the hill from MSRI to vari­ous greasy ham­burger joints in Berke­ley, and most of our ideas were ori­gin­ally scrawled on nap­kins at lunch. It was a time in both of our lives when new math­em­at­ics seemed to flow daily in an ef­fort­less way. As John has re­minded me nu­mer­ous times, life has nev­er been that easy for either of us be­fore or since.

In the mean­time, Don­ald­son had defined a slew of new in­vari­ants for 4-man­i­folds with \( b_2^+\ge 3 \) [e4], and over the next few years we turned our at­ten­tion to mak­ing some cal­cu­la­tions of these in­vari­ants and ap­ply­ing these cal­cu­la­tions to the study of al­geb­ra­ic or more gen­er­ally com­plex sur­faces. In par­tic­u­lar, we tried to sys­tem­at­ic­ally write down the im­plic­a­tions of Don­ald­son the­ory for com­plex sur­faces in [1]. One of our main ques­tions was the so-called \( (-1) \)-curve con­jec­ture, which states that for a com­plex sur­face \( S \) of non­neg­at­ive Kodaira di­men­sion (i.e., which is not a ra­tion­al or ruled sur­face), every ele­ment of \( H^2(S;\mathbb{Z}) \) which is the co­homo­logy class as­so­ci­ated to a smoothly em­bed­ded 2-sphere of self-in­ter­sec­tion \( -1 \) is the co­homo­logy class as­so­ci­ated to an ex­cep­tion­al curve. We were sub­sequently able to make par­tial com­pu­ta­tions of Don­ald­son in­vari­ants of simply con­nec­ted el­lipt­ic sur­faces with \( b_2^+ \ge 3 \), or, equi­val­ently, \( p_g > 0 \), the heart of the dis­crep­ancy between the smooth and to­po­lo­gic­al clas­si­fic­a­tion of al­geb­ra­ic sur­faces. This led to the fol­low­ing fi­nite­ness res­ult for al­geb­ra­ic sur­faces: the map

is fi­nite-to-one. This study also led to the first com­plete enu­mer­a­tion of Don­ald­son in­vari­ants, for \( K3 \) sur­faces [4]. The book [5] was then con­cerned with writ­ing up everything in de­tail and go­ing over the res­ults on nonsimply con­nec­ted el­lipt­ic sur­faces that could be ob­tained by clas­sic­al meth­ods. A by-product of this work was the state­ment that if \( S_1 \) and \( S_2 \) are two dif­feo­morph­ic com­plex sur­faces, then \( S_1 \) and \( S_2 \) have the same Kodaira di­men­sion, ex­cept pos­sibly if one of \( S_1 \), \( S_2 \) is a ra­tion­al sur­face and the oth­er is a sur­face of gen­er­al type. This was a key step to­ward prov­ing the Van de Ven con­jec­ture, that the Kodaira di­men­sion of a com­plex sur­face is a smooth in­vari­ant.

Sub­sequently, our in­terests di­verged as John began to use more power­ful \( C^\infty \) meth­ods to ana­lyze Don­ald­son in­vari­ants. For ex­ample, the pa­per [6] with Tom Mrowka gives a com­plete smooth clas­si­fic­a­tion of simply con­nec­ted el­lipt­ic sur­faces with \( b_2^+ \ge 3 \). (This work cul­min­ates with the pa­per of Fin­tushel and Stern [e6], which com­putes the full Don­ald­son series for simply con­nec­ted el­lipt­ic sur­faces with \( b_2^+ \ge 3 \) and relates the Kron­heimer–Mrowka ba­sic classes to the Seiberg–Wit­ten ba­sic classes.)

Then, in 1995, the Seiberg–Wit­ten re­volu­tion upen­ded gauge the­ory. Us­ing these new meth­ods, in a tour de force, John, along with Zoltán Szabó and Cliff Taubes, was able to prove the Thom con­jec­ture [7]. Re­turn­ing to the case of al­geb­ra­ic sur­faces, John and I were able to give to com­plete the proof of the \( (-1) \)-curve con­jec­ture and to give a short proof of the Van de Ven con­jec­ture, which had pre­vi­ously been es­tab­lished in [e5] build­ing on [5], as well as a more re­fined ver­sion, the smooth in­vari­ance of the pluri­gen­era [9]. (Strangely, there is no men­tion of this part of our work in Chapter IX in the book [e7].)

In 1996, we began work­ing on a dif­fer­ent circle of prob­lems mo­tiv­ated by phys­ics, jointly with Wit­ten and partly with Borel. John spent 1996–1997 at the In­sti­tute for Ad­vanced Study as part of a spe­cial year on quantum field the­ory. Wit­ten, who was mo­tiv­ated by \( F \)-the­ory, knew of our work on com­put­ing Don­ald­son in­vari­ants of el­lipt­ic sur­faces, which in­volved study­ing holo­morph­ic rank two vec­tor bundles or equi­val­ently holo­morph­ic prin­cip­al \( GL(2, \mathbb{C}) \)-bundles over sur­faces fibered over a curve with gen­er­al fiber an el­lipt­ic curve. He asked about the mod­uli spaces for more com­plic­ated struc­ture groups over an el­lipt­ic curve or more gen­er­ally over an el­lipt­ic­ally fibered al­geb­ra­ic vari­ety. The main case of in­terest for \( F \)-the­ory in­volves the struc­ture groups \( E_8 \) and \( E_8\times E_8 \). Earli­er work of Loo­ijenga [e1] had showed that there was a rich geo­met­ric pic­ture be­hind this ques­tion, for the groups \( E_6 \), \( E_7 \), and \( E_8 \), con­nec­ted with the de­form­a­tion the­ory of simple el­lipt­ic sin­gu­lar­it­ies and del Pezzo sur­faces, very spe­cial ex­amples of ra­tion­al sur­faces which in­clude the cu­bic sur­face in \( \mathbb{P}^3 \) (cor­res­pond­ing to the group \( E_6 \)). We stud­ied this prob­lem in [8], [10], [11], and [12]. A brief de­scrip­tion of the res­ults is as fol­lows.

Let \( G \) be an al­most simple com­plex al­geb­ra­ic group (i.e., its Lie al­gebra is a simple Lie al­gebra) and let \( K \) be the com­pact form of \( G \). Let \( \widetilde{G} \) be the uni­ver­sal cov­er of \( G \) and sim­il­arly for \( \widetilde{K} \). Thus \( \widetilde{G} \) has a fi­nite cen­ter \( Z \) which is iden­ti­fied with the cen­ter of \( \widetilde{K} \), and \( G \) and \( K \) are iso­morph­ic to quo­tients of \( \widetilde{G} \) and \( \widetilde{K} \), re­spect­ively, by the same sub­group of \( Z \). For an al­geb­ra­ic curve \( C \), one can con­struct the mod­uli space \( \mathfrak{M}_G(C) \) of holo­morph­ic semistable \( G \)-bundles over \( C \). Strictly speak­ing, \( \mathfrak{M}_G(C) \) para­met­rizes \( G \)-bundles up to a coars­er equi­val­ence re­la­tion than iso­morph­ism, namely S-equi­val­ence. However, for a gen­er­al ele­ment of \( \mathfrak{M}_G(C) \), S-equi­val­ence is the same as iso­morph­ism. There are three gen­er­al meth­ods for con­struct­ing semistable holo­morph­ic \( G \)-bundles over an el­lipt­ic curve \( E \):

1. By a the­or­em of Narasim­han–Se­shadri–Ramanath­an, S-equi­val­ence classes of semistable \( G \)-bundles on \( E \) are giv­en by ho­mo­morph­isms \( \rho\colon \pi_1(E) \to K \). Choos­ing a basis of \( \pi_1(E) \cong \mathbb{Z}\oplus \mathbb{Z} \) and lifts of the im­ages of the basis ele­ments to \( \widetilde{K} \), we ob­tain two ele­ments \( A, B \in \widetilde{K} \) such that \( ABA^{-1}B^{-1} = c \), where \( c \) is some ele­ment of \( Z \). We call the pair \( (A,B) \) an al­most com­mut­ing pair in \( \widetilde{K} \), and the de­scrip­tion of \( \mathfrak{M}_G(E) \) boils down to the de­scrip­tion of such pairs mod­ulo con­jug­a­tion, the prob­lem ori­gin­ally stud­ied by Loo­ijenga. The \( G \)-bundles \( \xi \) con­struc­ted by this meth­od sat­is­fy: \( \operatorname{Aut}_G(\xi) \) has max­im­al di­men­sion for \( G \)-bundles S-equi­val­ent to \( \xi \). Fi­nally, in case \( G \) and \( K \) are simply con­nec­ted, this con­struc­tion iden­ti­fies \( \mathfrak{M}_G(E) \) with the or­bi­fold \( (E\otimes_\mathbb{Z} \Lambda)/W \), where \( \Lambda \) is the co­root lat­tice of the group \( G \) and \( W \) is the Weyl group. Ac­cord­ing to Loo­ijenga’s the­or­em, \( (E\otimes_\mathbb{Z} \Lambda)/W \) is a weighted pro­ject­ive space with weights giv­en by the dual Coxeter num­bers of the group \( G \). (There is a some­what more com­plic­ated pic­ture if \( G \) is not simply con­nec­ted.)
2. We can try to re­duce the struc­ture group to a prop­er sub­group of \( G \). By a gen­er­al prin­ciple, an un­stable \( G \)-bundle over a curve \( C \) has a ca­non­ic­al re­duc­tion of struc­ture group to a para­bol­ic sub­group of \( G \). A key point is that, for an el­lipt­ic curve \( E \), if \( G \neq SL(n, \mathbb{C}) \), then there is an es­sen­tially unique “min­im­ally un­stable” \( G \)-bundle \( \xi_0 \) over \( E \), and its struc­ture group re­duces to a uniquely defined max­im­al para­bol­ic sub­group \( P \) of \( G \) (cor­res­pond­ing in the simply con­nec­ted case to a dis­tin­guished ver­tex in the Dynkin dia­gram for the root sys­tem of \( G \)). There is a dis­tin­guished \( \mathbb{C}^* \subseteq \operatorname{Aut}_G(\xi_0) \). The set of all “neg­at­ive weight” de­form­a­tions of \( \xi_0 \) can be glob­al­ized to an af­fine space \( \mathbb{A} \), with the ori­gin cor­res­pond­ing to \( \xi_0 \), and \( \mathbb{C}^* \) acts on \( \mathbb{A} \) with all weights strictly neg­at­ive. Thus the quo­tient \( (\mathbb{A}-\{0\})/\mathbb{C}^* \) is a weighted pro­ject­ive space para­met­riz­ing semistable \( G \)-bundles. The \( G \)-bundles \( \xi \) con­struc­ted by this meth­od sat­is­fy: \( \operatorname{Aut}_G(\xi) \) has min­im­al di­men­sion for \( G \)-bundles S-equi­val­ent to \( \xi \), mak­ing this con­struc­tion more suit­able for con­struct­ing uni­ver­sal bundles and in fam­il­ies. The main the­or­em of [12] is that the cor­res­pond­ing map
   \[ (\mathbb{A}-\{0\})/\mathbb{C}^* \to \mathfrak{M}_G(E) \cong (E\otimes_\mathbb{Z} \Lambda)/W \]

   is an iso­morph­ism. This res­ult gives a geo­met­ric in­ter­pret­a­tion to Loo­ijenga’s the­or­em.

3. To tie these meth­ods in with the work of Loo­ijenga in case \( G \) is simply con­nec­ted and of type \( E_6 \), \( E_7 \), \( E_8 \) (and also \( D_5 \), \( A_4 \), \( A_1\times A_2 \)), there is a third meth­od for con­struct­ing \( G \)-bundles via re­stric­tion. El­lipt­ic curves \( E \) arise nat­ur­ally as curves on del Pezzo sur­faces \( S \), smooth al­geb­ra­ic sur­faces \( S \) such that \( K_S^{-1} \) is ample, where \( K_S \) is the ca­non­ic­al line bundle on \( S \). We define the de­gree \( d \) of a del Pezzo sur­face \( S \) to be \( (c_1(K_S))^2 \). It is also nat­ur­al to con­sider gen­er­al­ized del Pezzo sur­faces \( S \), where \( S \) is al­lowed to have ra­tion­al double points and the ca­non­ic­al bundle \( K_S \) is re­placed by the du­al­iz­ing sheaf \( \omega_S \). It is well-known that, for del Pezzo sur­faces of de­gree \( d \le 3 \), there is a deep con­nec­tion between del Pezzo sur­faces \( S \) and groups of type \( E_r \), where \( r = 9- d \). One way to ex­plain this con­nec­tion is that the or­tho­gon­al com­ple­ment \( (c_1(K_S))^\perp \) in \( H^2(S;\mathbb{Z}) \) is a root lat­tice of type \( E_r \). Us­ing this fact, there is a “tau­to­lo­gic­al” \( \widehat{G} \)-bundle \( \Xi \) over \( S \), where
   \[ \widehat{G} = G\times_{\mathbb{Z}/d\mathbb{Z}}\mathbb{C}^* \]

   is an ap­pro­pri­ate con­form­al form of \( G \) and \( \mathbb{Z}/d\mathbb{Z} \) is iden­ti­fied with the cen­ter of \( G \). (Here, some care must be taken in the defin­i­tion of \( \Xi \) if \( S \) has ra­tion­al double points.) The con­struc­tion of semistable bundles over el­lipt­ic curves \( E \) in [14] is then roughly giv­en as fol­lows: Giv­en a triple \( (S,D, \varphi) \), where \( S \) is a gen­er­al­ized del Pezzo sur­face, \( D \) is a smooth curve on \( S \) defined by the van­ish­ing of a sec­tion of \( \omega_S^{-1} \), and \( \varphi\colon E \to D \) is an iso­morph­ism from the fixed el­lipt­ic curve \( E \) to \( D \), there is a ca­non­ic­al semistable \( G \)-bundle as­so­ci­ated to the \( \widehat{G} \)-bundle \( \varphi^*(\Xi|D) \). This gives an iden­ti­fic­a­tion of \( \mathfrak{M}_G(E) \) with the space of such triples \( (S,D, \varphi) \). By a the­or­em of Torelli type for the mixed Hodge struc­ture on \( S-D \), the space of triples \( (S,D, \varphi) \) is also iden­ti­fied with \( (E\otimes_\mathbb{Z} \Lambda)/W \), com­pat­ibly with the iso­morph­ism

   \[ \mathfrak{M}_G(E) \cong (E\otimes_\mathbb{Z} \Lambda)/W \]

   in (1) and (2).

Fi­nally, in a joint pa­per with Borel [13], we used these and oth­er ideas to clas­si­fy al­most com­mut­ing pairs and triples mod­ulo con­jug­a­tion in com­pact Lie groups and to veri­fy a con­jec­ture of Wit­ten on Chern–Si­mons in­vari­ants of flat \( K \)-bundles over a (real) 3-tor­us: these sat­is­fy a nu­mer­o­lo­gic­al prop­erty con­jec­tured by Wit­ten, for which he coined the term “clock­wise sym­metry”. Ac­cord­ing to John, Wit­ten and Borel had strik­ingly op­posed views on how best to ap­proach the sub­ject of com­pact Lie groups. Wit­ten pro­fessed not to know many gen­er­al res­ults about Lie groups and root sys­tems, but had an in­tim­ate know­ledge of every root sys­tem down to the minutest de­tail. Borel, on the oth­er hand, felt that no proof about root sys­tems was truly sat­is­fact­ory un­less it could be made clas­si­fic­a­tion in­de­pend­ent.

Our joint work was some­times seem­ingly ef­fort­less, some­times a long and hard struggle, but al­ways fun. John was fear­less in his will­ing­ness to tackle hard prob­lems and to ab­sorb whatever tech­niques were ne­ces­sary to solve them, and he con­tinu­ally pushed us to fol­low whatever paths we were on to their lo­gic­al con­clu­sions. It was my great priv­ilege to be on those paths with him for so many years.