My encounter with Seshadri and with the Narasimhan-Seshadri theorem
aa r X i v : . [ m a t h . HO ] F e b My encounter with Seshadriand with the Narasimhan-Seshadri theorem
Nitin NitsureC.S. Seshadri passed away in July 2020. The first part of this article contains somereminiscences from the early 1980s, when as a graduate student in the Tata Instituteof Fundamental Research (TIFR), Mumbai, I had the good fortune to learn fromhim. The second part is on the Narasimhan-Seshadri theorem and my encounterwith it.
I: Some recollections from the early 1980s.
I joined the School of Mathematics, TIFR, as a research scholar in August 1980.Seshadri taught us a five or six month long course titled ‘Geometry-Topology’. It istraditional in TIFR to have three courses in the first year, titled ‘Algebra’, ‘Analysis’and ‘Topology’. The title ‘Geometry’ was new, and so even before the course began,my seniors commented on this departure from the tradition.A few words about myself: till joining TIFR, my principal interest was theoreticalphysics, but I had taken an MSc degree in Mathematics to escape the physics labo-ratory and the hocus-pocus of the lectures. I had discovered that to learn MSc-levelmathematics, it is enough to sit at home and read by oneself the various excellenttextbooks such as Serge Lang’s ‘Algebra’ or Spanier’s ‘Algebraic Topology’, and asa result, I used spend more time in the physics department of the university, and goto the mathematics department mainly to write the examinations. When I joinedthe School of Mathematics of TIFR, my plan was to shift back to physics once I pickup enough advanced mathematics. I was keen to learn modern geometry, given itsimportance in physics, and so I was happy to find that there was going to be a courseon geometry. But I was not so happy with the idea of having to attend lectures –in the previous two years I had not followed any lecture courses, and had alwayspreferred reading books to attending lectures. I had not even heard Seshadri’s namebefore.Seshadri’s course was a revelation! He explained later (by which time he had leftTIFR) that the idea was to introduce the basic theory of manifolds, bundles, groupactions etc. from a Grothendieckian point of view, with emphasis on functorialproperties but without too much abstract machinery, and supplement it with somebasic algebraic topology. In his first lecture, he defined ‘separated topological spaces’as spaces in which any two distinct points have disjoint open neighbourhoods –which we of course recognized as Hausdorff spaces under a new name – and gavevarious exercises. The first one said that a topological space is separated if andonly if its diagonal ∆ X is closed in X × X . The second one said that if f, g : X → Y are continuous maps where Y is separated, then the subset defined by f ( x ) = g ( x ) is closed in X . Yet another one said that if some point in a topologicalgroup is a closed point then the group is separated, and so on. The whole timewas spent is giving about ten ‘Exercises’ followed by about ten more ‘ElementaryExercises’. This baffled us: the elementary exercises looked no easier than the1 itin Nitsure: A tribute to C S Seshadri. { x ∈ X | f ( x ) = g ( x ) } is simply the inverse image of the diagonal ∆ Y ⊂ Y × Y under ( f, g ) : X × X → Y × Y , and all the other exercises had similar easy arrow-theoretic solutions. There was no need to consider points and neighbourhoods, onceseparatedness was formulated in terms of the diagonal. I was hooked.The second lecture introduced proper maps and proper actions of topological groups.A map f : X → Y was defined to be proper if for all Z , the induced map f × id Z : X × Z → Y × Z is closed. A topological group action a : X × G → X was definedto be proper if the induced map ( p , a ) : X × G → X × X is proper. Once again,this was followed by a number of ‘Elementary Exercises’, such as if the action isproper then the quotient X/G is separated. I had not seen properness before, andthe definition of the properness of an action looked strange at first sight. But onceagain, solving the exercises revealed the meanings of these ideas, and reinforced thepower of arrow-theoretic arguments without recourse to points and neighbourhoods.Seshadri’s lectures were the high point of the week for me for the next few months.I said above that Seshadri’s course was a revelation. But so was Seshadri’s styleand persona. He was not exactly fluent as a speaker, and had to frequently consulthis notebook when he wrote on the board. This gave some much-needed extra timeto understand what he was trying to say, and sometimes (to my great delight) toanticipate what was coming. He was completely relaxed and clearly in love withthe material. The only times when he used to look flustered was when he arrivedto teach the class 30 minutes late instead of his usual 15 minutes late. After suchextra late arrivals, he would breathlessly say ‘oh my watch ...’, leaving the sentenceincomplete (therefore we could never quite figure out what was the matter with hiswatch). In fact, many mathematical sentences or even whole paragraphs were leftincomplete, with a few suggestive phrases, punctuated by a stream of ‘you know’s!A differential manifold was defined by drawing two disjoint circles, below them twointersecting circles and then by performing some kind of a dance with the chalk,showing by tapping on the board how a point in one of the neighbourhoods getsidentified with a point in the other neighbourhood. While performing this dance,he kept looking at us, saying ‘you know!’ and nothing more. (One of my batchmates was so overcome by this performance that he had to leave the room choking,doubled up with laughter. He ended up as an analytic number theorist – such isjustice.) Often, the rest of my day after Seshadri’s lecture would go into figuringout what he meant. Every lecture was a treat, in which fundamental and beautifulstuff was taken out of his magic notebook, and displayed to us. One day, he definedGrassmannians and the tautological vector bundles on them. The universal propertyof these bundles was thrown at us as a challenge, which I could prove only after ahard two or three days of struggle. I was anyway used to self-study: what I did notknow was what to study. So this course was just made for me! Later when timecame for teaching homology theory, Seshadri (who by then must have developedenough confidence in me) asked me to lecture in his place, while he sat in the backrow. That is when I discovered that though he was a woolly speaker, he was a sharplistener if and when he chose to pay attention. itin Nitsure: A tribute to C S Seshadri. itin Nitsure: A tribute to C S Seshadri.
II: The Narasimhan-Seshadri Theorem.
We know that Narasimhan and Seshadri were essentially self-taught as graduatestudents, who mastered huge portions of Seminaire Cartan and other latest suchmaterial as it arrived in TIFR in 1950s. They did this by running a seminar inwhich they lectured to each other. Occasional visits to TIFR by eminent mathe-maticians, and lecture courses by some of these visitors, gave them inputs about whatis important and what to study. Postdoctoral stints in Paris, mentored by Schwartz(Narasimhan) and Chevalley (Seshadri) had completed their transition from stu-dents to researchers. Even before they went to Paris they were aware (thanks toK.G. Ramanathan) of the paper by Weil [We] in which it is suggested that vectorbundles on compact Riemann surface X that are associated to unitary representa-tions of the fundamental group π ( X ) may have some unspecified important specialproperties. They began their investigation of such bundles after they came back toTIFR ( ∼ ρ : π ( X ) → U ( n ) of the fundamental groupof a compact Riemann surface X arose out of one domain of mathematics, andthe concept of a stable holomorphic vector bundle X arose from a very differentdomain. The theorem shows that not just the bundle E ρ on X associated to ρ is a stable holomorphic vector bundle, but every stable holomorphic vector bundleon X of rank n and degree 0 so arises up to isomorphism from exactly one suchirreducible unitary representation up to conjugacy of representations. There is asimilar statement (though a bit more complicated to state) for stable holomorphicvector bundles on X of a non-zero degree, which replaces representations of π ( X )by those representations of π ( X − x ) where x ∈ X which have a certain scalar localmonodromy around x .The concepts of stability of bundles and stability under group actions were definedby Mumford ( ∼ itin Nitsure: A tribute to C S Seshadri. itin Nitsure: A tribute to C S Seshadri. ∼ ∼ itin Nitsure: A tribute to C S Seshadri. ∼ ∼ itin Nitsure: A tribute to C S Seshadri. References [Ar] Artin, M. : Algebraization of formal moduli I. In
Global analysis, Papers inHonor of K. Kodaira , 21-71. Tokyo Univ press 1969.[A-B] Atiyah, M.F. and Bott, R. : Yang-Mills on Riemann Surfaces. Phil. Trans.R. Soc. Lond. 308 (1983) 523-615.[C] Corlette, K. : Flat G itin Nitsure: A tribute to C S Seshadri. ∼ ∼ ronnie/past-talks.html[Ni-9] Nitsure, N. : Curvature, torsion and the quadrilateral gaps. arXiv:1910.06615To appear in Proc. Indian Acad. Sci. Math. Sci. 131 (2021).[Se] Serre, J.-P. : Faiseaux Alg´ebriques Coh´erents. Annals of Math. 61 (1955)197-278.[S-1] Seshadri, C.S. : Vari´et´e de Picard d’une vari´et´e compl`ete. Ann. Mat. PuraAppl. (4) 57 (1962), 117-142.[S-2] Seshadri, C.S. : Space of unitary vector bundles on a compact Riemann surface.Ann. of Math. (2) 85 (1967) 303-336.[Sc] Schlessinger, M. : Functors of Artin rings. Trans. A.M.S. 130 (1968) 208-222.[Si-1] Simpson, C.T. : Harmonic bundles on noncompact curves. J. Amer. Math.Soc. 3 (1990), no. 3, 713-770.[Si-2] Simpson, C.T. : Higgs bundles and local systems. IH ´ES Sci. Publ. Math. 75(1992).[Si-3] Simpson, C.T. : Moduli of representations of the fundamental group of asmooth projective variety - I and II. IH ´ES Publ. Math. 79 (1994) 47-129 and 80(1994) 5-79.[U-Y-1] Uhlenbeck, K. and Yau, S.T. : On the existence of Hermitian-Yang-Millsconnections in stable vector bundles. Comm. Pure Appl. Math. 39 (1986), 257-293.[U-Y-2] Uhlenbeck, K. and Yau, S.T. : A note on our previous paper: ”On theexistence of Hermitian-Yang-Mills connections in stable vector bundles”. Comm.Pure Appl. Math. 42 (1989) 703-707.[We] Weil, A. : G´en´eralisation des fonctions ab´eliennes. J. Math. pures et appl. 17(1938) 47-87.[Wi] Witten, E : @witten271 reply to @raghumahajan and @scroll in, Twitter 2020.https://twitter.com/raghumahajan/status/1289453265868828673?lang=enronnie/past-talks.html[Ni-9] Nitsure, N. : Curvature, torsion and the quadrilateral gaps. arXiv:1910.06615To appear in Proc. Indian Acad. Sci. Math. Sci. 131 (2021).[Se] Serre, J.-P. : Faiseaux Alg´ebriques Coh´erents. Annals of Math. 61 (1955)197-278.[S-1] Seshadri, C.S. : Vari´et´e de Picard d’une vari´et´e compl`ete. Ann. Mat. PuraAppl. (4) 57 (1962), 117-142.[S-2] Seshadri, C.S. : Space of unitary vector bundles on a compact Riemann surface.Ann. of Math. (2) 85 (1967) 303-336.[Sc] Schlessinger, M. : Functors of Artin rings. Trans. A.M.S. 130 (1968) 208-222.[Si-1] Simpson, C.T. : Harmonic bundles on noncompact curves. J. Amer. Math.Soc. 3 (1990), no. 3, 713-770.[Si-2] Simpson, C.T. : Higgs bundles and local systems. IH ´ES Sci. Publ. Math. 75(1992).[Si-3] Simpson, C.T. : Moduli of representations of the fundamental group of asmooth projective variety - I and II. IH ´ES Publ. Math. 79 (1994) 47-129 and 80(1994) 5-79.[U-Y-1] Uhlenbeck, K. and Yau, S.T. : On the existence of Hermitian-Yang-Millsconnections in stable vector bundles. Comm. Pure Appl. Math. 39 (1986), 257-293.[U-Y-2] Uhlenbeck, K. and Yau, S.T. : A note on our previous paper: ”On theexistence of Hermitian-Yang-Mills connections in stable vector bundles”. Comm.Pure Appl. Math. 42 (1989) 703-707.[We] Weil, A. : G´en´eralisation des fonctions ab´eliennes. J. Math. pures et appl. 17(1938) 47-87.[Wi] Witten, E : @witten271 reply to @raghumahajan and @scroll in, Twitter 2020.https://twitter.com/raghumahajan/status/1289453265868828673?lang=en