Math. Ann. 231, 55--59 (1977) © by Springer-Verlag 1977
Two Theorems on Higher Dimensional Singularities
Stephen S.T. Yaul' 2
Institute for AdvancedStudy,Princeton, N.J. 08540, USA
§ 1. Introduction
Rational singularities ofsur faces have been studied by M. Artin [2], Brieskorn [4],
DuVal [6], Laufer [11], Lipman [12], and Tyurina [14]. Recently Burns [5] had a
natural generalization of Artin's ration surface singularities to higher dimension.
Let us recall the definition.
Definition 1.t. Let p be a normal isolated singularity in the n-dimensional analytic
space Vand let zr : M ~ Vbe a resolution of V.p is a rational singularity if R~zr.(d~),the
i-th direct image sheaf of the structure sheaf ~ on M, vanishes near p for i>0.
Equivalently, since W~t,(O)=0 at the regular points of V, p is rational if
dirlimHi(Tr-l(U),~)=0 for i>0, where U runs over a fundamental set of
neighborhoods ofp in V. Several examples were found of such singularities by Burns
[5], especially the Arnold singularities of [1], and all quotient singularities.
The condition for p to be rational is in fact independent of the choice of the
resolution by [9], Corollary 2, p. 153. In this paper the following two theorems are
proved.
Theorem A. Let p be a normal n-dimensional isolated singularity of V. Let N C V have
an admissible representation Q : N ~ A with Q(p)=0. Let U =Q- a(Izll 2 + ... + Iz.I ~ < r )
with r chosen sufficiently small so that Q is the only singularity of V in U. Then
direR"- lrc,(t~)p = dimF(U--p, f2")/L 2(U-p). (Here L 2 ( U - p ) denotes the set of all
square integrable holomorphic n forms on U - p , see p. 601 of [11].) In particular, if F
is a Gorenstein singularity, i.e. there exists a hotomorphic n-form e9 defined on a deleted
neighborhood of p in V, which is nowhere vanishing on this neighborhood, then p is
rational if and only if R n- ln,(tP)=0, i.e. R"- lzr,(d?) = 0 if and only if RiTt,(~)=O for
all i>0.
The last statement ofTheorem A is a trivial consequence o f the work of D. Burns
[5] and the first part of Theorem A.
1 Supported in part by a grant from NSF
2 Currentaddress: Departmentof Mathematics,Harvard University,Cambridge,MA02138, USA
56
St. S. T. Yau
- -
rtl
TheoremB. Let f ( z t , . . . , z , , z , + l ) - z , + l +al(z 1, .... z,)z,+
~-11 + ... +a,~(z 1.... ,z,) be
hotomorphic near (0, .... 0, 0). Let di be the order of the zero of ai(zt, ..., z,) at (0, .... 0),
di >=1. Let d = min(dfi), 1 < i < m. Suppose that
V = { z 1..... z,,, z,,+ 1):f(zl
....
, Zn, Zn+ 1) = 0 }
defined in a suitably small potydisc, has p =(0,...,0, O) as its only singularity. Let
: M ~ V be resolution of V. Then d i m H " - l ( M , O ) > ( m - 1 ) d - n .
F o r n = 2, Theorem A was proved by Laufer. As a consequence o f this, he gave a
necessary and sufficient condition for a surface singularity to be rational that does
not involve a priori knowledge of what a resolution o f p looks like. The p r o o f used
there cannot be generalized to higher dimension because C°°(0, n - 2) from 0 such
that 00 = 0 does not imply that 0 is a holomorphic (0, n - 2 ) form unless n = 2.
Instead, the p r o o f of Theorem A depends heavily on the properties of analytic
cover. Theorem B and its p r o o f are generalizations of Laufer's theorem [10] in the
two-dimensional case.
The author gratefully acknowledgesthe encouragement and help of Professor Henry B. Laufer
during the investigation of these results. We would also like to thank Professor B. Bennett, Professor
Gromov and Professor Kuga for their stimulatingconversationsabout Mathematics. Finally,we thank
referee for pointing out to us that one can also use Siu's Theorem [Theorem 4, p. 374, 13] to establish
the final step of the proof of Theorem A. However,our proof is more elementaryin the sense it does not
involve the machinery of gap sheaves.
§ 2. Proof of Theorem A and Theorem B
2.1. Proof of Theorem A. By Artin's algebraization theorem [3], we may assume for
a local question near p c V, that V i s a normal, affine algebraic variety. As such, by
[9], we m a y assume that we have a resolution zc : M ~ Vso that M is Zariski open in
a smooth projective variety. In this situation, we can apply the result of Grauert and
Riemenschneider [7], that Rin.I2 n = 0 for i > 0. Let R = n - I(U) and A = n - l(p). If T
is a strictly Levi pseudoconvex neighborhood of A (which may be obtained by
letting T = n - I(Bc~N) where B = {lzl 2 + . . . + lz, l2 <e}, assuming N is embedded
locally in tE~), H " - I ( T , (9) is finite dimensional by [8, Theorem IX, B. 6, pp. 2 6 8 - 270]. The restriction m a p H " - I(R, ( 9 ) ~ H " - I(T, (9) is an isomorphism by L e m m a 3.1
of [1i]. Hence H " - I(R, (9) is finite dimensional. So by Serre duality dim/-/"- t(R, (9)
= dim HI.(R, f2"). (2.2) o f [11] gives the exact sequence
0-, F,(R, O")--,F(R,a")~ FJR, a")~ H~,(R,O")~H~(R,a")~
. . . .
F.(R, O") = 0
and Hi(R, f~') = 0 by L e m m a 3.1 of [11].
Thus
H ~ , ( R , O " ) ~ F J R , O")/F(R, f2"). F(R, F2")=Lz(U-p) by Theorem 3.1 of [11], so
to finish the p r o o f we need that Fo~(R,O")~F(R-A, I 2 " ) ~ F ( U - p , F2"). By
Proposition 2.1 o f [11], we must show that every holomorphic n-form defined near
the boundary of U has an analytic continuation to U - p. It suffices to show that a
form defined on Q-l(r-~<lZl}Z+...+lznl2<r) extends to a form defined on
-l(r-e' < Izl I~ + ... + l z f < r) for some e ' > e. Then, choosing the greatest such e'
we can extend the form to Q - l ( 0 < l z , 1 2 + . . . +fz~12<r)=U-p.
Higher Dimensional Singularities
57
Let sE {r-e = ]za[2 + ... + Iz,[2}. Let L be a real hyperplane through s which is
tangent to the sphere r - e = Izl[2 + ... + ]z,I 2. F o r suitable ( a linear combination
of z 1..... z,, L = {lee[ =z} for some z > 1. Let
D=Q-l{(zl, z2 ..... z,) : r - e <[zl[ 2 + ... + [z,[ 2 < r - e + q , [e~[> z - ~/}
and let
E = O - I {(zl, z2, ..., z,) :[zll 2 + ... + [z,I 2 < r -
e+~t, levi > z - q }
where t/ is chosen small enough so that each c o m p o n e n t o f E m a y be covered
by a single coordinate system. T o finish the p r o o f o f the theorem it now suffices
to show that every h o l o m o r p h i c function on D can be extended to E. F o r since
E has a single coordinate system 0 ~ ~" so that h o l o m o r p h i c forms must extend
from O to E. Using the compactness of r - e = [z 1[2 + . . . + [z,12, we see that we may
extend forms to Q-l(r-e'<IzaI2+...+[z,[2<r) for some g > e provided that
extensions using different E's agree. But i f E and E' correspond to s and s', o(EnE') is
connected and meets r - e <[z~[2+ ... + ]z,[ 2 < r so that each c o m p o n e n t of Ec~E'
meets 0 - ~ ( r - ~ < Izl[2 + ... + [z,[2 < r). Then by the identity theorem the extensions
using E and E' agree.
Let h be a h o l o m o r p h i c function on D. We want to prove that h can be extended
to E. By shrinking D a little bit if necessary, we m a y assume that h is bounded on D.
Since 0 is proper, E is a Stein manifold. Let E 1 be a connected c o m p o n e n t orE. Since
0(E) is connected, ~(E 1) = o(E). It follows that Q:E1 --*0(E) is proper and onto. By [8,
T h e o r e m III, B. 7, p. 103], Q : E 1 -,0(E) is an analytic cover. As 0(D) is connected,
o(Elc~D ) =Q(D). So 0 : E 1nD~o(D) is also an analytic cover. By [8, T h e o r e m III,
B. 14, p. 105], h/EtnD satisfies a polynomial equation
h~+~(aioo)hi=O
on
ElnO
with a~eOo~D). By H a r t o g ' s theorem, the ai have unique holomorphic extensions
ai~(9~tg). Let V be the subvariety of E~ x C defined as follows:
V = {(p, z) :pE E 1, z z + ~ (a i oQ(p)zi = 0}.
By shrinking o(E) a little bit, we m a y assume that the roots o f this polynomial are
bounded, so the natural projection ~1 : V~E~ is proper. Let q~(p)=(p,h(p)) for
pe El c~D. Then q~ : El nD-~ E
We claim that ExnD is connected. We m a y write EanD = U u~ where u~ are
~tEA
connected c o m p o n e n t s on E~ c~D. If Ea n D is disconnected, then A has at least two
distinct indices. Since Q(D) is connected and Q : Ex c~D--.o(D) is proper, it follows that
O(u,) = 0(D). Hence 0 : w,-~o(D) is also an analytic cover. Let ~, fl be two distinct
d e m e n t s in A. Let x o ~ g(D), so that 0 - a(xo)c~ u , has 2, distinct points p~, ..., p~, and
0-X(x0)c~ u¢ has 2~ distinct points px, +~.... , pz, + ~. Let f e t~e, be such that f(p~)= i.
By [8, T h e o r e m III, B. 14, p. 105], there is a polynomial.
P(X) = X a ' +
~ a.rX~,
a~t~ ~(o)
such that P ( f ) = 0 in u~. N o w by H a r t o g ' s theorem the a~ have unique h o l o m o r p h i c
extensions a~eO¢~n) and then p(f)=fa,,+ y,(a~o#)f~tp~. But P ( f ) is zero on an
open set o f Et, so since E~ is connected, P ( f ) = 0 on all o f E r But then X ~*
58
St. S. T. Yau
+ ~a~(xo)X l has at least 2~ +2p roots, namely the values o f f at all points in
px~..... pa.+~, a contradiction o f the fact that 2a>0.
Let V' ~e the irreducible branch of V which contains q~(EanD). Since
0on1 :V--*Q(E) is proper, so ~ozq:V'--.Q(E) is proper. Observe that V' is an
irreducible Stein space. Thus as in the previous argument (Qo~)-~(0(D))~V'
=rc?a(DnE1)c~V ' is connected. But q~(DnE1) is both open and closed in
1:-I(Dc~EOc~V', so q~(DnE1)=n-~ l(DnE1)~ V'. Since n~ : V'~E~ is proper, the set
S=r~(S(V'))w~r{xeR(V'):rank~rc<n} is negligible in Ea. [Here we use the usual
convention: s(V') denotes the set of singular points of V' and R(V') denotes the set
of regular points of V'], and n : V ' - ~ - ~ ( S ) ~ E x - S is a k-sheeted covering map
for some 2. Since 2 = 1 over E 1c~D, 2 = 1 over E ~ - S. Thus every x e E~ - S has only
one inverse image point in V', For such x, define H(x) = z where re- ~(x)c~V' = {x, z}.
H is locally bounded and holomorphic on E ~ - S; and hence in E~ by the Riemann extension theorem). Further, H = h on Ea, so H is an extension of h to all
o f E 1. Q.E.D.
2.2. Proof of TheoremB. On E f ( z l ..... z,,z,+ 1)=0 so
Of dzl
Of dz, +
Of
Of dz, +1 =-O.
Oz~
+ ~z~ " "" + uz. dz" + oz.+ l
Thus
d z 2 / x ... A d z n + 1
af
=~r2
dz 3 ^ ... Adz,+ 1 Adz 1
Of
.....
~z 1
of
Of _
"'"
C3zn
dz 1 Adz z ^ ... Adz,
Of
Ozz
where ¢~ are either +1
~2. 2
~,+1
Of
or
-1.
Oz. +1
Since p is an isolated singularity,
df
dz 1
= 0 occurs only at p. Hence on V - p , o9 is a nowhere
Ozn + 1
vanishing holomorphic n form, Holomorphic n-forms 2 on V - p extend to M if and
only if S ,t A 2 is finite on V-- p near p [11, p, 603, Theorem 3.1].
Let D = d i m H " - ~ ( M , d ) ) = d i m H ° ( M - A , fP)/H°(M, fl"). Let a~ ..... a, be integers such that 0 ~ a~~ D. Then some nonzero linear combination ~ with complex
coefficients of
~...~"(o
~ a~<D
(2.1)
i=1
lies on H°(M, fP). So for some
~ = ~ l ~ 2 . . . ~ - g ( z I .... ,z.)a~eH°(M,~2 ")
polynomial g(z I ..... z.),
with z~Iz~2...~"a~ in
g(O,.... 0)4=0,
(2.1). Also,
n
~ . . . ~ " w e H ° ( M , O ") where ~, ai=D.
t--1
Q : (z 1..... z,, z,+ 1)~(zi .... , z,) express V as m-sheeted branched cover over the
(z~, ..., z,)-plane. The branch locus has zero Lebesgue measure. Let dA be Lebesgue
measure on the (z 1..... z,)-plane and let S denote integration near (0 .... ,0). Let
Higher Dimensional Singularities
Q-l(z 1..... z . ) = { ( z l , . . . , z , ,
59
z,+l,i), l <_i<m}. Since z~1... 4 " ogE H ° ( M , E2") where
~ ai=O.
i=1
J ~
df [Zll2"~"'lz"12""
i= 1 _ _ ( Z 1
2
dZ<~.
(2.2)
' ..., Zn, Zn+ 1,i)
g3fn+ l
Let r2=lzllzq-...-t-lZnl 2. For some constant C and for some suitable small
neighborhood of (0..... 0), Iz,+l,~(z 1..... z,)l <Crdl <i<_m. Thus for some constant K
dz-~--+(zt ..... z,+ ~,f) < K r t.Since trl21,= ~l+...+~,=oal!
~
D. . !., .,
1)d
lzll2,,...fznl2~,, O < a ~ D , l
(2.3)
<i<=n, from(2.2)and
(2.3),
r 2D
Hence 2(n- 1)- d - 2D < 2n. Q.E.D.
References
1. Arn••d• V. : N•rma• f•rms f•r functi•ns near degenerate critica• p•ints• the Wey• gr•ups •f Ak• Dk• E k,
and Lagrangian singularities. Func. Anal. App. 6 (1972)
2. Artin, M. : On isolated rational singularities of surfaces. Amer. J. Math. 88, 129--136 (1964)
3. Artin,M. : Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S. 36
(1969)
4. Brieskorn, E.: Rationale Singularit~iten Komplexer Flachen. Inv. Math. 4, 336--358 (1967/68)
5. Burns, D. :On Rational Singularities in Dimensions > 2. Math. Ann. 211, 237--244 (1974)
6. DuVal, P. : On isolated singularities of surfaces which do not affect the conditions of adjunction. I.
Proc. Cambridge Philos. Soc. 30, 453--459 (1934)
7. Grauert, H., Riemensehneider, O.: Verschwindungss~itze fiir analytische Kohomologiegruppen auf
komplexen R~iumen. Invent. Math. 11, 263--292 (1970)
8. Gunning, R., Rossi, H. : Analytic functions ofseveral variables. Englewood Cliffs, N.J. :Prentice-Hall
1965
9. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of charactersitie O.
I--II. Ann. Math. 79 (1964)
10. Laufer, H.: On minimally elliptic singularities. Amer. J. Math. (to appear)
11. Laufer, H. : On rational singularities. Amer. J. Math. 94, 597--608 (1972)
12. Lipman,J.: Rational singularities with applications to algebraic surfaces and unique factorizations.
Publ. Math. I.H.E.S. 36, 195--279 (1969)
13. Siu, Y.-T.: Absolute gap Sheaves and extension of coherent analytic sheaves. Trans. AMS 141,
361--376 (1969)
t4. Tyurina, G.N. : Absolute isolatedness of rational singularities and triple rational points. Func. Anal.
App. 2, 324---332 (1968)
Received November 18, 1976, and in revised form March 29, 1977