Surveys in Differential Geometry: Proceedings of the Conference on Geometry and Topology Held at Harvard University, April 27-29, 1990 (Supplement to the Journal of Differential Geometry, No. 1)

311 Pages·1991·18.64 MB·English

by  Shing-Tung Yau

s
VEYSIN
DIFFERENTIAL
GEOMETRY
R. Bott, Stable bundles revisited
G. D'Ambra & M. Gromov, Lectures on transformation
groups: geometry and dynamics 19
J. Kollar, Flips, flops, minimal models, etc. 113
R. M. Schoen, A report on some recent progress on non-
linear problems in geometry 201
E. Witten, Two-dimensional gravity and intersection theory
on moduli space 243
NUMBER 1 '1991
I • •
SUPPLEMENT TO Tt-:JE JOURNA( . ,
OF DIFFERENTIAL GEOMETRY
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EDITORIABLO ARD
JOURNALO F DIFFERENTIGAELO METRY
Editors-in-Chief
c.C .H SIUNG S.T .Y AU
LehigUhn iversity HarvarUdn iversity
BethlehePm ,A1 8015 CambridgMeA, 02138
Editors
ROBERT L.B RYANT H. BLAINEL AWSON,J R.
DukeU niversity StatUen iversoiftN ye w York
DurhamN,C 27706 StonByr ookN,Y 11794
SIMON K.D ONALDSON RICHARD M. SCHOEN
UniversoiftO yx ford StanfoUrndi versity
OxfordE,n glanOdX I3 LB StanforCdA, 9 4305
AssociEadtiet ors
ENRICAOR BARELLO MICHAEHL. F REEDMAN
UniversdietgalS it uddii R oma UniversoiftC ya lifornia
UniversiDtIap rA.:. oMro 2 LaJ ollCaA, 9 2093
00185R OI C;:l-HGEFUMMIO RI
JEFFC HEEG Iltoyf S ciences
StatUen i ,oyUan iversity
StonByr e !,o4y6a4 J,A PAN
Libraryo fC ongreCsast aloging-in-DPautbal ication
ConferenocneG eometrayn dT opolog(ysI t 1:9 90:L ehigUhn iversity)
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SURVEYSI N
DIFFERENTIAL
GEOMETRY
Proceedionfgt sh eC onference
onG eometrayn dT opology
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SponsorbeydL ehigUhn iversity
NUMBER 1 1991
SUPPLEMENTT O THE JOURNAL
OF DIFFERENTIAGLE OMETRY
PREFACE
As a service to the mathematics community, the editors of the Jour
nal of Differential Geometry decided to organize a conference, to be held
every three years, at which invited speakers would survey the field of dif
ferential geometry and related subjects. In choosing the speakers, we look
at geometry in a very broad way and hope geometers will go beyond the
classical perception of other fields. (Among the speakers at our first con
ference were people working in algebraic geometry, mathematical physics
and other fields.) We decided that the speakers should be prominent spe
cialists in their respective areas who are able to give a broad overview of
recent trends and make predictions and suggestions for future research.
We believe that these conferences will be useful gatherings for experts and
nonexperts alike, and in particular for those who are isolated from the
mainstream activity in geometry.
Our first conference, held at Harvard University from April 27-29,
1990, was made possible by the generous funding of the Journal of Dif
ferential Geometry. We are also grateful to Harvard University and the
faculty of the Mathematics Department who were strongly supportive of
this endeavor. Dean Michael Spence delivered the opening remarks com
mending our efforts to bring together geometry and other fields.
Central to the success of this conference were the speakers who prepared
not only two hours' worth of lectures, but also the manuscripts published
here. Their lectures were enthusiastically received by several hundred par
ticipants. We would like to express our gratitude to them for their efforts.
It is our hope that these conferences will become a tradition. We look
forward to the next conference, which will take place at Harvard University
in 1993.
S. T. Yau
H. Blaine Lawson, Jr.
SURVEYS IN DIFFERENTIAL GEOMETRY
1 (1991) 1-18
STABLE BUNDLES REVISITED
RAOUL BOTT
The topological classification of complex vector bundles over a Riemann
surface is of course very simple: they are classified by one integer c, (E) ,
corresponding to the first Chern class of E. On the other hand, a classi
fication in the complex-analytic-or algebro-geometric-category leads to
"continuous moduli" and subtle phenomena which have links with num
ber theory, gauge theory, and conformal field theory. I will try to report
briefly on some of these developments here.
The simplest instance of our problem occurs when dim E = 1, that
is, when we are dealing with complex analytic line bundles over M-a
compact Riemann surface of the genus g. Because line bundles form a
group under the tensor product L, L' ..,... L 0 L' , the set of isomorphism
classes of line bundles J(X) will in this case inherit an abelian group
structure, and the first fundamental theorem of the subject, going back to
Riemann, Abel, and Jacoby, asserts that J(M) can be given the structure
of a complex analytic abelian group. More precisely the first Chern class
gives rise to a homomorphism J(M) -+ Z -+ 0 whose kernel Jo(M) is a
complex analytic, and indeed algebraic, torus of dime = g :
0-+ Jo(M) -+ J(M) -+ Z -+ O.
A proof of this exact sequence on the complex analytic level is actually
quite easy, granted the basics of sheaf theory and Hodge theory. Indeed,
the exact sequence
O-+Ze-27+r1 C-+C * -+0
induces an analogous sequence on the corresponding sheafs of germs of
functions, and hence leads to an exact sequence in cohomology:
HO(M; ~.) -+ HI(M;~) -+ HI(M;~) -+ HI(M; ~.) ~ H2(M; ~).
But HI(X; ~*) is seen to be precisely J(M), by passing from a line bun
dle to its transition functions, while by the Dolbaux resolution, HI (M; ~)
::: HO, I (M) , that is, the vector space of type (0, 1) in HI (M ; JR) .
Received May 29, 1990.
RAOVLBOTT
2
Fm. aI I y, t h e us ·m th's sequence corIresponds. to c1 ' so that the ex. actness
o f t h e coh omoI o gY sequence together with the ObV. IO.U S computatIOn that
IfJ(M;~) = HO(M; C*) = C, leads to the descnptIOn of Jo(M) as the
quotient 1
Jo(M) = HO,I(M)/H (M; Z)
of a g-dimensional complex vector space by a lattice, i.e., a torus. The
algebraic structure on Jo(M) is much less trivial.
In any case, these classical results provide us with as beautiful an answer
as we could hope for: this Jacobian J(M) varies functorially with M, and
its own function theory is intimately related to that of M. In particular,
there is a natural map
M ~ J (M)
1
which maps a point P E M to the isomorphism class of the line bundle
Lp on M determined by p as a divisor on M. (In terms of a local
coordinate zp centered at p in M, Lp can also be described by the
data: let V = {Vo' VI} be the cover of M consisting of a small disc, Vo,
about p, and let VI = M - p. Let gu u = -L, so that the assignment
0' [ Zp
10 = zp' 11 == I defines a holomorphic section of Lp with precisely one
zero at p. Thus c1 (Lp) = 1 , and so p ---> Lp takes values in the component
J (M) of bundles with c = 1.) Iterating this map (g - I) times leads to
1 1
the diagram:
Mx .. ·xM ---> J x··· x J !!!. J (M)
~ ~ g-1
(g-l) (g-l)
1
M(g-l)
and so to a canonical map of the M(g-l) ,the (g - I)st symmetric power
of Minto Jg_ 1 (M) . The image of this arrow is now of codimension 1 in
Jg_1 (M) and so defines a divisor (the e-divisor), and hence a line bundle
Yo,
over Jg_1 (M).
In the terminology of our century the classical e-functions of Jacoby
of level k now appear as the space of holomorphic sections of the bundle
Yo
k over Jg_1 :
e-functions of level k 2'" HO(Jg_1 (M); .fek).
From the classical formulas one can then also compute the number of
e
such functions to be kg:
dim HO(Jg_1( M); .fek) = kg.
STABLE BUNDLES REVISITED 3
To summarize, for n = 1 , the classical theory teaches us that the moduli
space of line bundles is
A) a complex variety each of whose components is a torus of dim 2g
over R so that its Poincare series is given by
Pr{Jk(M)} = (1 + t)2g ,
and
B) the component Jg_I (M) carries a natural line bundle ~, with
e.
dimHo{Jg_I(M); ~k} =
The primary difficulty of extending these results to higher-dimensional
vector bundles is that the problem immediately becomes infinite dimen
sional. This comes about because bundles E of dim> 1 will in general
have many nontrivial automorphisms, and to properly deal with this situ
ation the Mumford notion of stability is indispensable. The condition is
as follows:
E is stable if and only if for any subbundle FeE,
(1) cl(F) cl(E)
dimF < dimE'
Let me illustrate the power of this condition, say for a bundle E of
dim 2. First, remark that E always admits a line subbundle LeE and
so fits into an exact sequence
O---'>L---'>E---'>Q---'>O
with Land Q line bundles.
Indeed, any rational section of E will determine such an L, and of
course the Chern classes of the three constituents are related by
CI (E) = cI (L) + cI (Q).
Now, if E is stable, we will have cl(L) < cl(E)/2 whence cl(Q) >
c (E)/2, and it follows that the sequence cannot split if E is stable!
I
Similarly a stable E can have no nontrivial automorphism. Indeed,
consider an automorphism
rp: E E.
---'>
The characteristic polynomial of rp is clearly constant because M is
compact. If the eigenvalues of rp differ, rp will split E into two line
bundles, which is ruled out by stability. Hence the eigenvalues are equal
to A, say, so that '" = rp - Al will have to be a nilpotent endomorphism:
",2 = O. Now if '" is not identically 0, and has constant rank, then we
4 RAOUL BOTT
have an exact sequence
o 'II
L EL 0
----t ----t ----t
where L = ker 1fI. But by stability c (L) < c (E)/2, which rules out this
I I
alternative. Hence IfI == A . 1. q.e.d.
The nonconstant rank case is eliminated similarly.
In any case, armed with this concept, the structure of vector bundles
now becomes tractable. First of all, Mumford shows that the isomorphism
classes of stable bundles do admit a natural structure as a smooth algebraic
variety In (M) , which varies functorially with M. The tangent space to
In(M) at E is given by
n I
TE(J (M) ~ H (M; EndE).
By Riemann-Roch and stability (dimHo = I!) we see that In(M)
has dimension n2 (g - 1) + 1. Just as for J = J I , these spaces fall into
components J;(M) , k = c (E) according to the topological type of E,
I
but these will not be isomorphic in general. In fact,
J;
is compact {:} nand k are relatively prime.
In short, for n > 1, the "true" analogues of the Jacobian J(M) are
the J,n(M) , with (n, r) = 1. When (n, r) i- 1, one usually compactifies
J,n(M) by semistable bundles (replace < by :S;) modulo a certain equiv
alence relation, to get a complete, but in general singular, variety, and in
the sequel J,n (M) will always denote this compactified version.
Actually this stability notion also helps to explain the structure of all
bundles over M. For instance, suppose E has dimension two but is not
semistable. Then one finds that E contains a unique line bundle LE C E
of maximal cI ' so that we can attach to E a unique exact sequence
o
LI E L2 O.
----t ----t ----t ----t
If we set l1i = cI (Li), then the bundles of this type are therefore natu
rally parametrized by the set
~: JJ1,(M) x JJ12(M) x HI(M; L; 0 LI), ul + u2 = cl(E); 1112: 112'
where the last factor measures the extension determined by E. But note
that this is no longer a true parametrization of isomorphism classes. For,
as is easily seen, two bundles whose extensions are multiples of each other,
1]1 = A1]2' Ai- 0 in HI (M; L; 0 LI) , will lead to isomorphic bundles.
Still this seems to be the best one can do while staying in a Hausdorff
framework, and it is therefore plausible to consider the disjoint union
.£,(2) = J}(M) Il~, 11 = (uI ' u2) with 11] +!1-2 = r, 1112: 112'
J1
STABLE BUNDLES REVISITED
as some sort of covering of the set of all complex 2-dimensional bundles
with c = rover M. In the algebraic category this data leads to what I
1
believe is called the "stack of bundles" over M.
In any case, this stratification plays an essential role in the approaches
that Harder [5], Harder & Narasimhan [6], and Atiyah & Bott [1] took
to the problem of computing H*(J,n). The Harder-Narasimhan method
is number-theoretical and has as its denouement the beautiful counting
formula which is a function field analogue (A. Weil, Tamagawa) of a cor
responding counting formula going back to Minkowski and C. L. Siegel in
the number field case:
'"' 1 1 (n1-l)(g-l)
L.IAutEI = q -1 q (M(2) ... ,"~(n).
E
Here on the left one is counting the isomorphism classes of vector bundles
(of fixed determinant!) over a curve of genus g, defined over a finite
field Fq , each vector bundle contributing the reciprocal of the number of
automorphisms it has. On the right the answer is given in terms of the
(-function of M:
n
2q(1 - wq-s)
r _ 1 I
'>M(s) - (1 _ q-S)( 1 _ q . q-S) ,
the wi(M) being intrinsically defined algebraic integers with IWil = q1/2 ,
and in terms of these {Wi} the number of rational points on the Jacobian
J(M) of M is given by:
2g
IJ(M)I = IT(1 - w).
1
Using these formulas it is an easy but beautiful computation to find
the contribution of each stratum in .#1,(2) to the left-hand side of the
Minkowski-Siegel-Weil formula. For ~(2) this then leads to the relation:
N
+ IJ 12 . ~ I _ I . 3q-3 2
(2) q_1 (_1)2 L. 2,+2-g - (q-l) q (M()'
q ,=0 q
and this relation in turn enabled Harder to compute IJ 121 by subtraction,
and hence, via the Weil conjectures, to obtain a hold on the Poincare poly
nomial of J2! Actually these computations were done before the solution
1
of the Weyl conjectures by Deligne and Grothendieck, so that at that time
they merely used this example to check these Weyl conjectures against the
computations Newstead [10] had made for p((JI2), using a topological
method which will be explained below.
6 RAOUL BOTT
Fi.r st, however, one partm. g comment on t he' " covenng " /rut(r 2) • It 'IS
unfortunately not true that the vector spaces HI (X; L; 0 Ll) in Cu are
constant in dimension: For large genus this vector space jumps as Ll and
L2 vary over J and J . A proper algebro-geometric model of the "set
U1 U2
of all bundles" should therefore, I think, be built out of the virtual bundles
with fiber Hl(X; L; 0 Ll) - HO(X; L; 0 Ll).
These bundles have a constant (even if negative!) dimension, and the
disjoint union
J?) Il Il
.Ji('2) V2Il· ..
= ~
now precisely corresponds to the left-hand side of (2), if we agree to count
the number of points of a virtual vector space, V, by qdim V
Indeed, with this understanding, the expression
cc
IJ21+ 2)~1
1
i=l
precisely goes over into the left-hand side of (2), of course multiplied by
IJI, because we have not fixed the determinant in the present discussion.
Now in topology it often happens that we are given a decomposition of
a space Y, by strata of the form:
Il Il ·Il
Y = Yo WI W2 .. Wk ••.
with the ~ vector bundles over Yi , so that Y is finally put together
by successively attaching the boundary sphere-bundle of ~ to what has
already been built. Indeed, any smooth function f on a manifold Y with
reasonable critical behavior induces such a decomposition. Here Yo is the
absolute minimum of f, and ~ is the "negative bundle" of f over the
critical set Yi of "index i". Thus these ~ are spanned by the direction
of steepest descent for f along Yi .
What is novel in the algebraic framework about our ~(2) is therefore
that the bundles V/l are virtual, and actually get more and more nega
tive as J1 -> ()(). In topology such a state of affairs is usually remedied
by "suspending" to convert the virtual bundles into honest ones. In the
present situation that would indicate that one has to suspend more and
more as the strata are added-in short, one seems to be building some
thing "down from ()()", a phenomenon often present in the heuristics of
modern physics.