Normal Two-Dimensional Singularities. (AM-71), Volume 71 9781400881741

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Annals of Mathematics Studies Number 71

NORMAL TWO-DIMENSIONAL SINGULARITIES

BY

HENRY B. LAUFER

PRINCETON UN IVERSITY PRESS AND U N IVERSITY OF TOKYO PRESS

PRINCETON, NEW JE R S E Y

1971

Copyright © 1971, by Princeton University Press A L L RIGH TS R ESER V ED

LC Card: 78-160261 ISBN: 0-691-08100-x AMS 1970: 32C40

Published in Japan exclusively by University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

To My Parents

PREFA CE This monograph is an outgrowth of a course given in 1969-70 at Princeton University. Its aim is to analytically describe and classify normal 2-dimensional singularities of complex spaces. By restricting considerations solely to dimension two, it is possible, in certain theorems, to get more detailed results than are known in the general case. The reader should have a good knowledge of several complex variables and some acquaintance with Riemann surfaces and covering spaces. I would like to thank the students who attended my course for their many helpful corrections and suggestions. I would also like to thank the secretaries at Fine Hall, especially Florence Armstrong and Elizabeth Epstein, for their aid in preparing the manuscript.

v ii

INTRODUCTION The detailed study of normal 2-dimensional singularities is much easier than the higher dimensional case primarily for two reasons. First, any normal 2-dimensional singularity p is isolated. Thus, in resolving p, §11, [H zl], we replace p by a compact analytic space A. Secondly, because p is 2-dimensional, A is 1-dimensional. The theory of compact Riemann surfaces gives a great deal of information about A and small neighborhoods of A. Since p is normal, p is determined by any small neighborhood U of A. It is U which we actually study. Let A = UA^ be the decomposition of A into irreducible components. Thus each A^ is a (possibly singular) Riemann surface. It is easy to reduce all considerations to the case where the A^ are non-singular, intersect transversely, and no three A^ meet at a point. Allowing regular points to also be “ resolved” , A comes from a resolution if and only if the intersection matrix of the A- in U is negative definite, §IV, [ M], [Gr2]. Moreover, it is easy to decide if two different A’s can possibly resolve the same singularity, §V, [Ho], [B], The next problem then is to get information about singularities with the same A and the same intersection matrix. This is done in §VI, [Gr2], [ H& R ] , where it becomes necessary to introduce infinitesmal neighbor­ hoods of A, or more precisely, analytic spaces with nilpotents having A as their underlying topological space. It is first shown that if A and A have suitable isomorphic infinitesmal neighborhoods, then A and A have formally equivalent neighborhoods. It is then shown that formal equiva­ lence implies actual equivalence. Using Riemann-Roch, it becomes possible to get an estimate, in terms of the genera of the A. and the ix

X

INTRODUCTION

intersection matrix, on which infinitesimal neighborhoods must be iso­ morphic in order for A and A to have isomorphic neighborhoods. Finally in §VII, [H & R ], we obtain a complete set of invariants for p, namely the C-algebra structure of 0 /m\ where m is the maximal ideal in 0 p and A is sufficiently large. Again, because of the dimension two condition, we can improve upon previously known results and get an estimate on A in terms of the genera of the A- and the intersection matrix. Thus the same degree of truncation suffices to determine all normal 2-dimensional singularities which have homeomorphic resolutions.

CONTENTS P R E FA C E ...........................................................................................

vii

IN TR O D U C T IO N ..............................................................................

ix

CHAPTER I RESOLUTION OF PLA N E CURVE SIN G U LARITIES. . . .

3

CHAPTER II RESOLUTION OF SINGULARITIES OF TWO DIMENSIONAI ANALYTIC S P A C E S ..................................................................

7

CHAPTER III NORMALIZATION OF TWO-DIMENSIONAL ANALYTIC S P A C E S ..................................................................

34

CHAPTER IV EXCEPTIONAL S E T S ..................................................................

47

CHAPTER V MINIMAL R E S O L U T IO N S ............................................................

72

CHAPTER VI EQUIVALENCE OF EM B ED D IN G S..........................................

93

§VIa. S p aces w ith N i l p o t e n t s ...................................................

95

§VIb. Cohomology in Sheaves of N on-A belian Groups . . .

106

§VIc. Form al Iso m o rp h ism s.........................................................

108

§VId. Form al E quivalence Im plies A ctual E quivalence . .

115

CHAPTER VII THE LOCAL RING S T R U C T U R E .............................................

134

B IB L IO G R A PH Y ..............................................................................

157

I N D E X .................................................................................................

159

xi

Normal Two-Dimensional Singularities

CHAPTER I RESOLUTION OF PLANE CURVE SINGULARITIES

Our primary tool for the study of singularities will be resolutions. Roughly speaking, in resolving singularities, we add more holomorphic functions and, if necessary, replace the singular points by larger sets in order to get a manifold. In this section we shall resolve the singularities of plane curves, i.e . hypersurfaces in 2-dimensional manifolds, via a canonical process. DE F I NI T I ON 1.1.

If V is an analytic space, a resolution of the singulari­

ties of V con sists of a manifold M and a proper analytic map that

77

77

: M -> V such

is biholomorphic on the inverse image of R, the regular points of

V, and such that

" ^(R) is dense in M.

77

DEFINITION 1.2. A quadratic transformation at a point p in a 2-dimensional manifold M consists of a new manifold 1VT and a map that

77

is biholomorphic on

and

77

such

is given near 77“^(p) as follows.

Let (x,y) be a coordinate system for a polydisc neighborhood A(0;r) = A of p, with p = (0,0). A' = 77'* (A) has two coordinate patches Ux = (u,v) and U2 = (u ',v ') with u '= i and v '= uv. U^ 0 U2 = lu 4= 0177(u,v) = (uv,v) and n (u ',v ') = ( v ',u 'v ') -

Thus A = i(x,y) ||x| < r j, |y| V is a resolution. Theorem 1.1 below says

that repeated quadratic transformations as above will always resolve the singularities of plane curves. T H E O R E M 1.1.

L e t V be a 1-dim en sion al su bvariety in a 2-dim ensional

m anifold M. T here e x is ts a m anifold M obtain ed from M by s u c c e s s iv e quadratic transform ations, n: M -»M, su ch that if R is the s e t of regular points on V, n : 77“ ^ (R) -> V is a resolu tion o f the sin gu larities o f V. L o c a lly , M is obtain ed from M by only a fin ite number of quadratic trans forma t ions. P ro o f: The singular points of V are isolated so the theorem is purely local in nature. We only have to resolve each singular point. F irst suppose that the origin 0 is a singularity of the irreducible subvariety V in C^.

By the local representation theorem (III. A. 10 of

G &R), we can choose coordinates near 0 so that p : (x,y)

x expresses

RESOLU TIO N O F P LA N E CU RVE SIN G U LA R ITIES

5

V — JOI (in some neighborhood of 0) as a connected (since V is irreducible) s-sheeted covering space of a punctured disc N — {01 in the x-plane. Since all s-sheeted coverings of N — {01 are analytically the same, p : V — {0} -> N — {0! is equivalent to p : U — {0} -> N — {0! via x = ts , where U is a disc in the t-plane. y is an analytic function of t on U — {01. Since the equivalence of p and p may be extended by mapping the origin to the origin, the Riemann removable singularity theorem insures that y is analytic in U. Hence V = { (x,y) |f(x,y) = 01 can be represented as the image of U under t -» (ts ,y (t)). y(t) has a power series expansion y(t) = amtm + am + 1

^

1

+ ••• > am 4 =

4

(ts »y( 0 ) is a one-to-one map.

No longer trying to preserve the covering map p, we now wish to find coordinates expressing V in a more useful manner as locally the image of a disc. Multiplying y by l/am and if necessary choosing a new parameter t and interchanging x and y, we may assume that V is locally given by the image of t -> (ts , t m 4- am+1 t m + 1 + . . . ) with m i s . s,m and those j such that aj =)= 0 are relatively prime, for if r =(= 1 divides s,m and all the j ’s, t -> (ts ,y (t)) cannot be one-to-one since it factors through t -> T = tr -» (T s /r,T m/r + . . . ) .

We may assume that m

is not a multiple of s, for if m = ks let ( x ', y ') = (x,y-x^). If s = 1, there is no singularity. What happens under a quadratic transformation? 9

V = i(x,y) I f(x,y) = a 10x + aQ1y + a20x + . . . = 0!. 1

2

2

(V) = i(u,v) I f *(u,v) = aj^uv + aQjv + a2oU v + . . . = 0!

U i( u ',v ') |a-^Qv' + 3 q ju 'v '+ a 2 Qu'2 + . . . = f * ( u ',v ') = 0 }. If k is the order of the zero of f(x,y) at (0,0), we may factor

out of

f*(u ,v ) and f * ( u ',v ') . Hence f*/v^ defines the subvariety V = 77_ 1 (V - {01). On M' — 7T~^ ({x = 0 )), u' = y/x and v '= x. Hence

(V — {0 }) is

given by t -> ( u ',v ') = (tm"s + am-fltm S+^ + ••• >*S) which extends to

6

NORMAL TWO-DIMENSIONAL SIN G U LA RITIES

0 -> (0,0). Thus u '= v '= 0 is the only possible singularity of V '. If m — s > s, perform another quadratic transformation at u' = v ' = 0. Eventually m — [is < s and we can choose new parameters for vQ*), t -> (ts , t m + . . . ) with s ' < s = m '. We can now perform more quadratic transformations. Eventually = 1 and we have a non-singular In the general case, the singularity of V at 0 will have a finite number of irreducible components. Perform quadratic transformations until these components are all desingularized. We now have to separate the components. As observed before, if two components intersect transversely (i.e. have distinct tangent planes), a single quadratic transformation at their point of intersection separates them. Suppose and V2 are non-singular and tangent at 0 = after an appropriate choice of coordinates, that (t,tm +

H V 2 . We may suppose, = (t,0) and V 2 =

+•••)• The proof is by induction on m. If m = 1, V-^

and V2 meet transversely. If m > 1, after a quadratic transformation, V ' = (t,0) and V ' = (t^ 01” 1 + X

+ . . . ) , which proves the induction

^

step .I We observe for use later the following. P R O P O S IT IO N 1.2.

If V is a non-singular 1-dim ensional su bvariety

near the origin of

and tt\ M'

is a quadratic transformation at

the origin, then n"*(V - { 0 )) and n"* ( 0 ) in tersect transversely.

CHAPTER II RESOLUTION OF SINGULARITIES OF TWO-DIMENSIONAL ANALYTIC SPACES

T H E O R E M 2 . 1 . Any 2 -d im en sion al an aly tic s p a c e has a resolu tion .

Theorem 2.1 is really a local theorem, but we do not know that yet. Anyway, we begin by proving a local version. P R O P O S IT IO N 2.2.

L e t p e s, a 2-dim en sion al an aly tic s p a c e .

Then

there is a reso lu tio n n : M -> V of som e n eighborhood V of p. P roof: We may assume that p is a singular point. S has a finite number V^, . . . ,Vg of irreducible components near p. It will suffice to resolve each

separately. We may assume that V = Vj is locally

given as an admissible presentation as in III. A. 10 of G & R , i.e. p : V -> { z p z 2 1 represents V as an analytic cover. Let B = loc D denote p(V(D) ), i.e . the image of the locus of the discriminant, which contains all points above which p may fail to be a covering map. B is the branch locus if p is thought of as a branched covering map. B is a plane curve. Choose a small enough neighborhood N of the origin so that 0 = (0,0) is the only possible singularity of B. By Theorem 1.1, with a finite number of quadratic transformations in the ( z j^ ^ p l a n e , we can resolve the singularity of B. Quadratic transformations do not depend on the coordi­ nate system, so we may assume that they are given by mappings of the form n : N'-> N, tt : (u,v)

(uv,v) = (z^ z^ ) and n : (u ',v ') -> (v',u v ') =

7

8

NORMAL TWO-DIMENSIONAL SIN G U LA R ITIES

(z^ z^ . If V is locally embedded in Cr , n may be extended to 7T: (u,v,z3 , . . . , zr) ^ (uv,v,z3 , . . . , zr) 77: ( u > ', z 3 , . . . , z f)-* (v ',u 'v ',z 3 , . . . ,z r) Let V ' =

77

"^ (V). V ' is a subvariety of 77”^ (Cr) and we have the follow­

ing commutative diagram:

( 2 . 1)

77

is biholomorphic except for

77

’ * ( p ’ * ( 0 ) ) = fr ‘ ^(0), a projective line.

Thus p'\ V '-* N 'represents V 'a s an analytic cover with 77“^ (B )= B 'a s branch locus. Thus when we resolve the singularities of B, we get su ccessive diagrams of the form of (2.1). We induce a change in V and get a new analytic cover with B ' =

77

“^“(B) =

77

"^ (0) U n ~^ (B- {0|) ( 7 7 ’ ^ (B -{0 | ) is

now a submanifold) as branch locus. We want to be able to choose local coordinates

such that the singularities of

77

"^ (B) are only of the

form \C\C2 = 0} • Hence more quadratic transformations are required. F irst we want the irreducible components (Bj i of B 't o intersect transversely. Since all of the B-' are non-singular, by Proposition 1.2, if j — j we perform a quadratic transformation co at a point q e B-' , a>“ (B-' - q) 1

J

J

(also denoted by B p and co~ (q) meet transversely. If B j and B £ , j 4= k, do not meet transversely at a point q, apply Theorem 1.1 to resolve the singularity at q. B j and B£ will then not meet at all. We now have all the irreducible components of B meeting transversely. It may still happen, however, that three or more B^ meet a single point q. A quadratic transformation at q separates the B^, which will now meet 77

”^(q) transversely at distinct points.

R ESO LU TIO N O F TWO-DIMENSIONAL SIN G U LA R ITIES

9

So far, by quadratic transformations we have brought V 't o the point where it is represented as an analytic cover in such a manner that locally the branch locus is either a submanifold or can be given by \C\C2 = 0! in some polydisc A. We shall resolve the latter case first. (A-{£^C 2 = 0 1 ) is a finite sheeted covering space of A —{£ 1^2 = 01 and each connected component has an irreducible subvariety as its closure (III. C. 20 of G & R ). These irreducible components are also represented by p ' as analytic covers with \C\C2 ~ 0! as branch locus. We shall resolve these irreducible singularities using the manifolds M(kp . . . , kg) below. Given k j , . . . , kg with the k^ integers such that kj > 2, M = M(k^,. . ., kg) will be covered by s+ 1 coordinate patches,

= ( u ^ ,v ® ) = C^,

0 < i < s , joined as follows. 1

u0 n u i = M 0 !

u

U1 D U 2 = i v '+ 0 ! x *

v "= i, v

u2 n u 3 = iu " 4 o i

u 2n

n

u 2n = l =

u

ki v =u v u " = v '\ '

=

*u (2 n > +

° !

u (2 n + 1 ) =

«2 „ +1 n « 2 „ +2 - lv(2" +1> + 01 v O - 2 ) .

v (2 n + 1 ) =

. ( ^ 2 ) . (v2n+l ) )k2w 2u(2w l )

A = {v = v '= 0 } U \u' = u " = 01 U { v " = v ' " = 0 ! analytic subset of M.

(U q fl U p

(u (2 n ) )k 2 n + l v (2 n )

U...

is a compact

— A = {uv 4 0} undergoes an automorphism

at each change of coordinates. Thus M = {uv 4 01 U A U { v ^ = 0[, if s is even and M = {uv 401 U A U {u = 01 U { u ^ - 01, if s is odd. Let us suppose that s is odd. If s is even, v ^ must be replaced by u ^ in the following arguments.

10

NORMAL TWO-DIMENSIONAL SIN G U LA R ITIES

v and v( s ) are in fact analytic functions on all of M. To see this, we must see that they are analytic in each coordinate patch. v(s) = (u( s - 1 ) ) ks v^s _ 1 ) = (u(s - 2 ))ks(v(s - 2 ))k s - l k s ' 1 = (u( s ~ 3 ) ) ks - 2 (ks - l ks ' 1)' ks ^v ( s - 3 ) ^ ks - l ks '^

_ (u(s ~4))ks —2^ks —l ks ' ^ ‘ ks (v(s—3)^ks —3^ks —2^ks —l ks “

—

(ks_iks- «

We must check that the exponents are all positive. In each coordinate patch, alternately dividing the first exponent by the second exponent and then dividing the second by the first, we get su ccessively, ks- ks - i - rU - 7• ksS —Z - 2 - r1 J s s —1

"S — — 7• ks - 3O - « k0 ^s~2

- ±1 k, -1 " k„

■

i.e . continued fractions, since k- > 2. An easy induction proof shows that these ratios are all greater than 1. Hence v ^ , and similarly v, is analytic on M. Moreover, v ^ = uav^ with a > b, a and b relatively prime and

If s is even,

= uav^ with, as before, a > b, a and b relatively

prime and a = U

b

1

_i_

k2 - k3

....

RESO LU TIO N O F TWO-DIMENSIONAL SIN G U L A R ITIES

11

If o and co are positive integers, (2 . 2 )

p : {uv ± Oj - (va , (uavb)w) = ( ^ ,C 2)

is a covering map onto ^ 1^2^ ^ analytic cover of C

maY be extended to make M an

with

as branch locus. This will be

exactly what we need in order to resolve the singularity currently under consideration. Thus, returning to the main line of our argument, we have a part W of V ' represented by p ' as an irreducible analytic cover in a polydisc neighborhood A of the origin with C\C2 ~ ^ as the branch locus. We first cla ssify p ' on (p ' T ^ i^ i^ 4 ^), where p ' is a covering map. The injection of A — \C\C2 4

r\

into C — \C\C2 =

*s a homotopy equiv­

alence. Thus analytic covering spaces of A — {^1^2 ~ naturally to analytic covering spaces of C

extend

— \C\C2 ~ ^ an(* ^ suffices

to classify the latter and then restrict back to A — \C\C2 =

r\

• Covering

spaces correspond to subgroups of the fundamental group. C — \C\C2 ~ may be continuously deformed onto |£]J = |^l = r\

771 ~ nl

a torus. Hence

“ ^1^2 = 0! ) = Z © Z. W is a finite sheeted covering space

and thus corresponds to a subgroup G C

77^ of finite index. Thus

G ^ Z e Z is determined by its two generators in n^ . Thus p ' is analytically equivalent to a covering p ' of the form ( 2 .3 )

with p '(£ ,rj) = ySy +

= 0 | -» C 2 — { ^ 2

p'-. C 2 - ^

0

= °!

with a , ft , y , 8 integers such that a 8 —

.

r\ Automorphisms of the covering space C — { £ 7 7 = 0} will not affect the validity of the cover and will enable us to put p ' into the form of ( 2 . 2 ). ( £ , 7 7 ) -» ( £ ' , 1 7 ') = ( £ , ^

77

) and (£ , 77) -> (£ "

v and p integers represent p ' as p '(€ ',r j')= and p '

" ) = ( £ 77 ^ , 7 7 ) with ~ p a f £ 'Y-q

" ,r j" ) = (£ " a “ ^ £ 7 7 " ^ , £

have the automorphisms ( £ , 77) h> ( 1

77

y)

respectively. We also ) and ( £ ,

77

77

) -> ( 7 7 , £ ) at our disposal.

12

NORMAL TWO-DIMENSIONAL SIN G U LA R ITIES

Using the Euclidean algorithm on a and /3 , as induced by appropriate automorphisms, we may put (2.3) into the form (omitting primes) p ': where a is the greatest common divisor of \a\ and |/3 |, and r > A > 0. If co is the greatest common divisor of r and A, we have p ': with a and b relatively prime and b > a > 0. Any ^ , a =(=0, can be written (uniquely) as a finite continued fraction of the form

_ _1_ k with k- > 2. Now restrict p ' to p'~^(A —

~

s ^

^orm

(2.2). Thus, after the appropriate restriction of p, there is a biholomorphic map n making the following diagram commutative, a = 0 corresponds to M (0) = C2 . M(bx , . . . , bg) D {uv 4= 0) J2* W - p y' 1 i^1C2 = 0 ! C C r

A

= \C\C2= 01

In local coordinates, n is locally bounded and hence by the Riemann removable singularity theorem extends to an analytic map n : M -> W giving the following commutative diagram: M --------- ---- ► W

\/

RESO LU TIO N OF TWO-DIMENSIONAL SIN G U LA R ITIES

p is a proper map since J|v| < 1 (

13

{|v^s ^| < 1 } is compact in M. Thus 77

is a proper map. Thus 77 is a resolution of W except possibly that n is not biholomorphic at manifold points of W which lie over the branch locus. Let us first determine the nature of 77 near the pre-image of a point q e W such that p '(q ) = ( q ^ ^ )

^1 4

^ anc* ^2 =

P

con sists of a distinct points, q ',q " , . . . , cfa\ on the v-axis in the Uq coordinate patch, p represents a neighborhood as an analytic cover with ^

® as branch locus. Also, p restricted to

is one-to-one on the branch locus. Since p ~ \ £ 1^2 = 0 ),

77

of each of the q ^

77

is biholomorphic off

(U N ^ ) w ill have exactly a connected components off

(p ')“ \ C 1C2 = ^)- Hence W has exactly a irreducible components above Some of these w ill meet at q. If W' is an irreducible component at q, there w ill be one N / which corresponds to W'. 77: N'-> W' is biholo­ morphic off the branch locus. Since p is one-to-one above the branch locus, 77 is also one-to-one above the branch locus. Thus 77 is a homeomorphism. Summarizing, the map 77 is obtained by breaking W up into irreducible components and then resolving each component by a homeomorphism. If q is a manifold point, there is only one irreducible component at q and 77“ ^ is a homeomorphism which is holomorphic except possibly above the branch locus. But then the Riemann removable singularity theorem shows that 77“ ^ is holomorphic. We must show that our locally defined 77^

->

agree for inter­

secting W- on V '. But according to the previous paragraph, whenever the branch locus of p ' : V / -> N ' is a submanifold (which is everywhere except for a discrete set), 77* is obtained by decomposing V ' into irreduc­ ible components and then resolving each component by a homeomorphism 77^. Given two different resolutions 77^ and 77-, tt^

o 77. is a homeomor-

phism and biholomorphic except on a proper subvariety. Thus 77£- 1 o 77j is biholomorphic on

(T Wj. The 77. patch together to give a manifold

X and 77: X -> V ' such that 77 is proper, holomorphic and biholomorphic

14

NORMAL TWO-DIMENSIONAL SIN G U LA R ITIES

on the x e X such that p ' ( n ( x ) ) is not in the branch locus. Also, except above the singular points of the branch locus, n is biholomorphic above the manifold points of V R ecall that fr

77

: V'-> V is proper, holomorphic and biholomorphic off

(p)» P has been assumed to be singular. Thus n

0 77 : X

V is a

resolution of the singularities of V .l We can now prove Theorem 2.1. Let S be the given analytic space. Consider the set of T of all points teS such that for some neighborhood V of t, a resolution 77 may be chosen so that for all qeV, 77 resolves each irreducible component of Vq by a homeomorphism. By the proof of Proposition 2.2, S-T is discrete. Moreover, the local resolutions near points in S-T will patch together with the resolutions near points in T to give a global resolution of S. I If 77: M ^ V is a resolution of a 2-dimensional analytic space con­ structed as in the proof of Theorem 2.1, then 77 “^(p) is O-dimensional except for a discrete set of points p in V. If then the irreducible components of A =

77

77

’ ^ (p) is 1-dimensional,

"^ (p) are just (singular) com­

pact Riemann surfaces. By Theorem 1.1, by performing quadratic trans­ formations at points of A, and thereby getting a different resolution for V, we can always find resolutions such that A con sists of non-singularly embedded Riemann surfaces which intersect transversely and such that no three intersect at a point. It is customary to represent A by its dual graph r as follows. Let jA-1 be the irreducible components of A. These Ai are the vertices of the graph. An edge connecting two vertices Ai and A- corresponds to a point of intersection of the Riemann surfaces Aand Aj. Each Riemann surface Ai represents a topological homology cla ss in M and thus has a well-defined self-intersection number A^’ A j, which may also be defined as the Chern c la ss of the normal bundle of the embedding. See Theorem 2.3 below.

RESO LU TIO N O F TWO-DIMENSIONAL SIN G U LA R ITIES

15

We may assume that M is oriented so that a transverse intersection of two submanifolds contributes + 1 to their intersection number. To each vertex A- of the graph T , we associate A- *A- and thus obtain a weighted graph, also denoted T . Thus the weighted graph associated to M (kj, . . . , kg) is

-kl

' k2 "k3

' ks

where each vertex is a projective line. We need not use Theorem 2.3 subsequently. It would always suffice to define A ‘ A as the Chern cla ss of the normal bundle rather than as the topological self-intersection number. T H E O R E M 2.3.

L e t N be the normal bundle o f a non-singular com pact

Reimann s u rfa ce A em bed d ed in the 2-dim en sion al m anifold M. Then A*A e q u a ls the Chern c la s s o f N. Theorem 2.3 follows immediately from Lemmas 2.4 and 2.5 below. I LEMMA 2.4. L e t D b e the 0-sectio n o f a line bundle N over a com pact Riemann s u rfa ce A. T hen c, the Chern c la s s o f N, eq u als D ‘D. P roof: If f is a non-trivial meromorphic section of N, which exists by [Gu, p. 107] , then c equals the algebraic sum of the number of zeros and poles of f. If we had an analytic section s : A

N, then s would be

homologous to D and to compute D*D, we could just count the number of intersections of s with D, i.e. the number of zeros of s. For f , look at a point q e A where f has a pole. We may choose local coordinates on N so that q = (0,0), N =

e

||z^|< 1}, D = \z2 - 0 }. f becomes

a meromorphic function z 2 = f(z p with a pole at z^. Thus we may choose the z^ coordinate so that f(z j)

, where v is the order of the pole of f, Z 1

lz ll 5

1

• On lz ll = £ > f(z l) = ~ • Thus we may get a piecewise e 2v differentiable section f on N by letting f = f for points where |z^|>£ and 6

16

NORMAL TWO-DIMENSIONAL SIN G U LA R ITIES

~ Z1 i i = - r r f° r lz ll e Zv

1

5

6

•

th is ^or

th$ P°le points, f is homologous to

D. Thus to compute D *D , we can algebraically sum the number of zeros a

i.t_

of f . For a v

order zero of f, f =

, |zjJ


6

and thus s till be a section. Thus a

order zero

of f contributes v to the number of zeros and poles. For a pole of f , —v zi v f ----- . As before, we approximate the anti-holomorphic function z / by £2 v an anti-holomorphic function with v simple zeros. A simple zero of an anti-holomorphic function contributes f and D. Thus a v

—1

to the intersection number of

order pole of f contributes —v to the number of

zeros of f . I LEM M A 2.5.

L e t A b e a non-singularly em bedded com pact Riemann

su rfa ce in the 2-dim en sion al m anifold M. L et D be the 0-sectio n o f N, the normal bundle to A. Then A . A = D •D. P roof: Let f be the section of N constructed

inLemma 2.4.

f meets

D transversely at a discrete set of points. Let us recall how N is defined. Let ^T be the tangent bundle to A. Let jyjT be the restriction to A of the tangent bundle to M. i : A -> M induces an injection of ^T into jyjT and N is defined as the quotient bundle, i.e . (2.4) is exact. (2.4)

0

- » a T 1 » mt £ n - » 0

We wish to show that (2.4) splits as a sequence of differentiable bundles, i.e . there is a C°° bundle map h : N -> MT such that p ° h is the identity, h will be complex linear on each fibre. (2.4)

yields (2.5), a new exact sequence of vector bundles. Exactness

is just exactness on each fibre and Horn is exact for vector spaces.

RESO LU TIO N O F TWO-DIMENSIONAL SIN G U LA R ITIES

(2.5)

17

0 -» Hom(N,AT ) -» Hom(N,v T ) -» Hom(N,N) -> 0

Using script letters, Ham , to denote the sheaf of germs of C00 sections of the vector bundle sequence (2.5), we get the following exact sheaf sequence: 0

--- ► Jiam (N,a T)

K